1 Introduction↩︎

1.1 The title↩︎

In the title of this paper, “new" (unsurprisingly) stands for something that is different from”old". Historically, quantum affine algebras \(U_q(\widehat{{\mathfrak{g}}})\) were first endowed with an old coproduct by Drinfeld and Jimbo ([1], [2], 0?), and then Drinfeld defined a new coproduct that was quite different from (and in a certain sense, orthogonal to) the old one. In the present paper, we work with quantum loop algebras in the greater generality of N?, Arbitrary?, in which one can define an analogue of Drinfeld’s new coproduct, but there is no sense in which the Drinfeld-Jimbo construction applies. Instead, we define a “new new" coproduct on quantum loop algebras that is different from (and in a certain sense, orthogonal to) the new one. The motivation behind this construction is that \[\Big(\text{new new coproduct} \Big) = \Big( \text{old coproduct} \Big)\] in the particular case of quantum affine algebras \(U_q(\widehat{{\mathfrak{g}}})\) for simple Lie algebras \({\mathfrak{g}}\). At a slightly deeper level, one can say that each word”new" in the title corresponds to rotating the loop root lattice by \(90^{\circ}\). Therefore, just like the Drinfeld-Jimbo coproduct differs from Drinfeld’s new coproduct by a rotation by \(90^{\circ}\), so does Drinfeld’s new coproduct from the new new coproduct we introduce in this paper.

1.2 Algebras↩︎

Fix a finite set \(I\), a field \({\mathbb{K}}\) of characteristic 0, and rational functions \[\label{eqn:zeta32intro} \zeta_{ij}(x) \in \frac{{\mathbb{K}}[x^{\pm 1}]}{(1-x)^{\delta_{ij}}}\tag{1}\] such that \[\label{eqn:assumption} \lim_{x \rightarrow \infty} \frac{\zeta_{ij}(x)}{\zeta_{ji}(x^{-1})} < \infty\tag{2}\] for all \(i, j \in I\). Under this assumption, we defined a quantum loop algebra in N?, Arbitrary? \[\label{eqn:quantum32loop32algebra32intro} \mathbf{U}= {\mathbb{K}}\Big \langle e_{i,d}, f_{i,d}, \varphi^+_{i,d'}, \varphi^-_{i,d'} \Big \rangle_{i \in I, d \in {\mathbb{Z}}, d' \geq 0} \Big/ \Big(\text{relations \eqref{eqn:rel32quantum321}-\eqref{eqn:rel32quantum329}}\Big)\tag{3}\] The motivating example of this construction arises from a simple Lie algebra \({\mathfrak{g}}\): we take \(I\) to be a set of simple roots, \({\mathbb{K}}= {\mathbb{C}}\) (hereafter fix \(q \in {\mathbb{C}}^* \backslash \sqrt[{\mathbb{N}}]{1}\)) and define \[\label{eqn:zeta32intro32particular} \zeta_{ij}(x) = \frac{(q^{-d_{ij}}-x)(-x)^{-\delta_{i>j}}}{(1-x)^{\delta_{ij}}}\tag{4}\] for some total order \(<\) on \(I\), where \[\label{eqn:cartan32matrix32intro} C = \left(c_{ij} = \frac{2d_{ij}}{d_{ii}} \in {\mathbb{Z}}\right)_{i,j \in I}\tag{5}\] is the Cartan matrix of \({\mathfrak{g}}\). In this particular case, the algebra \(\mathbf{U}\) is none other than \[\label{eqn:quantum32affine32intro} \Big( \mathbf{U}\text{ for the choice \eqref{eqn:zeta32intro32particular}} \Big) = U_q(L{\mathfrak{g}}) \cong U_q(\widehat{{\mathfrak{g}}})_{c=1}\tag{6}\] The isomorphism on the right was claimed by Drinfeld ([1]) and proved by Beck ([3], using work of [4]). However, the quantum loop algebras 3 are significantly more general than \(U_q(L{\mathfrak{g}})\). They include examples such as quantum affinizations of Kac-Moody Lie algebras, \(K\)-theoretic Hall algebras of doubled quivers, BPS algebras associated to toric Calabi-Yau threefolds, and Hall algebras of curves over finite fields (which have a long history of study by numerous authors, but we refer to [5], N?, Symmetric?, N?, Wheel?, N?, Reduced? respectively for a treatment in the language of the present paper).

1.3 Simple modules↩︎

In the context of the present paper, loop weights \[{\boldsymbol{\psi}}= \left(\psi_i(z) = \sum_{d=0}^{\infty} \frac{\psi_{i,d}}{z^d} \in {\mathbb{K}}[[z^{-1}]]^\times \right)_{i \in I}\] are \(I\)-tuples of power series in \({\mathbb{K}}\). We are interested in studying simple modules \[\label{eqn:simple32intro} `` \;\mathbf{U}\curvearrowright L({\boldsymbol{\psi}}) \;"\tag{7}\] which are generated by a single vector \(|\varnothing\rangle\) modulo the relations \[e_{i,d} \cdot |\varnothing\rangle= 0\] \[\varphi^+_{i,d'} \cdot |\varnothing\rangle= \psi_{i,d'} |\varnothing\rangle\] for all \(i \in I\), \(d \in {\mathbb{Z}}\), \(d' \geq 0\). However, as is already apparent in the situation of quantum affine algebras 6 (studied in [6], [7]), such modules are not acted on by the entire quantum loop algebra \(\mathbf{U}\). Instead, we construct triangular decompositions \[\label{eqn:triangular32intro} \mathbf{U}= {\mathcal{A}}^{\geq {\boldsymbol{p}}} \otimes {\mathcal{A}}^{\leq {\boldsymbol{p}}}\tag{8}\] for any \({\boldsymbol{p}}\in {\mathbb{R}^I}\), generalizing the construction of N?, Cat?, N?, Char? (which in turn generalizes the triangular decomposition of \(U_q(\widehat{{\mathfrak{g}}})\) into positive and negative Borel subalgebras, in the particular case of 6 ). For any loop weight \({\boldsymbol{\psi}}\), we will define a simple module \[\label{eqn:simple32actual32intro} {\mathcal{A}}^{\geq {\boldsymbol{p}}} \curvearrowright L({\boldsymbol{\psi}})\tag{9}\] in Section 4, thus generalizing the main construction of N?, Cat?, which itself generalizes [7]. More systematically, we consider the following analogue of the main construction of loc. cit.: a module \[{\mathcal{A}}^{\geq {\boldsymbol{p}}} \curvearrowright V\] is said to be in category \({\mathcal{O}}\) if it is diagonalizable with respect to the finite Cartan subalgebra generated by \(\{\varphi_{i,0}^+\}_{i \in I}\), with finite-dimensional eigenspaces that are non-zero only when the eigenvalues lie in a certain union of cones (see Definition 6). The prime example of a module in category \({\mathcal{O}}\) is the simple module \(L({\boldsymbol{\psi}})\) of 9 if all the constituent power series of \({\boldsymbol{\psi}}= (\psi_i(z))_{i\in I}\) are expansions of rational functions. Such loop weights \({\boldsymbol{\psi}}\) will be called rational.

1.4 Coproducts↩︎

However, what was missing from N?, Cat?, N?, Char? was a notion of tensor product of modules, and this is the main construction of the present paper.

The completion \(\stackrel{\mathsf{H}}{\otimes}\) is such that the coproduct ?? gives rise to a well-defined \({\mathcal{A}}^{\geq {\boldsymbol{p}}}\)-module structure on \(V \otimes W\) for any modules \({\mathcal{A}}^{\geq {\boldsymbol{p}}} \curvearrowright V,W\) in category \({\mathcal{O}}\).

For any \({\boldsymbol{p}}\), the subalgebra \({\mathcal{A}}^{\geq {\boldsymbol{p}}} \subset \mathbf{U}\) contains the positive loop Cartan subalgebra \[\label{eqn:half32loop32cartan} {\mathbb{K}}[\varphi^+_{i,d}]_{i \in I, d \geq 0}\tag{10}\] which is commutative. We will consider those modules \({\mathcal{A}}^{\geq {\boldsymbol{p}}} \curvearrowright V\) in category \({\mathcal{O}}\), that decompose into generalized eigenspaces for the subalgebra 10 \[\label{eqn:decompose32intro} V = \bigoplus_{\text{loop weights }{\boldsymbol{\psi}}} V_{{\boldsymbol{\psi}}}\tag{11}\] such that every \(\varphi^+_{i,d}\) acts via the generalized eigenvalue \(\psi_{i,d}\) on \(V_{\boldsymbol{\psi}}\) (the decomposition 11 exists on general grounds if \({\mathbb{K}}\) is algebraically closed). Following [8], we define the \(q\)-character as \[\label{eqn:q-character32intro} \chi_q(V) = \sum_{\text{loop weights }{\boldsymbol{\psi}}} \dim_{{\mathbb{K}}} \left( V_{{\boldsymbol{\psi}}} \right) [{\boldsymbol{\psi}}]\tag{12}\] for various formal symbols \([{\boldsymbol{\psi}}]\) associated to loop weights. If we define the product of these formal symbols component-wise (i.e. \([{\boldsymbol{\psi}}][{\boldsymbol{\psi}}'] = [{\boldsymbol{\psi}}{\boldsymbol{\psi}}']\)), then we show in Proposition 17 that \(q\)-characters are multiplicative with respect to tensor products, thus generalizing a result of [8] for quantum affine algebras. More broadly speaking, the coproduct \(\Delta_{{\boldsymbol{p}}}\) makes category \({\mathcal{O}}\) into a tensor category, and the \(q\)-character gives an (injective, on general grounds) ring homomorphism from the Grothendieck group of category \({\mathcal{O}}\) to the ring of formal linear combinations of the symbols \([{\boldsymbol{\psi}}]\).

1.5 R-matrices↩︎

For any \({\boldsymbol{p}}\in {\mathbb{R}^I}\), Theorem 8 proves that the decomposition \[\label{eqn:triangular32intro322} \mathbf{U}= {\mathcal{A}}^{\geq {\boldsymbol{p}}} \otimes {\mathcal{A}}^{\leq {\boldsymbol{p}}}\tag{13}\] is a particular case of Drinfeld’s quantum double construction, with respect to the coproduct \(\Delta_{{\boldsymbol{p}}}\) and the pairing 151 . Thus, there exists a universal \(R\)-matrix \[\label{eqn:intro32r-matrix321} _{\bar{{\boldsymbol{p}}}}{\mathcal{R}}_{{\boldsymbol{p}}} \in \mathbf{U}\;\widehat{\otimes} \;\mathbf{U}, \qquad \left(_{\bar{{\boldsymbol{p}}}}{\mathcal{R}}_{{\boldsymbol{p}}}\right) \cdot \Delta_{{\boldsymbol{p}}}(-) = \Delta_{{\boldsymbol{p}}}^{\text{op}}(-) \cdot \left(_{\bar{{\boldsymbol{p}}}}{\mathcal{R}}_{{\boldsymbol{p}}}\right)\tag{14}\] where we define completions \(\widehat{\otimes}\) and \(\bar{\otimes}\) in Subsection 3.7. More generally, we will construct the following objects for all \({\boldsymbol{p}}^1,{\boldsymbol{p}}^2 \in {\mathbb{R}^I}\) \[\begin{align} &_{{\boldsymbol{p}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} \in \mathbf{U}\;\bar{\otimes} \;\mathbf{U}, \qquad \left( _{{\boldsymbol{p}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} \right) \cdot \Delta_{{\boldsymbol{p}}^1}(-) = \Delta_{{\boldsymbol{p}}^2}(-) \cdot \left( _{{\boldsymbol{p}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} \right) \tag{15}\\ &_{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} \in \mathbf{U}\;\widehat{\otimes} \;\mathbf{U}, \qquad \left( _{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} \right) \cdot \Delta_{{\boldsymbol{p}}^1}(-) = \Delta_{{\boldsymbol{p}}^2}^{\text{op}}(-) \cdot \left( _{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} \right) \tag{16} \end{align}\] The existence of \(_{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1}\) is subject to the usual caveats involving the Cartan subalgebra, that the reader may find recalled in Subsection 3.9. We may evaluate the above \(R\)-matrices in tensor products of modules as follows. Generalizing a phenomenon that has long been known for quantum affine algebras with the Drinfeld-Jimbo coproduct, we show in Proposition 24 that for rational loop weights \({\boldsymbol{\psi}}\) which are regular\(^{\neq 0}\) (i.e. the rational functions \(\psi_i(z)\) are regular and non-zero at \(z=0\) for all \(i \in I\)), the action \({\mathcal{A}}^{\geq {\boldsymbol{p}}} \curvearrowright L({\boldsymbol{\psi}})\) extends to an action \[\label{eqn:extended32action32intro} \mathbf{U}\curvearrowright L({\boldsymbol{\psi}})\tag{17}\] which is independent of \({\boldsymbol{p}}\). Thus, 17 is an example of a module \[\label{eqn:integrable32intro} \mathbf{U}\curvearrowright V\tag{18}\] in category \({\mathcal{O}}\), namely one whose weight spaces are finite-dimensional and non-zero only in a finite union of cones. For any two modules \(\mathbf{U}\curvearrowright V,W\) in category \({\mathcal{O}}\), the coproducts \(\Delta_{{\boldsymbol{p}}}\) and \(\Delta_{{\boldsymbol{p}}}^{\text{op}}\) for various \({\boldsymbol{p}}\in{\mathbb{R}^I}\) give rise to module structures \[\label{eqn:tensor32action32intro} \mathbf{U}\curvearrowright V \otimes_{{\boldsymbol{p}}} W \qquad \text{and} \qquad \mathbf{U}\curvearrowright V \otimes^{\text{op}}_{{\boldsymbol{p}}} W\tag{19}\] on the tensor product \(V \otimes W\). The tensors 15 and 16 produce \(\mathbf{U}\)-intertwiners \[\label{eqn:intro32r-matrix32in32rep321} _{{\boldsymbol{p}}^2}R_{{\boldsymbol{p}}^1}(u): V^u \otimes_{{\boldsymbol{p}}^1} W \rightarrow V^u \otimes_{{\boldsymbol{p}}^2} W[u^{\pm 1}]\tag{20}\] \[\label{eqn:intro32r-matrix32in32rep322} _{\bar{{\boldsymbol{p}}}^2}R_{{\boldsymbol{p}}^1}(u): V^u \otimes_{{\boldsymbol{p}}^1} W \rightarrow V^u \otimes^{\text{op}}_{{\boldsymbol{p}}^2} W((u))\tag{21}\] where \(V^u\) refers to twisting the action of \(V\) by powers of \(u\) (see Subsection 4.11 for details). This opens the door to asking, in the current generality of \(\mathbf{U}\), a host of interesting questions that have been studied for \(U_q(\widehat{{\mathfrak{g}}})\): factorization of \(R\)-matrices, calculation of transfer matrices, relation to XXZ Hamiltonians, Baxter’s equations, the Bethe ansatz, relations to \(W\)-algebras etc ([8]–[12]).

1.6 Techniques↩︎

Let us discuss our approach in more technical detail. We will heavily use the shuffle algebra incarnation of quantum loop algebras, namely \[\mathbf{U}\cong {\mathcal{S}}^+ \otimes {\mathbb{K}}[\varphi^+_{i,d}, \varphi^-_{i,d}]_{i \in I, d\geq 0} \otimes {\mathcal{S}}^-\] where \({\mathcal{S}}^\pm\) are defined in Subsection 2.3. We also recall (the natural generalization of) Drinfeld’s new topological coproduct \(\Delta\) on \(\mathbf{U}\), and the pairing \[\label{eqn:pair32shuffle32intro} {\mathcal{S}}^+ \otimes {\mathcal{S}}^- \xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{22}\] with respect to which \(\mathbf{U}\) is a Drinfeld double. The coproduct and pairing above are the essential inputs that we use to define the coproducts \(\Delta_{{\boldsymbol{p}}}\) on \({\mathcal{A}}^{\geq {\boldsymbol{p}}}\) and \({\mathcal{A}}^{\leq {\boldsymbol{p}}}\), and to show that \(\mathbf{U}\) is the Drinfeld double of the algebras \({\mathcal{A}}^{\geq {\boldsymbol{p}}}\) and \({\mathcal{A}}^{\leq {\boldsymbol{p}}}\) for any \({\boldsymbol{p}}\in {\mathbb{R}^I}\). With this in mind, our approach produces explicit formulas for \(\Delta_{{\boldsymbol{p}}}\) only inasmuch as one has explicit answers to the following problems.

Problem 1. Find explicit descriptions of \({\mathcal{S}}^\pm\) as sets of polynomials satisfying some collection of “wheel conditions", e.g. 201 (see the general discussion of N?, Arbitrary?).

Problem 2. Find explicit descriptions of the pairing 22 , see Remark 3.

So far, Problem 1 has been solved in the following particular cases: quantum affine algebras ([13]), quantum affinizations of simply-laced Kac-Moody Lie algebras (N?, Symmetric?), \(K\)-theoretic Hall algebras of doubled quivers (N?, Wheel?) and BPS algebras associated to toric Calabi-Yau threefolds (N?, Reduced?). Problem 2 has also been solved in the latter three settings in the works referenced above; we will provide a solution to this problem in the case of quantum affine algebras in Lemma 3. However, a general solution to Problems 1 and 2 is open and would be extremely interesting.

1.7 Acknowledgements↩︎

I would like to thank David Hernandez and Alexander Tsymbaliuk for many great conversations on quantum loop algebras and their representation theory. I gratefully acknowledge the support of the Swiss National Science Foundation grant 10005316.

2 Quantum loop and shuffle algebras↩︎

We recall the definition of quantum loop algebras in the generality of Subsection 1.2, as well as their shuffle algebra incarnation. Special emphasis will be placed on tools and techniques which will be used in later sections, such as the bialgebra structure, Drinfeld double, and various completions of the aforementioned algebras.

2.1 Basic notations↩︎

The set \({\mathbb{N}}\) is henceforth thought to contain 0. We fix a finite set \(I\), a field \({\mathbb{K}}\) of characteristic 0 and a collection of rational functions \(\{\zeta_{ij}(x)\}_{i,j \in I}\) as in Subsection 1.2. The abelian group \({\mathbb{Z}^I}\) plays the role of the root lattice for our quantum algebras, with \[\boldsymbol{\varsigma}^i = \underbrace{(0,\dots,0,1,0,\dots,0)}_{1 \text{ on }i\text{-th spot}}\] playing the role of simple roots. For any \({\boldsymbol{m}}= (m_i)_{i \in I}, {\boldsymbol{n}}= (n_i)_{i \in I} \in {\mathbb{R}^I}\), we let \[\label{eqn:dot32product} {\boldsymbol{m}}\cdot {\boldsymbol{n}}= \sum_{i \in I} m_i n_i\tag{23}\] For \({\boldsymbol{m}},{\boldsymbol{n}}\in {\mathbb{R}^I}\), we will write \({\boldsymbol{m}}\leq {\boldsymbol{n}}\) if \(m_i \leq n_i\) for all \(i \in I\). We will also use the notation \({\boldsymbol{0}} = (0,\dots,0)\) and \({\boldsymbol{1}}= (1,\dots,1)\). For any \({\boldsymbol{m}}= (m_i)_{i \in I} \in {\mathbb{R}^I}\), we write \[\label{eqn:coordinate32sum} |{\boldsymbol{m}}| = \sum_{i \in I} m_i\tag{24}\] Because of assumption 2 , we can write for all \(i,j \in I\) \[\label{eqn:magnitude} \zeta_{ij}(x) = \frac{c_{ij} x^{\#_{ij}+\delta_{ij}} + \dots + c'_{ij} x^{-\#_{ji}}}{(x-1)^{\delta_{ij}}}\tag{25}\] for various scalars \(c_{ij}, c_{ij}' \in {\mathbb{K}}^*\) and integers \(\#_{ij}\), such that \(c_{ij} x^{\#_{ij}}\) is the leading order term of \(\zeta_{ij}(x)\) as \(x \rightarrow \infty\). The numbers \(\#_{ij}\) give rise to a bilinear form \[\label{eqn:euler32form} \langle {\boldsymbol{m}}, {\boldsymbol{n}}\rangle = \sum_{i,j \in I} m_i n_j \#_{ij}\tag{26}\] for all \({\boldsymbol{m}}= (m_i)_{i \in I}, {\boldsymbol{n}}= (n_i)_{i \in I}\). The following scalars will come up in Section 3 \[\label{eqn:scalars} \gamma_{{\boldsymbol{m}}, {\boldsymbol{n}}} = \prod_{i,j \in I} \left[ (-1)^{\delta_{ij}}\frac{c_{ij}}{c_{ji}'} \right]^{m_i n_j} \in {\mathbb{K}}^*\tag{27}\]

2.2 The pre-quantum loop algebra↩︎

The following notion is motivated by the setting of 6 , in which it \(q\)-deforms the Lie bracket on \({\mathfrak{g}}[t^{\pm 1}]\) for simple \({\mathfrak{g}}\).

We also consider the negative part of the pre-quantum loop algebra \[\mathbf{\widetilde{U}}^-= \mathbf{\widetilde{U}}^{+,\text{op}}\] with generators denoted by \(f_{i,d}\) instead of \(e_{i,d}\). They satisfy the relations \[\label{eqn:rel32quad32opp} f_i(z) f_j(w) \zeta_{ij} \left( \frac{z}{w} \right) = f_j(w) f_i(z) \zeta_{ji} \left(\frac{w}{z}\right)\tag{28}\] for all \(i,j \in I\), where \(f_i(z) = \sum_{d \in {\mathbb{Z}}} \frac{f_{i,d}}{z^d}\).

2.3 The shuffle algebra↩︎

The following construction generalizes the trigonometric version (due to [14]) of the Feigin-Odesskii elliptic shuffle algebras of [15]. Consider the vector space of Laurent polynomials in arbitrarily many variables \[\label{eqn:big32shuffle} {\mathcal{V}}= \bigoplus_{{\boldsymbol{n}}\in {\mathbb{N}^I}} {\mathcal{V}}_{{\boldsymbol{n}}}, \quad \text{where} \quad {\mathcal{V}}_{(n_i \geq 0)_{i \in I}} = {\mathbb{K}}[z_{i1}^{\pm 1},\dots,z_{in_i}^{\pm 1}]^{\text{sym}}_{i \in I}\tag{29}\] Above, “sym" refers to Laurent polynomials which are color-symmetric, i.e. symmetric in the variables \(z_{i1},\dots,z_{in_i}\) for each \(i \in I\) separately. The vector space \({\mathcal{V}}\) is called the big shuffle algebra when endowed with the following shuffle product: \[\label{eqn:mult} E( z_{i1}, \dots, z_{i n_i})_{i \in I} * E'(z_{i1}, \dots,z_{i n'_i})_{i \in I} =\qquad{(3)}\] \[\textrm{Sym} \left[ \frac{E(z_{i1}, \dots, z_{in_i}) E'(z_{i,n_i+1}, \dots, z_{i,n_i+n'_i})}{{\boldsymbol{n}}! {\boldsymbol{n}}'!} \prod_{i,j \in I} \mathop{\prod_{1 \leq a \leq n_i}}_{n_j < b \leq n_j+n_j'} \zeta_{ij} \left( \frac{z_{ia}}{z_{jb}} \right) \right]\] The word”Sym" in ?? denotes symmetrization with respect to the \[({\boldsymbol{n}}+{\boldsymbol{n}}')! := \prod_{i\in I} (n_i+n'_i)!\] permutations of the variables \(\{z_{i1}, \dots, z_{i,n_i+n'_i}\}\) for each \(i\) independently. Let \[{\mathcal{V}}^+ = {\mathcal{V}}\qquad \text{and} \qquad {\mathcal{V}}^- = {\mathcal{V}}^{\text{op}}\] The reason for formula ?? is to ensure that there exist algebra homomorphisms \[\label{eqn:tupsilon} \widetilde{\Upsilon}^\pm : \mathbf{\widetilde{U}}^\pm\rightarrow {\mathcal{V}}^{\pm}\tag{30}\] given by sending \(e_{i,d}\) and \(f_{i,d}\) (respectively) to \(z_{i1}^d \in {\mathcal{V}}_{\boldsymbol{\varsigma}^i} = {\mathcal{V}}^{\text{op}}_{\boldsymbol{\varsigma}^i}\), for all \(i \in I\), \(d \in {\mathbb{Z}}\).

Elements of \({\mathcal{S}}^\pm\) will be called shuffle elements. By definition, any \(E \in {\mathcal{S}}^+\) can be written as a linear combination of shuffle elements of the form \[\label{eqn:spherical} E = \text{Sym} \left[ \nu(z_1,\dots,z_n) \prod_{1 \leq a < b \leq n} \zeta_{i_ai_b} \left(\frac{z_a}{z_b} \right) \right]\tag{34}\] as \(\nu\) goes over all Laurent polynomials. In formula 34 , we consider any \(i_1,\dots,i_n \in I\) and use the notation \(z_a\) as a placeholder for the variable \(z_{i_a\bullet_a}\), where \(\bullet_1,\dots,\bullet_n\) denote the minimal positive integers such that \(\bullet_a < \bullet_b\) if \(a<b\) and \(i_a = i_b\).

2.4 Extended algebras↩︎

To make \(\mathbf{\widetilde{U}}^\pm\), \({\mathcal{V}}^\pm\), \(\mathbf{U}^\pm \cong {\mathcal{S}}^\pm\) into bialgebras, we need to extend them by introducing commuting loop Cartan elements. The vector spaces \[\begin{align} &\mathbf{\widetilde{U}}^\geq= \mathbf{\widetilde{U}}^+\otimes {\mathbb{K}}[\varphi^+_{i,d}]_{i \in I, d \geq 0} \tag{35} \\ &\mathbf{\widetilde{U}}^\leq= {\mathbb{K}}[\varphi^-_{i,d}]_{i \in I, d \geq 0} \otimes \mathbf{\widetilde{U}}^-\tag{36} \end{align}\] can be made into algebras by imposing the commutation relations \[\begin{align} &\varphi^+_i(z) e_j(w) = e_j(w) \varphi^+_i(z) \frac{\zeta_{ij} \left(\frac{z}{w} \right)}{\zeta_{ji} \left(\frac{w}{z} \right)} \tag{37} \\ &f_j(w)\varphi^-_i(z) = \varphi^-_i(z) f_j(w) \frac{\zeta_{ij} \left(\frac{z}{w} \right)}{\zeta_{ji} \left(\frac{w}{z} \right)} \tag{38} \end{align}\] for all \(i,j \in I\), where \[\varphi^+_i(z) = \sum_{d=0}^{\infty} \frac{\varphi^+_{i,d}}{z^d} \qquad \text{and} \qquad \varphi^-_i(z) = \sum_{d=0}^{\infty} \varphi^-_{i,d} z^d\] Note that the assumption 2 implies that \(\frac{\zeta_{ij} (x)}{\zeta_{ji} (x^{-1})}\) is regular and non-zero at both \(x = 0\) and \(x = \infty\), for all \(i,j \in I\). With this in mind, one interprets the relations in 37 (respectively 38 ) by expanding them as power series in negative (respectively positive) powers of \(\frac{z}{w}\). Similarly, we define \[\begin{align} &{\mathcal{V}}^{\geq} = {\mathcal{V}}^+ \otimes {\mathbb{K}}[\varphi^+_{i,d}]_{i \in I, d \geq 0} \tag{39} \\ &{\mathcal{V}}^{\leq} = {\mathbb{K}}[\varphi^-_{i,d}]_{i \in I, d \geq 0} \otimes {\mathcal{V}}^- \tag{40} \end{align}\] and make them into algebras by imposing the commutation relations \[\begin{align} &\varphi^+_i(y) E(z_{j1},\dots,z_{jn_j})_{j \in I} = E(z_{j1},\dots,z_{jn_j})_{j \in I} \varphi^+_i(y) \prod_{j \in I} \prod_{a=1}^{n_j}\frac{\zeta_{ij} \left(\frac{y}{z_{ja}} \right)}{\zeta_{ji} \left(\frac{z_{ja}}{y} \right)} \tag{41} \\ &F(z_{j1},\dots,z_{jn_j})_{j \in I} \varphi^-_i(y) = \varphi^-_i(y) F(z_{j1},\dots,z_{jn_j})_{j \in I} \prod_{j \in I} \prod_{a=1}^{n_j}\frac{\zeta_{ij} \left(\frac{y}{z_{ja}} \right)}{\zeta_{ji} \left(\frac{z_{ja}}{y} \right)} \tag{42} \end{align}\] for all \(E \in {\mathcal{V}}^+\), \(F \in {\mathcal{V}}^-\) and \(i \in I\). Because the homomorphisms \(\widetilde{\Upsilon}^\pm\) respect all the constructions above, we obtain natural algebra structures on \[\begin{align} &\mathbf{U}^\geq= \mathbf{U}^+\otimes {\mathbb{K}}[\varphi^+_{i,d}]_{i \in I, d \geq 0} \tag{43} \\ &\mathbf{U}^\leq= {\mathbb{K}}[\varphi^-_{i,d}]_{i \in I, d \geq 0} \otimes \mathbf{U}^-\tag{44} \end{align}\] and \[\begin{align} &{\mathcal{S}}^{\geq} = {\mathcal{S}}^+ \otimes {\mathbb{K}}[\varphi^+_{i,d}]_{i \in I, d \geq 0} \tag{45} \\ &{\mathcal{S}}^{\leq} = {\mathbb{K}}[\varphi^-_{i,d}]_{i \in I, d \geq 0} \otimes {\mathcal{S}}^- \tag{46} \end{align}\] which are isomorphic to each other with respect to 33 .

2.5 Gradings and completions↩︎

All algebras in this paper are graded by \[\deg = (\text{hdeg}, \text{vdeg}) \in {\mathbb{Z}^I}\times {\mathbb{Z}}\] with the \({\mathbb{Z}^I}\) component called horizontal degree (denoted by hdeg) and the \({\mathbb{Z}}\) component called vertical degree (denoted by vdeg). Explicitly, we have \[\begin{align} &\deg e_{i,d} = (\boldsymbol{\varsigma}^i, d) \\ &\deg f_{i,d} = (-\boldsymbol{\varsigma}^i, d) \\ &\deg \varphi^+_{i,d} = (0, d) \\ &\deg \varphi^-_{i,d} = (0, -d) \\ &\deg E = ({\boldsymbol{n}}, \text{homogeneous degree of }E) \\ &\deg F = (-{\boldsymbol{n}}, \text{homogeneous degree of }F) \end{align}\] for any \(E \in {\mathcal{V}}_{{\boldsymbol{n}}}\) and \(F \in {\mathcal{V}}^{\text{op}}_{{\boldsymbol{n}}}\) which are homogeneous in all their variables. Write \[\begin{align} &{\mathcal{V}}^+ = \bigoplus_{{\boldsymbol{n}}\in {\mathbb{N}^I}} {\mathcal{V}}_{{\boldsymbol{n}}} = \bigoplus_{({\boldsymbol{n}},d) \in {\mathbb{N}^I}\times {\mathbb{Z}}} {\mathcal{V}}_{{\boldsymbol{n}},d} \tag{47} \\ &{\mathcal{V}}^- = \bigoplus_{{\boldsymbol{n}}\in {\mathbb{N}^I}} {\mathcal{V}}_{-{\boldsymbol{n}}} = \bigoplus_{({\boldsymbol{n}},d) \in {\mathbb{N}^I}\times {\mathbb{Z}}} {\mathcal{V}}_{-{\boldsymbol{n}},d} \tag{48} \end{align}\] for the graded summands of \({\mathcal{V}}^\pm\) (and analogously for \({\mathcal{S}}^\pm, \mathbf{\widetilde{U}}^\pm, \mathbf{U}^\pm\)). For any algebra \(A\) which is graded by \({\mathbb{Z}^I}\times {\mathbb{Z}}\), we may define the completions \[\begin{align} &A \stackrel{\mathsf{V}}{\otimes}A = \bigoplus_{({\boldsymbol{n}},d) \in {\mathbb{Z}^I}\times {\mathbb{Z}}} (A \stackrel{\mathsf{V}}{\otimes}A)_{{\boldsymbol{n}},d} \tag{49} \\ &A \stackrel{\mathsf{H}}{\otimes}A = \bigoplus_{({\boldsymbol{n}},d) \in {\mathbb{Z}^I}\times {\mathbb{Z}}} (A \stackrel{\mathsf{H}}{\otimes}A)_{{\boldsymbol{n}},d} \tag{50} \end{align}\] where \((A \stackrel{\mathsf{V}}{\otimes}A)_{{\boldsymbol{n}},d}\) (respectively \((A \stackrel{\mathsf{H}}{\otimes}A)_{{\boldsymbol{n}},d}\)) consists of infinite sums of tensors \(x \otimes y\) where all but finitely many summands have the property that \(\text{vdeg }x \geq N\) and \(\text{vdeg }y \leq -N\) (respectively \(|\text{hdeg }x| \leq -N\) and \(|\text{hdeg }y| \geq N\), recall the notation 24 ) for any \(N \in {\mathbb{N}}\). It is a straightforward exercise, which we leave to the reader, to check that 49 and 50 are indeed algebras, i.e. the product of infinite sums of tensors is a well-defined infinite sum of tensors in the sense above.

2.6 The topological coproduct↩︎

Our reason for defining the extended algebras in the previous Subsection is to make them into bialgebras. We start with the natural generalization of Drinfeld’s new coproduct on quantum affine algebras \[\label{eqn:coproduct32tu} \Delta : \mathbf{\widetilde{U}}^\geq\rightarrow \mathbf{\widetilde{U}}^\geq\stackrel{\mathsf{V}}{\otimes}\mathbf{\widetilde{U}}^\geq\qquad \text{and} \qquad \Delta : \mathbf{\widetilde{U}}^\leq\rightarrow \mathbf{\widetilde{U}}^\leq\stackrel{\mathsf{V}}{\otimes}\mathbf{\widetilde{U}}^\leq\tag{51}\] given by \[\label{eqn:coproduct32h} \Delta(\varphi^\pm_i(z)) = \varphi^\pm_i(z) \otimes \varphi^\pm_i(z)\tag{52}\] \[\label{eqn:coproduct32e} \Delta(e_i(z)) = \varphi^+_i(z) \otimes e_i(z) + e_i(z) \otimes 1\tag{53}\] \[\label{eqn:coproduct32f} \Delta(f_i(z)) = 1 \otimes f_i(z) + f_i(z) \otimes \varphi^-_i(z)\tag{54}\] Similarly, the following coproducts \[\label{eqn:coproduct32v} \Delta : {\mathcal{V}}^{\geq} \rightarrow {\mathcal{V}}^{\geq} \stackrel{\mathsf{V}}{\otimes}{\mathcal{V}}^{\geq} \qquad \text{and} \qquad \Delta : {\mathcal{V}}^{\leq} \rightarrow {\mathcal{V}}^{\leq} \stackrel{\mathsf{V}}{\otimes}{\mathcal{V}}^{\leq}\tag{55}\] are natural generalizations of those of N?, Shuffle?: \(\Delta(\varphi^\pm_i(z)) = \varphi^\pm_i(z) \otimes \varphi^\pm_i(z)\) and \[\begin{align} \Delta(E) = \sum_{{\boldsymbol{0}} \leq {\boldsymbol{m}}\leq {\boldsymbol{n}}} \frac{\prod^{j \in I}_{m_j < b \leq n_j} \varphi^+_j(z_{jb}) E(z_{i1},\dots , z_{im_i} \otimes z_{i,m_i+1}, \dots, z_{in_i})}{\prod^{i \in I}_{1\leq a \leq m_i} \prod^{j \in I}_{m_j < b \leq n_j} \zeta_{ji} \left( \frac{z_{jb}}{z_{ia}} \right)} \tag{56} \\ \Delta(F) = \sum_{{\boldsymbol{0}} \leq {\boldsymbol{m}}\leq {\boldsymbol{n}}} \frac{F(z_{i1},\dots , z_{im_i} \otimes z_{i,m_i+1}, \dots, z_{in_i}) \prod^{j \in I}_{1 \leq b \leq m_j} \varphi^-_j(z_{jb})}{\prod^{i \in I}_{1\leq a \leq m_i} \prod^{j \in I}_{m_j < b \leq n_j} \zeta_{ij} \left( \frac{z_{ia}}{z_{jb}} \right)} \tag{57} \end{align}\] for all \(E \in {\mathcal{V}}_{{\boldsymbol{n}}}\), \(F \in {\mathcal{V}}_{-{\boldsymbol{n}}}\). To make sense of the right-hand side of formulas 56 and 57 , we expand the denominator as a power series in the range \(|z_{ia}| \ll |z_{jb}|\), and place all the powers of \(z_{ia}\) to the left of the \(\otimes\) sign and all the powers of \(z_{jb}\) to the right of the \(\otimes\) sign (for all \(i,j \in I\), \(1 \leq a \leq m_i\) and \(m_j < b \leq n_j\)). It is easy to see that the maps 30 respect the coproducts, and thus descend to \[\label{eqn:coproduct32u} \Delta : \mathbf{U}^\geq\rightarrow \mathbf{U}^\geq\stackrel{\mathsf{V}}{\otimes}\mathbf{U}^\geq\quad \text{ and } \quad \Delta : \mathbf{U}^\leq\rightarrow \mathbf{U}^\leq\stackrel{\mathsf{V}}{\otimes}\mathbf{U}^\leq\tag{58}\] \[\label{eqn:coproduct32s} \Delta : {\mathcal{S}}^{\geq} \rightarrow {\mathcal{S}}^{\geq} \stackrel{\mathsf{V}}{\otimes}{\mathcal{S}}^{\geq} \qquad \text{and} \quad \;\;\Delta : {\mathcal{S}}^{\leq} \rightarrow {\mathcal{S}}^{\leq} \stackrel{\mathsf{V}}{\otimes}{\mathcal{S}}^{\leq}\tag{59}\] which match each other under the isomorphisms \(\mathbf{U}^\geq\cong {\mathcal{S}}^{\geq}\) and \(\mathbf{U}^\leq\cong {\mathcal{S}}^{\leq}\).

2.7 The pairing↩︎

Consider the following notation for all rational functions \(G\): \[\label{eqn:contour32integral} \int_{|z_1| \gg \dots \gg |z_n|} G(z_1,\dots,z_n)\tag{60}\] denotes the constant term in the expansion of \(G\) as a power series in \[\frac{z_2}{z_1}, \dots, \frac{z_n}{z_{n-1}}\] This notation is motivated by the fact that when \({\mathbb{K}}= {\mathbb{C}}\), one could compute the constant term as the contour integral of \(G(z_1,\dots,z_n)\prod_{a=1}^n \frac{dz_a}{2\pi i z_a}\) over concentric circles centered at the origin. We define \(\int_{|z_1| \ll \dots \ll |z_n|} G(z_1,\dots,z_n)\) analogously.

In the right-hand sides of ?? and ?? , we implicitly identify \[\label{eqn:relabeling} z_a \quad \text{with} \quad z_{i_a\bullet_a}, \quad \forall a \in \{1,\dots, n\}\tag{61}\] where \(\bullet_1,\dots,\bullet_n\) are the minimal positive integers such that \(\bullet_a < \bullet_b\) if \(a < b\) and \(i_a = i_b\). Note that the pairings ?? and ?? are non-zero only on elements of opposite degree in \({\mathbb{Z}^I}\times {\mathbb{Z}}\). It was shown in N?, Arbitrary? that there exist descended pairings \[\label{eqn:pair32descended} \mathbf{U}^+\otimes {\mathcal{S}}^- \xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\qquad \text{and} \qquad {\mathcal{S}}^+ \otimes \mathbf{U}^-\xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{62}\] which are non-degenerate and coincide under the isomorphisms 33 .

2.8 Doubles↩︎

In what follows, we will use Sweedler’s notation for coproducts \[\Delta(a) = a_1 \otimes a_2 \quad \text{instead of the more precise} \quad \Delta(a) = \sum_k a_{1,k} \otimes a_{2,k}\] Suppose we have bialgebras \(A\) and \(B\) over the field \({\mathbb{K}}\). A pairing \[\label{eqn:bialgebra32pairing} A \otimes B \xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{63}\] is called a bialgebra pairing if it satisfies \[\begin{align} &\Big \langle a,b'b'' \Big \rangle = \Big \langle \Delta(a), b' \otimes b'' \Big \rangle = \langle a_1,b'\rangle \langle a_2,b''\rangle\tag{64} \\ &\Big \langle a' a'' ,b \Big \rangle = \Big \langle a' \otimes a'', \Delta^{\text{op}}(b) \Big \rangle = \langle a',b_2\rangle \langle a'',b_1\rangle\tag{65} \end{align}\] for all \(a,a',a'' \in A\) and \(b,b',b'' \in B\) (\(\Delta^{\text{op}}\) is the opposite coproduct). The notion above is sometimes called a skew-bialgebra pairing, as it identifies the coproduct on \(A\) with the dual of the product on \(B\) and the opposite coproduct on \(B\) with the dual of the product on \(A\). Whenever we have bialgebras \(A\) and \(B\) with a bialgebra pairing 63 as above, the Drinfeld double construction makes the vector space \[\label{eqn:drinfeld32double} D = A \otimes B\tag{66}\] into a bialgebra which contains \(A = A \otimes 1\) and \(B = 1 \otimes B\) as sub-bialgebras. Indeed, the multiplication in \(D\) is governed by the relation \[\label{eqn:drinfeld32double32relation} a_1 b_1 \langle a_2,b_2 \rangle = \langle a_1,b_1\rangle b_2a_2, \qquad \forall a \in A = A \otimes 1, \;b \in B = 1 \otimes B\tag{67}\] Practically, 67 allows one to take an arbitrary product of \(a\)’s and \(b\)’s and convert it into a linear combination of products of the form \(ab\), which are elements of 66 .

If the pairing 63 is non-degenerate, then it induces injective maps \[\label{eqn:hooks} A \hookrightarrow B^* \qquad \text{and} \qquad B \hookrightarrow A^*\tag{68}\] If \(A\) and \(B\) are finite-dimensional over \({\mathbb{K}}\), a non-degenerate pairing is perfect, i.e. the inclusions in 68 are actually isomorphisms. The algebras considered in the present paper are certainly not finite-dimensional, but we will often encounter algebras with finite-dimensional graded summands. In this case, we make the convention that all our duals will always be considered in the graded sense, i.e. \[\text{if } A = \bigoplus_{({\boldsymbol{n}},d) \in {\mathbb{Z}^I}\times {\mathbb{Z}}} A_{{\boldsymbol{n}},d}, \quad \text{then we define } A^* = \bigoplus_{({\boldsymbol{n}},d) \in {\mathbb{Z}^I}\times {\mathbb{Z}}} A_{{\boldsymbol{n}},d}^*\] In this case, a non-degenerate pairing (which respects the grading, by which we mean that it only pairs non-trivially elements of opposite degrees) between graded bialgebras with finite dimensional graded summands is always perfect.

2.9 Universal \(R\)-matrices↩︎

One of the big motivations for the introduction of Drinfeld doubles is that they are endowed with a universal \(R\)-matrix \[{\mathcal{R}}\in D \otimes D\] which satisfies the properties \[\label{eqn:universal321} {\mathcal{R}}\cdot \Delta(-) = \Delta^{\text{op}}(-) \cdot {\mathcal{R}}\tag{69}\] \[\label{eqn:universal322} \left( \Delta \otimes \text{Id}_D \right)({\mathcal{R}}) = {\mathcal{R}}_{13} {\mathcal{R}}_{23}\tag{70}\] \[\label{eqn:universal323} \left( \text{Id}_D \otimes \Delta \right)({\mathcal{R}}) = {\mathcal{R}}_{13} {\mathcal{R}}_{12}\tag{71}\] (in the right-hand side of the expressions above, we write \({\mathcal{R}}_{12}, {\mathcal{R}}_{13}, {\mathcal{R}}_{23}\) for the tensors in \(D \otimes D \otimes D\) which are equal to \({\mathcal{R}}\) on the two indices in the subscript and 1 on the third index). Indeed, we have the following general result.

2.10 The quantum loop algebra↩︎

We conclude this Section by assembling all the ingredients above into the hodge-podge that is the definition of quantum loop algebras. We work with any \((I,{\mathbb{K}},\zeta_{ij}(x))\), as in Subsection 1.2. Recall the pairing \[\label{eqn:quantum32pairing321} \mathbf{U}^+\otimes \mathbf{U}^-\xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{77}\] induced by 62 and the isomorphisms 33 . It is straightforward to show that one can extend 77 to a bialgebra pairing \[\label{eqn:quantum32pairing322} \mathbf{U}^\geq\otimes \mathbf{U}^\leq\xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{78}\] by requiring for all \(i,j \in I\) that \[\label{eqn:pairing32h39s} \Big \langle \varphi^+_i(z), \varphi^-_j(w) \Big \rangle = \frac{\zeta_{ij}\left(\frac{z}{w} \right)}{\zeta_{ji}\left(\frac{w}{z} \right)}\tag{79}\] (and that all pairings between \(\varphi^\pm\) on one hand and \(e,f\) on the other hand vanish).

Explicitly, \(\mathbf{U}\) has generators \(\{e_{i,d}, f_{i,d}, \varphi^+_{i,d'}, \varphi^-_{i,d'} \}_{i \in I, d \in {\mathbb{Z}}, d' \geq 0}\) modulo relations \[\label{eqn:rel32quantum321} e_i(z) e_j(w) \zeta_{ji} \left(\frac{w}{z}\right) = e_j(w) e_i(z) \zeta_{ij} \left( \frac{z}{w} \right)\tag{81}\] \[\label{eqn:rel32quantum322} f_i(z) f_j(w) \zeta_{ij} \left( \frac{z}{w} \right) = f_j(w) f_i(z) \zeta_{ji} \left(\frac{w}{z}\right)\tag{82}\] \[\label{eqn:rel32quantum323} \Big(\text{any element of Ker }\widetilde{\Upsilon}^\pm \Big) = 0\tag{83}\] \[\label{eqn:rel32quantum324} \left[\varphi^+_{i}(z), \varphi^+_{j}(w)\right] = \left[\varphi^+_{i}(z), \varphi^-_{j}(w)\right] = \left[\varphi^-_{i}(z), \varphi^-_{j}(w)\right] = 0\tag{84}\] \[\begin{align} &\varphi^+_i(z) e_j(w) = e_j(w) \varphi^+_i(z) \frac{\zeta_{ij} \left(\frac{z}{w} \right)}{\zeta_{ji} \left(\frac{w}{z} \right)} \tag{85} \\ &\varphi^-_i(z) e_j(w) = e_j(w) \varphi^-_i(z) \frac{\zeta_{ij} \left(\frac{z}{w} \right)}{\zeta_{ji} \left(\frac{w}{z} \right)} \tag{86} \\ &f_j(w)\varphi^+_i(z) = \varphi^+_i(z) f_j(w) \frac{\zeta_{ij} \left(\frac{z}{w} \right)}{\zeta_{ji} \left(\frac{w}{z} \right)} \tag{87} \\ &f_j(w)\varphi^-_i(z) = \varphi^-_i(z) f_j(w) \frac{\zeta_{ij} \left(\frac{z}{w} \right)}{\zeta_{ji} \left(\frac{w}{z} \right)} \tag{88} \end{align}\] \[\label{eqn:rel32quantum329} \Big[e_{i}(z), f_{j}(w)\Big] = \delta_{ij} \delta \left(\frac{z}{w} \right) \Big( \varphi^-_j(w) - \varphi^+_i(z)\Big)\tag{89}\] for all \(i,j \in I\), where \(\delta(x) = \sum_{d \in {\mathbb{Z}}} x^d\) is a formal series. Similarly, we define \[\label{eqn:double32shuffle} {\mathcal{S}}= {\mathcal{S}}^{\geq} \otimes {\mathcal{S}}^{\leq}\tag{90}\] as a Drinfeld double with respect to the coproducts 59 and the bialgebra pairing \[\label{eqn:pair32shuffle32extended} {\mathcal{S}}^{\geq} \otimes {\mathcal{S}}^{\leq} \xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{91}\] induced by ?? and 79 . Clearly, we have \({\mathcal{S}}\cong \mathbf{U}\) and so we henceforth identify \[\label{eqn:identify} {\mathcal{S}}= \mathbf{U}\tag{92}\]

2.11 Cartan subalgebras↩︎

The subalgebra \[\label{eqn:loop32cartan} {\mathbb{K}}[\varphi_{i,d}^+, \varphi_{i,d}^-]_{i \in I, d \geq 0} \subset \mathbf{U}\tag{93}\] is called the loop Cartan subalgebra, and its subalgebras \[{\mathbb{K}}[\varphi_{i,d}^+]_{i \in I, d \geq 0} \qquad \text{and} \qquad {\mathbb{K}}[\varphi_{i,d}^-]_{i \in I, d \geq 0}\] are called the positive and negative loop Cartan, respectively. The leading terms \(\varphi_{i,0}^\pm\) will act by non-zero elements of \({\mathbb{K}}\) on all (homogeneous) elements of all modules considered in the present paper, and so it makes sense to assume \[\label{eqn:power32series} \varphi_{i}^\pm(z) = \kappa_i^\pm \exp \left(\sum_{d=1}^{\infty} \frac{p_{i,\pm d}}{d z^{\pm d}}\right)\tag{94}\] In other words, we simply assume that all the \(\varphi_{i,d}^\pm\) are multiples of \(\varphi_{i,0}^\pm = \kappa_i^\pm\), which could be ensured (for example) by inverting the latter elements. The subalgebra \[{\mathbb{K}}[\kappa_i^{+}, \kappa_i^{-}]_{i \in I} \subset \mathbf{U}\] is called the (finite) Cartan subalgebra. An easy consequence ¹ of 52 is \[\label{eqn:coproduct32k} \Delta(\kappa_i^\pm) = \kappa_i^\pm \otimes \kappa_i^\pm\tag{95}\] \[\label{eqn:coproduct32p} \Delta(p_{i,\pm d}) = p_{i,\pm d} \otimes 1 + 1 \otimes p_{i,\pm d}\tag{96}\] for all \(i \in I\) and \(d \geq 1\). Moreover, 79 implies \[\label{eqn:pairing32k} \langle \kappa_i^+, \kappa_j^- \rangle = \gamma_{\boldsymbol{\varsigma}^i,\boldsymbol{\varsigma}^j}\tag{97}\] \[\label{eqn:pairing32p} \langle p_{i,d}, p_{j,-d} \rangle = d \alpha_{ij}^{(d)}\tag{98}\] for all \(i,j \in I\) and \(d \geq 1\), where \(\alpha_{ij}^{(d)} \in {\mathbb{K}}\) are defined by the power series expansion \[\label{eqn:notation} \frac{\zeta_{ij}(x)}{\zeta_{ji}\left(x^{-1}\right)} = \gamma_{\boldsymbol{\varsigma}^i,\boldsymbol{\varsigma}^j} \exp \left(\sum_{d=1}^{\infty} \frac{\alpha_{ij}^{(d)}}{dx^d} \right)\tag{99}\] (see 27 for the definition of \(\gamma_{\boldsymbol{\varsigma}^i,\boldsymbol{\varsigma}^j}\)). All pairings between \(\kappa\)’s and \(p\)’s other than 97 and 98 vanish for degree reasons. We will call the zeta functions 25 fully factored if their numerators are fully factored into linear terms, i.e. \[\label{eqn:fully32factored} \zeta_{ij}(x) = \frac{c_{ij} x^{-\#_{ji}}(x-s_{ij|1})\dots (x-s_{ij|\#_{ij}+\#_{ji}+\delta_{ij}})}{(x-1)^{\delta_{ij}}}\tag{100}\] for all \(i,j \in I\) and various \(s_{ij|b} \in {\mathbb{K}}^*\) (the fully factored condition is automatically satisfied if \({\mathbb{K}}\) is algebraically closed). In this case, an easy consequence of 99 is \[\label{eqn:consequence} \alpha_{ij}^{(d)} = \sum_{b=1}^{\#_{ij}+\#_{ji}+\delta_{ij}} (s_{ji|b}^{-d} - s_{ij|b}^d)\tag{101}\]

3 Slopes and coproducts↩︎

We begin by recalling the discussion of slopes in shuffle algebras and quantum loop algebras, following [16], N?, Shuffle?, N?, R-matrix?, which will allow us to construct the subalgebras \[{\mathcal{A}}^{\geq {\boldsymbol{p}}}, {\mathcal{A}}^{\leq {\boldsymbol{p}}} \subset \mathbf{U}\] for any \({\boldsymbol{p}}\in {\mathbb{R}^I}\). We then define topological coproducts \(\Delta_{{\boldsymbol{p}}}\) on \({\mathcal{A}}^{\geq {\boldsymbol{p}}}\) and \({\mathcal{A}}^{\leq {\boldsymbol{p}}}\), and show that \(\mathbf{U}\) is their Drinfeld double with respect to a pairing induced by 78 . We call \(\Delta_{{\boldsymbol{p}}}\) thus defined “new new" coproducts, and in the particular case of quantum affine algebras 6 , we show that \(\Delta_{{\boldsymbol{0}}}\) matches the Drinfeld-Jimbo coproduct. We also deduce various factorizations of universal \(R\)-matrices.

3.1 Slopes↩︎

Our references for slope subalgebras are N?, R-matrix? and N?, Char?; while both these works are written for particular choices of \((I,{\mathbb{K}},\zeta_{ij}(x))\), all the results therein actually hold in our current level of generality. Recall the notation 23 26 .

If moreover the limits in ?? ?? are all 0, then we will say that \(E\) and \(F\) therein have slopes \(<{\boldsymbol{p}}\) or \(>{\boldsymbol{p}}\), respectively. We will write \[\label{eqn:slopes} {\mathcal{S}}^\pm_{\geq {\boldsymbol{p}}}, \;{\mathcal{S}}^\pm_{\leq {\boldsymbol{p}}}, \;{\mathcal{S}}^\pm_{> {\boldsymbol{p}}}, \;{\mathcal{S}}^\pm_{< {\boldsymbol{p}}}\tag{102}\] for the subsets of elements of \({\mathcal{S}}^\pm\) of slope \(\geq {\boldsymbol{p}}\), \(\leq {\boldsymbol{p}}\), \(> {\boldsymbol{p}}\), \(< {\boldsymbol{p}}\), respectively. It is elementary to show that all the sets which appear in 102 are actually subalgebras of \({\mathcal{S}}^\pm\) with respect to the shuffle product. As proved in N?, R-matrix?, the coproduct of Subsection 2.6 interacts with the subalgebras defined above as follows \[\begin{align} &\Delta ({\mathcal{S}}^+_{\geq {\boldsymbol{p}}} ) \subset {\mathcal{S}}^\geq_{\geq {\boldsymbol{p}}} \stackrel{\mathsf{V}}{\otimes}{\mathcal{S}}^+ \tag{103} \\ &\Delta ({\mathcal{S}}^+_{\leq {\boldsymbol{p}}} ) \subset {\mathcal{S}}^\geq \stackrel{\mathsf{V}}{\otimes}{\mathcal{S}}^+_{\leq {\boldsymbol{p}}} \tag{104} \\ &\Delta ({\mathcal{S}}^-_{\leq {\boldsymbol{p}}} ) \subset {\mathcal{S}}^-_{\leq {\boldsymbol{p}}} \stackrel{\mathsf{V}}{\otimes}{\mathcal{S}}^\leq \tag{105} \\ &\Delta ({\mathcal{S}}^-_{\geq {\boldsymbol{p}}} ) \subset {\mathcal{S}}^- \stackrel{\mathsf{V}}{\otimes}{\mathcal{S}}^\leq_{\geq {\boldsymbol{p}}} \tag{106} \end{align}\] where \({\mathcal{S}}^\geq_{\geq {\boldsymbol{p}}}\) and \({\mathcal{S}}^\leq_{\geq {\boldsymbol{p}}}\) are obtained from \({\mathcal{S}}^+_{\geq {\boldsymbol{p}}}\) and \({\mathcal{S}}^-_{\geq {\boldsymbol{p}}}\) by adding loop Cartan elements as in 45 and 46 .

3.2 Slope subalgebras↩︎

If a shuffle element \(X\) simultaneously has slope \(\leq {\boldsymbol{p}}\) and \(\geq {\boldsymbol{p}}\), then \(\text{vdeg }X = {\boldsymbol{p}}\cdot \text{hdeg }X\). The set of such elements is denoted by \[\label{eqn:slope32subalgebra} {\mathcal{B}}_{{\boldsymbol{p}}}^\pm = \bigoplus_{{\boldsymbol{n}}\in {\mathbb{N}^I}\text{ s.t. } {\boldsymbol{p}}\cdot {\boldsymbol{n}}\in {\mathbb{Z}}} {\mathcal{B}}_{{\boldsymbol{p}}|\pm {\boldsymbol{n}}}\tag{107}\] and is called a slope subalgebra. Recall the notation \(\kappa^\pm_i = \varphi_{i,0}^\pm\). The vector spaces \[\begin{align} &{\mathcal{B}}_{{\boldsymbol{p}}}^\geq = {\mathcal{B}}_{{\boldsymbol{p}}}^+ \otimes {\mathbb{K}}[\kappa^+_i]_{i \in I} \tag{108} \\ &{\mathcal{B}}_{{\boldsymbol{p}}}^\leq = {\mathbb{K}}[\kappa^-_i]_{i \in I} \otimes {\mathcal{B}}_{{\boldsymbol{p}}}^- \tag{109} \end{align}\] are subalgebras of \({\mathcal{S}}^\geq\) and \({\mathcal{S}}^\leq\), respectively, as long as we impose the following commutation relations derived from taking the leading order terms of 41 42 \[\label{eqn:cartan32commutation32plus} \kappa^+_i X = X \kappa^+_i \gamma_{\boldsymbol{\varsigma}^i, \text{hdeg }X}\tag{110}\] \[\label{eqn:cartan32commutation32minus} \kappa^-_i X = X \kappa^-_i \gamma_{\text{hdeg }X, \boldsymbol{\varsigma}^i}^{-1}\tag{111}\] for any \(X \in {\mathcal{S}}\) (recall that the scalars \(\gamma\) are defined in 27 ). We can further make the vector spaces 108 and 109 into bialgebras, using \(\Delta_{{\boldsymbol{p}}}(\kappa_i^\pm) = \kappa_i^\pm \otimes \kappa_i^\pm\) and \[\begin{align} &\Delta_{\boldsymbol{p}}(E) = \sum_{{\boldsymbol{0}} \leq {\boldsymbol{m}}\leq {\boldsymbol{n}}} (\kappa^+_{{\boldsymbol{n}}-{\boldsymbol{m}}} \otimes 1) (\text{value of the limit \eqref{eqn:slope32e32geq}}) \tag{112} \\ &\Delta_{\boldsymbol{p}}(F) = \sum_{{\boldsymbol{0}} \leq {\boldsymbol{m}}\leq {\boldsymbol{n}}} (\text{value of the limit \eqref{eqn:slope32f32leq}}) (1 \otimes \kappa^-_{{\boldsymbol{m}}}) \tag{113} \end{align}\] for all \(E \in {\mathcal{B}}^+_{\boldsymbol{p}}\) and \(F \in {\mathcal{B}}^-_{\boldsymbol{p}}\), where we write \(\kappa^\pm_{{\boldsymbol{m}}} = \prod_{i \in I} (\kappa^\pm_i)^{m_i}\) for all \({\boldsymbol{m}}\in {\mathbb{N}^I}\).

3.3 Infinite slope↩︎

We will also define slope \(\boldsymbol{\infty}\) versions of slope subalgebras, \[\label{eqn:slope32infinity} {\mathcal{B}}^{\geq}_{\boldsymbol{\infty}} = {\mathbb{K}}[\varphi^+_{i,d}]_{i \in I, d\geq 0} \qquad \text{and} \qquad {\mathcal{B}}^{\leq}_{\boldsymbol{\infty}} = {\mathbb{K}}[\varphi^-_{i,d}]_{i \in I, d\geq 0}\tag{114}\] \[\label{eqn:slope32infinity322} {\mathcal{B}}^{+}_{\boldsymbol{\infty}} = {\mathbb{K}}[p_{i,d}]_{i \in I, d> 0} \qquad \text{and} \qquad {\mathcal{B}}^{-}_{\boldsymbol{\infty}} = {\mathbb{K}}[p_{i,-d}]_{i \in I, d> 0}\tag{115}\] with the notation as in Subsection 2.11. The coproduct \(\Delta_{\boldsymbol{\infty}}\) is then defined as restriction of \(\Delta\) to the above commutative subalgebras. We assume that \[\label{eqn:pairing32slope32infinity} {\mathcal{B}}^\geq_{\boldsymbol{\infty}} \otimes {\mathcal{B}}^\leq_{\boldsymbol{\infty}} \xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{116}\] is non-degenerate, which boils down to the condition that the \(|I| \times |I|\) matrix with entries RHS of 98 is invertible for all \(d\), and that the scalars 27 are linearly independent in both \({\boldsymbol{m}}\in {\mathbb{N}^I}\) and \({\boldsymbol{n}}\in {\mathbb{N}^I}\) separately. If the latter condition fails, all is not lost; the usual solution is to enlarge the finite Cartan subalgebra, i.e. to add new symbols \(\kappa_j^\pm\) which interact with \(X \in {\mathcal{S}}\) according to 110 111 , for newly chosen scalars \[\Big\{ \gamma_{\boldsymbol{\varsigma}^j,{\boldsymbol{n}}} \Big\}_{{\boldsymbol{n}}\in {\mathbb{Z}^I}} \qquad \text{and} \quad \Big\{ \gamma_{{\boldsymbol{n}},\boldsymbol{\varsigma}^j}^{-1} \Big\}_{{\boldsymbol{n}}\in {\mathbb{Z}^I}}\] that are additive in \({\boldsymbol{n}}\) and sufficiently generic.

3.4 Factorizations↩︎

Slope subalgebras are important because they are the building blocks of shuffle algebras, as we will now recall. A parameterized curve \[{\mathbb{R}}\rightarrow {\mathbb{R}^I}, \quad t \mapsto {\boldsymbol{p}}(t) = (p_i(t))_{i \in I}\] will be called catty-corner if \[\label{eqn:catty-corner321} t_1 < t_2 \quad \text{implies} \quad p_i(t_1) < p_i(t_2)\tag{117}\] and \[\label{eqn:catty-corner322} \lim_{t \rightarrow \pm \infty} p_i(t) = \pm \infty\tag{118}\] for all \(i \in I\). Note that condition 117 is stronger than \({\boldsymbol{p}}(t_1) < {\boldsymbol{p}}(t_2)\), because the latter just means \({\boldsymbol{p}}(t_1) \leq {\boldsymbol{p}}(t_2)\) and \({\boldsymbol{p}}(t_1) \neq {\boldsymbol{p}}(t_2)\). We may also define catty-corner curves on bounded intervals of \({\mathbb{R}}\), in which case condition 118 would be dropped.

By the same token as in Proposition 7, we have \[\label{eqn:factorization322} \bigotimes^{\rightarrow }_{t \in [t_1,t_2]} {\mathcal{B}}^\pm_{{\boldsymbol{p}}(t)} \xrightarrow{\sim} {\mathcal{S}}^\pm_{\geq {\boldsymbol{p}}(t_1)} \cap {\mathcal{S}}^\pm_{\leq {\boldsymbol{p}}(t_2)}\tag{119}\] for any \(t_1 \leq t_2\), as well as the natural analogues of 119 when some endpoints of the intervals may be open instead of closed; while the proof of these statements given in N?, R-matrix? is written in the particular case of \(K\)-theoretic Hall algebras of double quivers, the argument therein is completely general. Thus, we obtain a host of factorizations of the so-called wedge subalgebras ² \[{\mathcal{S}}^\pm_{[{\boldsymbol{p}}^1 , {\boldsymbol{p}}^2]} = {\mathcal{S}}^\pm_{\geq {\boldsymbol{p}}^1} \cap {\mathcal{S}}^\pm_{\leq {\boldsymbol{p}}^2}\] for any \({\boldsymbol{p}}^1 \leq {\boldsymbol{p}}^2\) in \({\mathbb{R}^I}\), as well as their natural analogues when some endpoints of the intervals may be open instead of closed (the word “host" is due to the fact that there is a great degree of freedom in choosing a catty-corner curve joining \({\boldsymbol{p}}^1\) and \({\boldsymbol{p}}^2\)). The following result is easy, so we leave it as an exercise to the reader.

In a nutshell, the condition that a shuffle element \(X\) lies in either \({\mathcal{S}}^\pm_{\geq {\boldsymbol{p}}}\) or \({\mathcal{S}}^\pm_{\leq {\boldsymbol{p}}}\) imposes a bound (either lower or upper) on the degrees of all of its variables; then fixing \((\text{hdeg }X,\text{vdeg }X)\) leads to a finite number of monomials that may appear in \(X\). As a consequence of Lemma 2, the restrictions of the pairing ?? to \[\label{eqn:perfect32pairing321} {\mathcal{B}}^+_{{\boldsymbol{p}}} \otimes {\mathcal{B}}^-_{{\boldsymbol{p}}} \rightarrow {\mathbb{K}}, \qquad {\mathcal{S}}^+_{[{\boldsymbol{p}}^1 , {\boldsymbol{p}}^2]} \otimes {\mathcal{S}}^-_{[{\boldsymbol{p}}^1 , {\boldsymbol{p}}^2]} \rightarrow {\mathbb{K}}\tag{122}\] \[\label{eqn:perfect32pairing322} {\mathcal{S}}^+_{\geq {\boldsymbol{p}}} \otimes {\mathcal{S}}^-_{\geq {\boldsymbol{p}}} \rightarrow {\mathbb{K}}, \qquad \qquad {\mathcal{S}}^+_{\leq {\boldsymbol{p}}} \otimes {\mathcal{S}}^-_{\leq {\boldsymbol{p}}} \rightarrow {\mathbb{K}}\tag{123}\] are perfect for all \({\boldsymbol{p}}\) and \({\boldsymbol{p}}^1 \leq {\boldsymbol{p}}^2 \in {\mathbb{R}^I}\); this is because Proposition 7 implies that the pairings above inherit their non-degeneracy from that of 91 , and then we invoke the last sentence of Subsection 2.8 to go from non-degenerate to perfect.

3.5 More factorizations↩︎

As a consequence of Proposition 7, for any \({\boldsymbol{p}}\in {\mathbb{R}^I}\) we have an isomorphism induced by multiplication \[\label{eqn:two32factor} {\mathcal{S}}^{\pm}_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \otimes {\mathcal{S}}^{\pm}_{[{\boldsymbol{p}},\boldsymbol{\infty}]} \xrightarrow{\sim} {\mathcal{S}}^\geq\tag{124}\] (recall 120 121 ). Formula 124 allows us to uniquely write any \(E \in {\mathcal{S}}^\geq\), \(F \in {\mathcal{S}}^\leq\) as \[E = \sum_{k} c^+_k \cdot A^+_k * B^+_k \quad \text{and} \quad F = \sum_{k} c^-_k \cdot A^-_k * B^-_k\] for any bases \(\{A^\pm_k\}\) of \({\mathcal{S}}^{\pm}_{(-\boldsymbol{\infty},{\boldsymbol{p}})}\) and \(\{B_{k}^\pm\}\) of \({\mathcal{S}}^\pm_{[{\boldsymbol{p}},\boldsymbol{\infty}]}\), and various coefficients \(c^\pm_k \in {\mathbb{K}}\). Moreover, given elements \(E\) and \(F\) as above, formula ?? implies that \[\label{eqn:pair32slopes32advanced} \langle E,F \rangle = \sum_{k,\ell} c^+_k c^-_{\ell} \cdot \langle A^+_k, A^-_{\ell} \rangle \langle B^+_k, B^-_{\ell} \rangle\tag{125}\] In particular, if \((E, F) \in {\mathcal{S}}^+_{[{\boldsymbol{p}},\boldsymbol{\infty}]} \times {\mathcal{S}}^-_{(-\boldsymbol{\infty},{\boldsymbol{p}})}\) or \((E, F) \in {\mathcal{S}}^+_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \times {\mathcal{S}}^-_{[{\boldsymbol{p}},\boldsymbol{\infty}]}\), then \[\label{eqn:epsilon} \langle E, F \rangle = \varepsilon(E) \varepsilon(F)\tag{126}\] with \(\varepsilon\) denoting the counit. By 125 and the perfectness of the pairings 122 123 , we may uniquely define for any \(E \in {\mathcal{S}}^\geq\) and \(F \in {\mathcal{S}}^\leq\) the elements \[\begin{align} &[E]_{\geq {\boldsymbol{p}}} \in {\mathcal{S}}^+_{[{\boldsymbol{p}},\boldsymbol{\infty}]} \quad \;\text{ by} \quad \langle [E]_{\geq {\boldsymbol{p}}}, Y \rangle = \langle E, Y \rangle, \;\forall Y \in {\mathcal{S}}^-_{[{\boldsymbol{p}},\boldsymbol{\infty}]} \tag{127} \\ &[F]_{< {\boldsymbol{p}}} \in {\mathcal{S}}^-_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \quad \text{by} \quad \langle X, [F]_{< {\boldsymbol{p}}} \rangle = \langle X, F \rangle, \;\forall X \in {\mathcal{S}}^+_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \tag{128} \\ &[F]_{\geq {\boldsymbol{p}}} \in {\mathcal{S}}^-_{[{\boldsymbol{p}},\boldsymbol{\infty}]} \quad \;\text{ by} \quad \langle X, [F]_{\geq {\boldsymbol{p}}} \rangle = \langle X, F \rangle, \;\forall X \in {\mathcal{S}}^+_{[{\boldsymbol{p}},\boldsymbol{\infty}]} \tag{129} \\ &[E]_{< {\boldsymbol{p}}} \in {\mathcal{S}}^+_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \quad \text{by} \quad \langle [E]_{<{\boldsymbol{p}}}, Y \rangle = \langle E, Y \rangle, \;\forall Y \in {\mathcal{S}}^-_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \tag{130} \end{align}\] ³ Moreover, if we write the coproduct as \(\Delta(E) = E_1 \otimes E_2\), \(\Delta(F) = F_1 \otimes F_2\), then \[\begin{align} &E = [E_1]_{< {\boldsymbol{p}}} [E_2]_{\geq {\boldsymbol{p}}} \tag{131} \\ &F = [F_2]_{< {\boldsymbol{p}}} [F_1]_{\geq {\boldsymbol{p}}} \tag{132} \end{align}\] for all \(E \in {\mathcal{S}}^\geq\), \(F \in {\mathcal{S}}^\leq\). Formula 131 is proved by pairing both sides with an arbitrary \(XY\) where \(X \in {\mathcal{S}}^-_{(-\boldsymbol{\infty},{\boldsymbol{p}})}\), \(Y \in {\mathcal{S}}^-_{[{\boldsymbol{p}},\boldsymbol{\infty}]}\) and then evaluating the left-hand side with 64 and the right-hand side with 125 . Formula 132 is proved similarly.

3.6 The half subalgebras↩︎

Consider any \({\boldsymbol{p}}\in {\mathbb{R}^I}\). The straightforward generalization of N?, Cat? yields subalgebras \[\begin{align} &{\mathcal{A}}^{\geq {\boldsymbol{p}}} = {\mathcal{S}}^+_{[{\boldsymbol{p}},\boldsymbol{\infty}]} \otimes {\mathcal{S}}^-_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \tag{133} \\ &{\mathcal{A}}^{\leq {\boldsymbol{p}}} = {\mathcal{S}}^-_{[{\boldsymbol{p}},\boldsymbol{\infty}]} \otimes {\mathcal{S}}^+_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \tag{134} \end{align}\] of \({\mathcal{S}}\), which provide an isomorphism of vector spaces (cf. N?, Cat?) \[\label{eqn:a32triangular} {\mathcal{A}}^{\geq {\boldsymbol{p}}} \otimes {\mathcal{A}}^{\leq {\boldsymbol{p}}} \xrightarrow{\sim} {\mathcal{S}}= \mathbf{U}\tag{135}\] Note that the loop Cartan subalgebras \({\mathcal{B}}_{\boldsymbol{\infty}}^\geq\) and \({\mathcal{B}}_{\boldsymbol{\infty}}^\leq\) are contained in \({\mathcal{A}}^{\geq {\boldsymbol{p}}}\) and \({\mathcal{A}}^{\leq {\boldsymbol{p}}}\), respectively. The initial motivation for this construction is that in the case of simple Lie algebras \({\mathfrak{g}}\) (i.e. the quantum loop algebra 6 ), the subalgebras \({\mathcal{A}}^{\geq {\boldsymbol{0}}}\) and \({\mathcal{A}}^{\leq {\boldsymbol{0}}}\) correspond to the Borel subalgebras of the quantum affine algebra \(U_q(\widehat{{\mathfrak{g}}})\) under the Drinfeld-Beck isomorphism, see Subsection 3.10. However, the importance of this construction for our purposes lies in the following strengthening of Theorem 1.

Thus, there are as many topological coproducts on \(\mathbf{U}\) as there are elements \({\boldsymbol{p}}\in {\mathbb{R}^I}\) (beside the Drinfeld new coproduct \(\Delta\), which morally corresponds to \({\boldsymbol{p}}= \boldsymbol{\infty}\)). In geometric situations such as critical \(K\)-theoretic Hall algebras of quivers, we expect our coproducts to match the coproducts defined by [17], [18] using the FRT formalism and stable envelopes. In the particular case of preprojective \(K\)-theoretic Hall algebras of quivers that was developed in [19], the coincidence between our construction and that of loc. cit. follows from [20]. Finally, when \(\mathbf{U}\) is defined using the zeta functions 4 for an affine Lie algebra \({\mathfrak{g}}\), it would be interesting to compare the coproduct ?? with the coproduct on quantum toroidal algebras defined in [21] using double affine braid group actions.

We observe that the coproduct \(\Delta_{{\boldsymbol{p}}}\) preserves degrees. To see this, consider for instance any \(\textcolor{red}{E} \textcolor{blue}{F}\) whose coproduct is \(\textcolor{red}{E_1} \textcolor{blue}{Y} \otimes \textcolor{red}{X} \textcolor{blue}{F_1}\) as in 137 , and note that \[\text{deg}(\textcolor{red}{E} \textcolor{blue}{F}) - \text{deg}(\textcolor{red}{E_1} \textcolor{blue}{Y}) - \text{deg}(\textcolor{red}{X} \textcolor{blue}{F_1}) = \text{deg}(\textcolor{red}{E_2}) + \text{deg}(\textcolor{blue}{F_2}) - \text{deg}(\textcolor{red}{X}) - \text{deg}(\textcolor{blue}{Y})\] The quantity above is equal to 0 because of 141 and the fact that the pairing \(\langle \cdot, \cdot \rangle\) is non-zero only on elements of opposite degrees. The latter is also the reason why the pairing 151 is also non-zero only on elements of opposite degrees.

3.7 Universal \(R\)-matrices↩︎

Let us now consider universal \(R\)-matrices as in Subsection 2.9 for the algebra \({\mathcal{S}}= \mathbf{U}\). Since this algebra has numerous coproducts and corresponding structures of a Drinfeld double, we may define partial universal \(R\)-matrices as follows. We will encounter the notation \[\begin{align} &{\mathcal{S}}\;\widehat{\otimes} \;{\mathcal{S}}= \prod_{({\boldsymbol{n}},d) \in {\mathbb{Z}^I}\times {\mathbb{Z}}} {\mathcal{S}}_{{\boldsymbol{n}},d} \;\widehat{\otimes} \;{\mathcal{S}}_{-{\boldsymbol{n}},-d} \\ &{\mathcal{S}}\;\bar{\otimes} \;{\mathcal{S}}= \prod_{({\boldsymbol{n}},d) \in {\mathbb{Z}^I}\times {\mathbb{Z}}} {\mathcal{S}}_{{\boldsymbol{n}},d} \;\otimes \;{\mathcal{S}}_{-{\boldsymbol{n}},-d} \end{align}\] where \[\label{eqn:hat} {\mathcal{S}}_{{\boldsymbol{n}},d} \;\widehat{\otimes} \;{\mathcal{S}}_{-{\boldsymbol{n}},-d} = \left \{ \sum_{k} A_k \otimes B_k \right \}\tag{159}\] in which the sum can go over infinitely many \(A_k \in {\mathcal{S}}_{{\boldsymbol{n}},d}\) and \(B_k \in {\mathcal{S}}_{-{\boldsymbol{n}},-d}\), but for any \(N \in {\mathbb{N}}\), all but finitely many \(k\) have the property that \(A_k = A_k'A_k''\) and \(B_k = B_k' B_k''\) with \(|\text{hdeg }A_k'| \geq N, |\text{hdeg }A_k''| \leq - N, |\text{hdeg }B_k'| \leq - N, |\text{hdeg }B_k''|\geq N\).

For any \({\boldsymbol{p}}^1, {\boldsymbol{p}}^2\) in \({\mathbb{R}^I}\), we may consider \[\begin{align} &_{{\boldsymbol{p}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} = {{\mathcal{R}}_{{\boldsymbol{p}}^2}^{-1}} \cdot {{\mathcal{R}}_{{\boldsymbol{p}}^1}} \in {\mathcal{S}}\;\bar{\otimes} \;{\mathcal{S}}\tag{173} \\ &_{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} = {_{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}} \cdot {{\mathcal{R}}_{{\boldsymbol{p}}^1}} \in {\mathcal{S}}\;\widehat{\otimes} \;{\mathcal{S}}\tag{174} \end{align}\] and so ?? ?? imply \[\label{eqn:intertwine323} ({_{{\boldsymbol{p}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1}}) \cdot \Delta_{{\boldsymbol{p}}^1}(-) = \Delta_{{\boldsymbol{p}}^2}(-) \cdot ( {_{{\boldsymbol{p}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1}} )\tag{175}\] \[\label{eqn:intertwine324} ( {_{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1}}) \cdot \Delta_{{\boldsymbol{p}}^1}(-) = \Delta^{\text{op}}_{{\boldsymbol{p}}^2}(-) \cdot ( {_{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1}} )\tag{176}\]

3.8 Factorizations of \(R\)-matrices↩︎

To further refine the constructions above, consider for any \({\boldsymbol{p}}\in {\mathbb{R}^I}\) the canonical tensor \[\label{eqn:slope32r-matrix} {\mathcal{P}}_{{\boldsymbol{p}}} \in {\mathcal{B}}^{+}_{{\boldsymbol{p}}} \;\bar{\otimes} \;{\mathcal{B}}^{-}_{{\boldsymbol{p}}} \subset {\mathcal{S}}\;\bar{\otimes} \;{\mathcal{S}}\tag{177}\] of the first pairing in 122 . We then use ?? to obtain the identities \[\begin{align} &{\mathcal{R}}_{{\boldsymbol{p}}^1} = \prod_{t \in (-\infty,t_1)}^{\rightarrow} {\mathcal{P}}_{{\boldsymbol{p}}(t)}^{\text{op}}\tag{178} \\ &_{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}= \prod_{t \in [t_2,\infty]}^{\rightarrow} {\mathcal{P}}_{{\boldsymbol{p}}'(t)} \tag{179} \end{align}\] for any catty-corner curves \({\boldsymbol{p}}: (-\infty,t_1) \rightarrow {\mathbb{R}^I}\) and \({\boldsymbol{p}}' : [t_2,\infty] \rightarrow {\mathbb{R}^I}\) with \({\boldsymbol{p}}(t_1) = {\boldsymbol{p}}^1\) and \({\boldsymbol{p}}'(t_2) = {\boldsymbol{p}}_2\). In 178 , we set \({\mathcal{P}}^{\text{op}} = \text{swap}({\mathcal{P}})\). If we put the formulas above together, we obtain the following factorization of the universal \(R\)-matrix 174 \[\label{eqn:factorization32r-matrix323} _{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} = \prod_{t \in [t_2,\infty]}^{\rightarrow} {\mathcal{P}}_{{\boldsymbol{p}}'(t)} \prod_{t \in (-\infty,t_1)}^{\rightarrow} {\mathcal{P}}_{{\boldsymbol{p}}(t)}^{\text{op}}\tag{180}\] When \({\boldsymbol{p}}^1={\boldsymbol{p}}^2\), 180 generalizes the well-known formulas for factorizations of \(R\)-matrices of [23]–[27], see also [17]–[20], N?, Stable? for a geometric incarnation.

3.9 Infinite slope revisited↩︎

Because of the description of the universal \(R\)-matrix \({_{{\boldsymbol{p}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1}}\) as the canonical tensor of the pairing ?? , all of its homogeneous summands are finite sums (this statement actually holds for all \({\boldsymbol{p}}^1,{\boldsymbol{p}}^2 \in {\mathbb{R}^I}\)). This is not true for \({_{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1}}\), whose homogeneous summands lie in the completion \[{\mathcal{S}}\;\widehat{\otimes} \;{\mathcal{S}}\] defined in 159 . However, there is a bigger problem than the necessity of completions, which we will now address. While the slope \(R\)-matrices 177 are well-defined for all \({\boldsymbol{p}}\in {\mathbb{R}^I}\), when \({\boldsymbol{p}}= \boldsymbol{\infty}\) we define \[\label{eqn:infinite32factorization} {\mathcal{P}}_{\boldsymbol{\infty}} = {\mathcal{P}}'_{\boldsymbol{\infty}} {\mathcal{P}}''_{\boldsymbol{\infty}}\tag{181}\] where \({\mathcal{P}}'_{\boldsymbol{\infty}}\) and \({\mathcal{P}}''_{\boldsymbol{\infty}}\) are the canonical tensors of \[\label{eqn:first32pairing} {\mathbb{K}}[\kappa^+_i]_{i \in I} \otimes {\mathbb{K}}[\kappa^-_i]_{i \in I} \xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{182}\] and \[\label{eqn:second32pairing} {\mathbb{K}}[p_{i,d}]_{i \in I, d\geq 1} \otimes {\mathbb{K}}[p_{i,-d}]_{i \in I, d \geq 1} \xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}\tag{183}\] respectively. And while the canonical tensor \({\mathcal{P}}_{\boldsymbol{\infty}}''\) of 183 is well-defined (subject to the non-degeneracy assumption in Subsection 3.3) and rather easy to compute, there is no reasonable canonical tensor of the pairing 182 . The workaround for this issue is the usual one: replace the ground field \({\mathbb{K}}\) with \({\mathbb{K}}[[\hbar]]\), and replace \[\kappa^\pm_i \quad \text{by} \quad e^{\hbar H^\pm_i}\] for primitive elements \(\{H^+_i, H^-_i\}_{i \in I}\). If the \(|I| \times |I|\) matrix with coefficients \(\langle H^+_i, H^-_j \rangle\) is invertible, then the pairing \[\label{eqn:third32pairing} {\mathbb{K}}[[\hbar]][H^+_i]_{i \in I} \otimes {\mathbb{K}}[[\hbar]][H^-_i]_{i \in I} \xrightarrow{\langle \cdot, \cdot \rangle} {\mathbb{K}}((\hbar))\tag{184}\] has a canonical tensor, which serves as a replacement for the factor \({\mathcal{P}}'_{\boldsymbol{\infty}}\) in 181 . We will ignore this issue in what follows, as both the problem and its workaround are well-known and similar in the case at hand to the classic cases such as \(U_q({\mathfrak{sl}}_2)\).

3.10 Quantum affine algebras↩︎

Let \({\mathfrak{g}}\) be a simple finite-dimensional Lie algebra, with a set of simple roots \(\{\alpha_i\}_{i \in I}\) and associated Cartan matrix \[C = \left(c_{ij} = \frac{d_{ij}}{d_{i}} \in {\mathbb{Z}}\right)_{i,j \in I}\] where we abbreviate \(d_{ij} = (\alpha_i, \alpha_j)\) and \(d_i = \frac{(\alpha_i, \alpha_i)}{2}\). We work over the field \({\mathbb{K}}= {\mathbb{C}}\), fix \(q \in {\mathbb{C}}^*\) not a root of unity, and consider the rational functions \[\label{eqn:zeta32particular} \zeta_{ij}(x) = \frac{(q^{-d_{ij}}-x)(-x)^{-\delta_{i>j}}}{(1-x)^{\delta_{ij}}}\tag{185}\] with respect to an arbirary total order \(<\) on \(I\). Then the corresponding quantum loop algebra is denoted by \[\label{eqn:quantum32loop32algebra} U_q(L{\mathfrak{g}})= \Big(\mathbf{U}\text{ for the choice \eqref{eqn:zeta32particular}} \Big)\tag{186}\] ⁵ Drinfeld constructed an isomorphism \[U_q(\widehat{{\mathfrak{g}}})_{c=1} = {\mathbb{C}}\Big \langle e_i,f_i, \kappa_i, \kappa_i^{-1}, e_0, f_0 \Big \rangle_{i \in I} \Big/ \Big(\text{relations}\Big)\] \[\label{eqn:drinfeld32iso} \xrightarrow{\Xi} U_q(L{\mathfrak{g}})\Big/ \Big(\kappa_i^+ \kappa_i^- = 1 \Big)_{i \in I}\tag{187}\] by sending for all \(i \in I\) \[\label{eqn:send} \kappa_i^{\pm 1} \mapsto \kappa_i^\pm, \qquad e_i \mapsto e_{i,0}, \qquad f_i \mapsto f_{i,0}\tag{188}\] \[\begin{align} &e_{0} \mapsto c (\kappa_{\boldsymbol{\theta}}^+)^{-1} \textcolor{blue}{F} \quad \;\;\text{ where } \quad \textcolor{blue}{F} = [f_{i_1,0},[f_{i_2,0},\dots,[f_{i_{k},0}, f_{j,1}]_q\dots]_q]_q \tag{189} \\ &f_{0} \mapsto c' (\kappa_{\boldsymbol{\theta}}^-)^{-1} \textcolor{red}{E'} \quad \text{ where } \quad \textcolor{red}{E'} = [e_{i_1,0},[e_{i_2,0},\dots,[e_{i_{k},0}, e_{j,-1}]_q\dots]_q]_q \tag{190} \end{align}\] where \(c,c'\) are non-zero constants, we write \[\kappa_{\boldsymbol{\theta}}^\pm = \prod_i (\kappa_i^\pm)^{\theta_i}\] with \(\boldsymbol{\theta}\in {\mathbb{N}^I}\) being the longest root and \([x,y]_q = xy - q^{(\text{hdeg }x, \text{hdeg }y)} yx\). In the formulas above, \(i_1,\dots,i_k,j \in I\) are such that \[\boldsymbol{\varsigma}^{i_1}+\dots+\boldsymbol{\varsigma}^{i_k} + \boldsymbol{\varsigma}^j = \boldsymbol{\theta}\] and \(\boldsymbol{\varsigma}^{i_a}+\dots+\boldsymbol{\varsigma}^{i_k} +\boldsymbol{\varsigma}^j\) is a root \(\forall a\). It is clear that \(\deg \textcolor{blue}{F} = (-\boldsymbol{\theta},1)\) and \(\deg \textcolor{red}{E'} = (\boldsymbol{\theta},-1)\). We proved in N?, Cat? that the isomorphism 187 sends \[\begin{align} &\Xi \left(U_q(\widehat{{\mathfrak{b}}}^+)_{c=1} \right) = {\mathcal{A}}^{\geq {\boldsymbol{0}}} \tag{191} \\ &\Xi \left(U_q(\widehat{{\mathfrak{b}}}^-)_{c=1} \right) = {\mathcal{A}}^{\leq {\boldsymbol{0}}} \tag{192} \end{align}\] where \(U_q(\widehat{{\mathfrak{b}}}^+)\) and \(U_q(\widehat{{\mathfrak{b}}}^-)\) denote the Borel subalgebras of \(U_q(\widehat{{\mathfrak{g}}})\) generated by \(\{e_i,e_0,\kappa_i\}_{i \in I}\) and \(\{f_i, f_0,\kappa_i^{-1}\}_{i \in I}\), respectively.

In principle, following the proof above would lead to explicit formulas for \(\Delta_{{\boldsymbol{0}}}\) akin to the formulas for the Drinfeld-Jimbo coproduct obtained in [28] for \({\mathfrak{sl}}_n\) (or for general \({\mathfrak{g}}\), following in the footsteps of [29]).

3.11 The pairing explicit↩︎

We conclude this section with an explicit formula for the pairing ?? in the case of the quantum affine algebra \(U_q(\widehat{{\mathfrak{g}}})\) associated to a simple Lie algebra \({\mathfrak{g}}\), which will provide an answer to Problem 2 of Subsection 1.6 in this particular case. The contents of the present Subsection will not be used anywhere else in this paper, but we include them to fill a gap in the literature. We keep the notation as in Subsection 3.10, and we recall from [13] that \[\label{eqn:wheel} {\mathcal{S}}^\pm = \left\{E \in {\mathcal{V}}^\pm \text{ s.t. } E\Big|_{z_{i1} = z_{j1} q^{d_{ij}}, z_{i2} = z_{j1} q^{d_{ij}+d_{ii}}, \dots, z_{i, 1-c_{ij}} = z_{j1} q^{-d_{ij}}} = 0, \forall i \neq j\right\}\tag{201}\] (if \(E\) above does not have enough variables to make the specialization, the vanishing condition is vacuous). The conditions on \(E\) in 201 are called wheel conditions and were introduced by [14], following the seminal work of [15]. The goal of Problem 1 in Subsection 1.6 is to find descriptions of \({\mathcal{S}}^\pm\) for general \((I,{\mathbb{K}},\zeta_{ij}(x))\) that are similar to 201 above. As for Problem 2, we will provide in Lemma 3 an analogue of N?, Symmetric?, N?, Wheel?, N?, Reduced?.

In the upcoming Lemma, we will use the notation \[\label{eqn:iterated32residue} \underset{(z_1,\dots,z_n) = (w q^{n-1}, \dots, w q^{1-n})}{\text{Res}} = \underset{z_1 = z_2 q^2}{\text{Res}} \; \underset{z_2 = z_3 q^2}{\text{Res}} \dots \;\underset{z_{n-1}=z_nq^2}{\text{Res}}\tag{202}\] followed by relabeling the variable \(z_n\) as \(w q^{1-n}\). Moreover, \[\int_{|w| = r} G(w)\] denotes the contour integral of \(G(w)\frac{dw}{2\pi i w}\) over the circle of radius \(r\) centered at the origin. In all subsequent contour integrals, we assume that \(|q|>1\), although this is not important due to our result being purely algebraic. Write \(q_i = q^{d_i}\) for all \(i \in I\).

3.12 A generalization↩︎

The contents of the previous subsection may be generalized as follows. Following a suggestion of David Hernandez, we call a Kac-Moody Lie algebra \({\mathfrak{g}}\) strongly symmetrizable if \[\label{eqn:strongly32symmetrizable} d_{ij} \in \{ 0, - \max(d_i,d_j) \}\tag{211}\] for all \(i\neq j\). This definition includes all finite and affine type Lie algebras, except for affine \(A_1\) (thus, the discussion in the present subsection does not apply to affine \(A_1\), in which case one must instead follow the treatment of N?, Symmetric?). For a strongly symmetrizable Kac-Moody Lie algebra \({\mathfrak{g}}\), Lemma 3 holds as stated, where \[\label{eqn:def} {\mathcal{S}}^\pm := \left\{E \in {\mathcal{V}}^\pm \text{ s.t. } E\Big|_{z_{i1} = z_{j1} q^{d_{ij}}, z_{i2} = z_{j1} q^{d_{ij}+d_{ii}}, \dots, z_{i, 1-c_{ij}} = z_{j1} q^{-d_{ij}}} = 0, \forall i \neq j\right\}\tag{212}\] Thus, we are in the situation of N?, Wheel?: Lemma 3 provides a non-degenerate pairing between the subalgebra \({\mathcal{S}}^\pm\) of 212 and the a priori smaller subalgebra \(\text{Im }\widetilde{\Upsilon}^\mp \subseteq {\mathcal{V}}^\mp\) linearly spanned by the elements 34 . Then the argument of N?, Wheel? carries through and implies that these two algebras coincide \[\Big( {\mathcal{S}}^\pm \text{ of \eqref{eqn:def}} \Big) = \Big(\text{Im }\widetilde{\Upsilon}^\pm \text{ of Definition \ref{def:shuffle}} \Big)\] This gives a complete description of the shuffle algebra associated to a strongly symmetrizable Kac-Moody Lie algebra \({\mathfrak{g}}\). As shown in [13], the wheel conditions 212 are dual to the so-called Drinfeld-Serre relations \[S_{ij}^+ = \sum_{k=0}^{1-c_{ij}} (-1)^k {1-c_{ij} \choose k}_{q_i} \text{Sym}_{z_1,\dots,z_{1-c_{ij}}} e_i(z_1) \dots e_i(z_k) e_j(w) e_i(z_{k+1}) \dots e_i(z_{1-c_{ij}})\] \[S_{ij}^- = \sum_{k=0}^{1-c_{ij}} (-1)^k {1-c_{ij} \choose k}_{q_i} \text{Sym}_{z_1,\dots,z_{1-c_{ij}}} f_i(z_1) \dots f_i(z_k) f_j(w) f_i(z_{k+1}) \dots f_i(z_{1-c_{ij}})\] Following the general principle laid out in N?, Arbitrary?, this implies that \[\text{Ker }\widetilde{\Upsilon}^\pm = \left( S_{ij}^\pm \right)_{i\neq j}\] We conclude that the full set of relations in the quantum loop algebra 186 associated to any strongly symmetrizable Kac-Moody Lie algebra consists of \[\label{eqn:drinfeld-serre} S_{ij}^\pm = 0, \quad \forall i \neq j\tag{213}\] together with 81 , 82 , 84 , 85 , 86 , 87 , 88 , 89 . In the particular case of quantum toroidal algebras of type other than \(A_1\), this proves that such algebras possess triangular decompositions in terms of their Drinfeld positive and negative halves, and that the usual Hopf pairing between the two halves is non-degenerate (i.e. the aforementioned set of relations is maximal so that the resulting algebra keeps its usual Hopf algebra structure and Hopf pairing); see N?, Arbitrary? for details.

4 Modules and tensor products↩︎

We will now introduce simple modules for the subalgebras \({\mathcal{A}}^{\geq {\boldsymbol{p}}} \subset {\mathcal{S}}= \mathbf{U}\) defined in the previous Section, and we will use the coproducts \(\Delta_{{\boldsymbol{p}}}\) to define tensor products. Our main reference will be N?, Cat?; although loc. cit. pertains to a particular choice of \((I,{\mathbb{K}},\zeta_{ij}(x))\), most proofs therein are completely general, and we will provide alternative proofs for those results where the generalization is not straightforward.

4.1 Borel category \({\mathcal{O}}\)↩︎

The following definitions are natural generalizations of classic constructions for quantum affine algebras 6 . After finite-dimensional type 1 \(U_q(\widehat{{\mathfrak{g}}})\)-modules were classified in [6], it was recognized in [7] that one can obtain more by restricting to the Borel subalgebra. In light of 191 , we generalize this by choosing any \({\boldsymbol{p}}\in {\mathbb{R}^I}\) and considering modules \[\label{eqn:module} {\mathcal{A}}^{\geq {\boldsymbol{p}}} \curvearrowright V\tag{214}\] for which the action of the Cartan subalgebra \(\{\kappa^+_i = \varphi^+_{i,0}\}_{i \in I}\) is diagonalizable: \[\label{eqn:weight32decomposition} V = \bigoplus_{\boldsymbol{\lambda}= (\lambda_i)_{i \in I} \in ({\mathbb{K}}^*)^I} V_{\boldsymbol{\lambda}}\tag{215}\] where \[V_{\boldsymbol{\lambda}} = \Big\{ v \in V \Big| \kappa^+_i \cdot v = \lambda_i v, \forall i \in I \Big\}\] Above, \(\boldsymbol{\lambda}\) will be referred to as weights. Because of the commutation relation \[\label{eqn:commute} \kappa^+_i X = X \kappa^+_i \gamma_{\boldsymbol{\varsigma}^i, \text{hdeg }X}\tag{216}\] for all \(i \in I\) and \(X \in {\mathcal{A}}^{\geq {\boldsymbol{p}}}\) (see 110 ), it is easy to see that algebra elements interact with the weight space decomposition according to the rule \[\label{eqn:r32action} X : V_{\boldsymbol{\lambda}} \rightarrow V_{\boldsymbol{\lambda}\boldsymbol{\gamma}^{\text{hdeg }X}}\tag{217}\] where \(\boldsymbol{\gamma}^{{\boldsymbol{n}}}\) denotes the weight with \(i\)-th component \(\gamma_{\boldsymbol{\varsigma}^i, {\boldsymbol{n}}}\) for all \({\boldsymbol{n}}\in {\mathbb{Z}^I}\). In the right-hand side of 217 we multiply weights component-wise, so \(\boldsymbol{\lambda}\boldsymbol{\mu}= (\lambda_i\mu_i)_{i \in I}\) for all weights \(\boldsymbol{\lambda}= (\lambda_i)_{i \in I}\) and \(\boldsymbol{\mu}= (\mu_i)_{i \in I}\).

In order for the above definition to be meaningful, we assume throughout that \[\label{eqn:generic} \boldsymbol{\gamma}^{{\boldsymbol{n}}} \neq {\boldsymbol{1}}\tag{218}\] for all \({\boldsymbol{n}}\in {\mathbb{Z}^I}\backslash {\boldsymbol{0}}\), which allows us to define the partial order \[\label{eqn:order} \boldsymbol{\lambda}\boldsymbol{\gamma}^{{\boldsymbol{n}}} > \boldsymbol{\lambda}, \quad \forall \text{ weight }\boldsymbol{\lambda}, \;\forall \;{\boldsymbol{n}}\in {\mathbb{N}^I}\backslash {\boldsymbol{0}}\tag{219}\] If condition 218 is not satisfied (e.g. in the important case of quantum toroidal \({\mathfrak{gl}}_1\) studied in [9]) then there is a fix following [30]: one adds additional finite Cartan elements \(\kappa_j^+\) and imposes relations 216 for various constants \(\gamma_{\boldsymbol{\varsigma}^j, {\boldsymbol{n}}}\) that are multiplicative in \({\boldsymbol{n}}\). If the aforementioned constants are chosen generic enough, then we can ensure 218 for all \({\boldsymbol{n}}\in {\mathbb{Z}^I}\backslash {\boldsymbol{0}}\).

4.2 Loop weights↩︎

Beside the finite Cartan subalgebra generated by \(\kappa^+_i = \varphi^+_{i,0}\), we have the (positive) loop Cartan subalgebra \[{\mathcal{B}}^{\geq}_{\boldsymbol{\infty}} = {\mathbb{K}}[\varphi^+_{i,d}]_{i \in I, d \geq 0}\] in \({\mathcal{A}}^{\geq {\boldsymbol{p}}}\) for any \({\boldsymbol{p}}\in {\mathbb{R}^I}\). The \(\varphi^+_{i,d}\)’s give an infinite family of commuting operators in any module \(V\) in category \({\mathcal{O}}\), so we may consider their joint generalized eigenspaces \[\label{eqn:loop32weight32decomposition} V = \bigoplus_{{\boldsymbol{\psi}}= (\psi_i(z))_{i \in I} \in ({\mathbb{K}}[[z^{-1}]]^\times)^I} V_{{\boldsymbol{\psi}}}\tag{220}\] ⁷ where \[V_{{\boldsymbol{\psi}}} = \Big\{ v \in V \Big| \left( \varphi^+_{i,d} - \psi_{i,d}\text{Id}_V \right)^N v = 0, \forall i \in I, d \geq 0 \text{ and }N \text{ large enough} \Big\}\] Above, \[\label{eqn:loop32weight} {\boldsymbol{\psi}}= \left(\psi_i(z) = \sum_{d=0}^{\infty} \frac{\psi_{i,d}}{z^d} \right)_{i \in I}\tag{221}\] will be referred to as loop weights, by analogy with the classic situation of quantum affine algebras. The vector spaces \(V_{\boldsymbol{\psi}}\) are finite-dimensional for any \(V\) in category \({\mathcal{O}}\), so we may define the \(q\)-character following [8] \[\label{eqn:q-character} \chi_q(V) = \sum_{\text{loop weights }{\boldsymbol{\psi}}} \dim_{{\mathbb{K}}}(V_{{\boldsymbol{\psi}}}) [{\boldsymbol{\psi}}]\tag{222}\] where \([{\boldsymbol{\psi}}]\) are formal symbols. Recall that \([{\boldsymbol{\psi}}][{\boldsymbol{\psi}}'] = [{\boldsymbol{\psi}}{\boldsymbol{\psi}}']\) with respect to the component-wise multiplication of \(I\)-tuples of power series.

4.3 Polynomiality of the theta series↩︎

Inspired by the work of Huafeng Zhang ([31], which originally established Theorem 18 below for quantum affine algebras), we expand on the proof of Proposition 17 in the following sense. Fix \(i \in I\) and assume that there exists a Cartan series \[\label{eqn:def32t} T_i(x) \in \mathbf{U}[[x^{-1}]]\tag{224}\] (initially defined for quantum affine algebras in [12]) which is group-like for 58 \[\label{eqn:coproduct32t} \Delta(T_i(x)) = T_i(x) \otimes T_i(x)\tag{225}\] and commutes with the positive and negative halves of the quantum loop algebra according to the following analogues of 41 and 42 \[\begin{align} &T_i(x) E(z_{j1},\dots,z_{jn_j})_{j \in I} = E(z_{j1},\dots,z_{jn_j})_{j \in I} T_i(x) \prod_{a=1}^{n_i} \left(1- \frac{z_{ia}}{x} \right) \tag{226} \\ &F(z_{j1},\dots,z_{jn_j})_{j \in I} T_i(x) = T_i(x) F(z_{j1},\dots,z_{jn_j})_{j \in I} \prod_{a=1}^{n_i} \left(1- \frac{z_{ia}}{x} \right) \tag{227} \end{align}\] When the zeta functions 1 have sufficiently generic coefficients (which happens for instance in the case of quantum affine algebras), it is well-known that \(T_i(x)\) can be written as an appropriate product of the power series \(\{\varphi^+_j(xa)\}_{j \in I, a \in {\mathbb{K}}^*}\). Even if the aforementioned genericity fails (which happens for instance in the case of quantum toroidal algebras), one can still formally add the series \(T_i(x)\) to the algebra \(\mathbf{U}\), all the while imposing 225 , 226 , 227 .

4.4 Simple modules↩︎

For any loop weight \({\boldsymbol{\psi}}\), we will define a horizontally graded simple module \[\label{eqn:simple} {\mathcal{A}}^{\geq {\boldsymbol{p}}} \curvearrowright L^{{\boldsymbol{p}}}({\boldsymbol{\psi}})\tag{230}\] generated by a single vector \(|\varnothing\rangle\) modulo the relations \[\begin{align} &\varphi^+_i(z) \cdot |\varnothing\rangle= \psi_i(z) |\varnothing\rangle, \quad \forall i \in I \tag{231} \\ &E \cdot |\varnothing\rangle= 0, \;\;\quad \qquad \qquad \forall E \in {\mathcal{S}}^+_{\geq {\boldsymbol{p}}} \quad \text{of hdeg}>{\boldsymbol{0}} \tag{232} \end{align}\] The construction of this simple module is quite standard: define the Verma module \[\label{eqn:verma32action} W^{{\boldsymbol{p}}}({\boldsymbol{\psi}}) = {\mathcal{S}}^-_{< {\boldsymbol{p}}} |\varnothing\rangle\tag{233}\] with \(F \in {\mathcal{S}}^-_{<{\boldsymbol{p}}}\) acting by left multiplication and \(E \in {\mathcal{S}}^+_{\geq {\boldsymbol{p}}}\) acting by \[\begin{gather} E \cdot F |\varnothing\rangle\stackrel{\eqref{eqn:dd1}}= \langle E_1, F_1 \rangle \langle E_3, S(F_3) \rangle F_2 E_2 \cdot |\varnothing\rangle\\ \stackrel{\eqref{eqn:epsilon}}= \langle E_2, S(F_2) \rangle F_1 E_1 \cdot |\varnothing\rangle\stackrel{\text{\eqref{eqn:simple32relation321}-\eqref{eqn:simple32relation322}}}= \langle E\psi, S(F_2) \rangle F_1|\varnothing\rangle\label{eqn:action32verma} \end{gather}\tag{234}\] where for \(E = E(z_{i1},\dots,z_{in_i})_{i \in I}\), we write \[E \psi = E(z_{i1},\dots,z_{in_i})_{i \in I}\prod_{i \in I} \prod_{a=1}^{n_i} \psi_i(z_{ia})\] While the expression \(E\psi\) is a power series in the variables \(z_{ia}^{-1}\), only finitely many terms in the expansion pair non-trivially with any given \(S(F_2)\), since the vertical degree of the latter is bounded above for any \(F\). The following result is quite standard, and it can be construed as the definition of \(L^{{\boldsymbol{p}}}({\boldsymbol{\psi}})\).

4.5 Rational loop weights↩︎

The following description of the maximal \(-{\mathbb{N}^I}\backslash {\boldsymbol{0}}\) horizontally graded \({\mathcal{A}}^{\geq {\boldsymbol{p}}}\)-submodule of \(W^{{\boldsymbol{p}}}({\boldsymbol{\psi}})\) is proved as in N?, Cat?: \[\text{Ker }\pi = J^{{\boldsymbol{p}}}({\boldsymbol{\psi}}) |\varnothing\rangle\] where \[\label{eqn:j32psi} J^{{\boldsymbol{p}}}({\boldsymbol{\psi}}) = \left\{F \in {\mathcal{S}}^-_{(-\boldsymbol{\infty},{\boldsymbol{p}})} \Big| \;\langle E \psi, S(F) \rangle = 0 , \;\forall E \in {\mathcal{S}}^+_{\geq {\boldsymbol{p}}} \right \}\tag{235}\] Thus, we conclude the following description of the underlying vector space of simple modules \[\label{eqn:simple32modules} L^{{\boldsymbol{p}}}({\boldsymbol{\psi}}) = {\mathcal{S}}^-_{<{\boldsymbol{p}}} \Big/ J^{{\boldsymbol{p}}}({\boldsymbol{\psi}})\tag{236}\] This will allow us to determine which simple modules \(L^{{\boldsymbol{p}}}({\boldsymbol{\psi}})\) lie in category \({\mathcal{O}}\), by analogy with [7].

4.6 Decompositions↩︎

A particular case of 235 occurs for the loop weight \[z^{-{\mathbf{r}}} = (z^{-r_i})_{i \in I}\] for any \({\mathbf{r}}= (r_i)_{i \in I} \in {\mathbb{Z}^I}\), namely \[\label{eqn:j32r} J^{{\boldsymbol{p}}}(z^{-{\mathbf{r}}}) = \left\{F \in {\mathcal{S}}^-_{<{\boldsymbol{p}}} \;\Big| \;\langle Ez^{-{\mathbf{r}}}, S(F) \rangle = 0 , \;\forall E \in {\mathcal{S}}^+_{\geq {\boldsymbol{p}}} \right \}\tag{240}\] The corresponding simple module 236 has underlying vector space \[\label{eqn:graded32modules} L^{{\boldsymbol{p}}}(z^{-{\mathbf{r}}}) = {\mathcal{S}}^-_{<{\boldsymbol{p}}} \Big/ J^{{\boldsymbol{p}}}(z^{-{\mathbf{r}}})\tag{241}\] which inherits a grading from the vertical degree of \({\mathcal{S}}^-\) (generalizing the grading constructed in [12]). Moreover, it is clear that shuffle elements \(F\) of large enough vertical degree vanish in \(L^{{\boldsymbol{p}}}(z^{-{\mathbf{r}}})\). We will also encounter the following vector space \[\label{eqn:j32neq} J^{\neq 0}({\boldsymbol{\psi}}) = \left\{F \in {\mathcal{S}}^- \;\Big| \;\langle E\psi, S(F) \rangle = 0, \forall E \in {\mathcal{S}}_{\geq N{\boldsymbol{1}}} \text{ for }N\text{ large enough} \right\}\tag{242}\] where in the right-hand side, \(N\) is allowed to be large enough in comparison to the horizontal degree of \(F\). As in formula 239 , we have \(F \in J^{\neq 0}({\boldsymbol{\psi}})\) if and only if \[\label{eqn:r4} \int_{1 \ll |z_1| \ll \dots \ll |z_n|} \frac{z_1^{d_1}\dots z_n^{d_n} F(z_1,\dots,z_n)}{\prod_{1\leq a < b \leq n} \zeta_{i_bi_a} \left(\frac{z_b}{z_a} \right)} \prod_{a=1}^n \psi_{i_a}(z_a) = 0\tag{243}\] for all \(i_1,\dots,i_n \in I\) and all \(d_1,\dots,d_n\) large enough.

While the quotient \[\label{eqn:neq32modules} L^{\neq 0}({\boldsymbol{\psi}}) = {\mathcal{S}}^- \Big/ J^{\neq 0}({\boldsymbol{\psi}})\tag{244}\] is not a \({\mathcal{A}}^{\geq {\boldsymbol{p}}}\)-module, we will see in the next Subsection that it is a module for a shifted version of the quantum loop algebra \(\mathbf{U}\). Recall the coproduct 57 , and let us write it as follows \[\label{eqn:delta32prime} \Delta(F) = \sum_{{\boldsymbol{0}} \leq {\boldsymbol{m}}\leq {\boldsymbol{n}}} F'(z_{i1},\dots , z_{im_i}) \otimes F''(z_{i,m_i+1}, \dots, z_{in_i}) \prod_{i \in I} \prod_{a=1}^{m_i} \varphi^-_i(z_{ia})\tag{245}\] (in other words, we absorb the denominator of 57 in the sum of tensors \(F' \otimes F''\), at the cost of allowing infinite sums). For any loop weight \({\boldsymbol{\psi}}= (\psi_i(z))_{i \in I}\), let \({\mathbf{r}}= \boldsymbol{ord }{\boldsymbol{\psi}}\) denote the \(I\)-tuple of orders of the poles of the functions \(\psi_i(z)\) at \(z=0\).

4.7 Factorization of \(q\)-characters↩︎

Since the decomposition ?? preserves the action of the positive loop Cartan subalgebra, it implies an equality of \(q\)-characters \[\label{eqn:decomposition32q-char} \chi_q(L^{\boldsymbol{p}}({\boldsymbol{\psi}})) = \chi_q(L^{\boldsymbol{p}}(z^{-\boldsymbol{ord }{\boldsymbol{\psi}}})) \cdot \chi_q(L^{\neq 0}({\boldsymbol{\psi}}))\tag{258}\] As shown in N?, Char?, the first term above can be completely calculated in terms of the graded dimensions of slope subalgebras, as follows. We will not reprove the result below, as it follows the proof of Theorem 1.3 of loc. cit. almost verbatim.

In ?? , we identify the weight \([\boldsymbol{\gamma}^{{\boldsymbol{n}}}]\) with a constant loop weight, i.e. an \(I\)-tuple of constant power series whose constant term is given by the scalars \(\gamma_{\boldsymbol{\varsigma}^i,{\boldsymbol{n}}} \in {\mathbb{K}}^*\). Because of this, we note that ?? is simply a character and not a \(q\)-character: it does not actually depend on loop weights, only on the underlying weights.

4.8 Eigenspaces↩︎

As for the second term in 258 , it was shown in N?, Cat? that \[\label{eqn:q-character32non-zero} \chi_q(L^{\neq 0}({\boldsymbol{\psi}})) = [{\boldsymbol{\psi}}] \sum_{{\boldsymbol{n}}= (n_i)_{i \in I} \in {\mathbb{N}^I}} \sum_{\boldsymbol{x}\in ({\mathbb{K}}^*)^{({\boldsymbol{n}})}} \dim_{{\mathbb{K}}}(L^{\neq 0}({\boldsymbol{\psi}})_{\boldsymbol{x}}) \prod_{i \in I} \prod_{a=1}^{n_i} A_{i,x_{ia}}^{-1}\tag{259}\] where the sum runs over \(\boldsymbol{x}= (x_{i1},\dots,x_{in_i})_{i \in I} \in ({\mathbb{K}}^*)^{({\boldsymbol{n}})} = \prod_{i \in I} ({\mathbb{K}}^*)^{n_i}/S_{n_i}\), and \[\label{eqn:fm} A_{i,x}^{-1} = \left[ \frac{\zeta_{ij} \left(\frac{x}{z} \right)}{\zeta_{ji} \left(\frac{z}{x} \right)} \right]_{j \in I}\tag{260}\] The right-hand side of 259 features the finite-dimensional vector spaces \[\label{eqn:l32x} L^{\neq 0}({\boldsymbol{\psi}})_{\boldsymbol{x}} = {\mathcal{S}}_{-{\boldsymbol{n}}} \Big / J^{\neq 0}({\boldsymbol{\psi}})_{\boldsymbol{x}}\tag{261}\] where \(J^{\neq 0}({\boldsymbol{\psi}})_{\boldsymbol{x}}\) denotes the set of \(F(z_{i1},\dots,z_{in_i})_{i \in I} \in {\mathcal{S}}_{-{\boldsymbol{n}}}\) such that \[\label{eqn:j32x} \underset{z_n = x_n}{\text{Res}} \dots \underset{z_1 = x_1}{\text{Res}} \frac{F(z_1,\dots,z_n)(\text{any monomial in the }z_a)}{\prod_{1\leq a < b \leq n} \zeta_{i_ai_b} \left(\frac{z_a}{z_b} \right)} \prod_{a=1}^n \psi_{i_a}(z_a) = 0\tag{262}\] for all \(i_1,\dots,i_n \in I\) such that \(\boldsymbol{\varsigma}^{i_1}+\dots+\boldsymbol{\varsigma}^{i_n} = {\boldsymbol{n}}\) and all orderings \(x_1,\dots,x_n\) of the coordinates of \(\boldsymbol{x}= (x_{i1},\dots,x_{in_i})_{i \in I}\) such that \(x_a = x_{i_a\bullet_a}\) for various \(\bullet_a \geq 1\) (see 61 for the meaning of \(F(z_1,\dots,z_n)\) in formula 262 ). Thus, we realize the \(q\)-character 259 as the dimension of certain explicit vector spaces. More generally, \[\label{eqn:eigenspace32decomposition} L^{\neq 0}({\boldsymbol{\psi}}) = \bigoplus_{{\boldsymbol{n}}\in {\mathbb{N}^I}} L^{\neq 0}({\boldsymbol{\psi}})_{{\boldsymbol{n}}}, \qquad L^{\neq 0}({\boldsymbol{\psi}})_{{\boldsymbol{n}}} = \bigoplus_{\boldsymbol{x}\in ({\mathbb{K}}^*)^{({\boldsymbol{n}})}} L^{\neq 0}({\boldsymbol{\psi}})_{\boldsymbol{x}}\tag{263}\] where each summand in the RHS is the generalized eigenspace of \(L^{\neq 0}({\boldsymbol{\psi}})\) on which \[\begin{align} &\kappa_i^+ \text{ acts by } \ell_i \gamma_{\boldsymbol{\varsigma}^i,-{\boldsymbol{n}}} \tag{264} \\ &p_{i,d} \text{ acts by } y_{i,d} - \sum_{j \in I} \alpha_{ij}^{(d)}(x_{j1}^d+\dots+x_{jn_j}^d) \tag{265} \end{align}\] where we set \(\psi_i(z) = \ell_i \exp \left(\sum_{d=1}^{\infty} \frac{y_{i,d}}{z^d} \right)\) and recall the notation in Subsection 2.11. In particular, formula 265 follows from the fact that the loop Cartan elements \(p_{i,d}\) act on the shuffle algebra \({\mathcal{S}}^-\) by multiplication with appropriate linear combinations of color-symmetric Laurent polynomials, which descend to \(L^{\neq 0}({\boldsymbol{\psi}})_{{\boldsymbol{n}}}\) as the formulas \[\label{eqn:p32acts32again} p_{i,d} \left(F \text{ mod }J^{\neq 0}({\boldsymbol{\psi}})_{{\boldsymbol{n}}} \right) = F \cdot \left(y_{i,d} - \sum_{j \in I} \alpha_{ij}^{(d)}(z_{j1}^d+\dots+z_{jn_j}^d) \right)\text{ mod }J^{\neq 0}({\boldsymbol{\psi}})_{{\boldsymbol{n}}}\tag{266}\] for any \(F = F(z_{i1},\dots,z_{in_i})_{i \in I} \in {\mathcal{S}}_{-{\boldsymbol{n}}}\).

4.9 Shifted quantum loop algebras↩︎

In the present Subsection, we adapt the material of [32], which in turn generalizes the representation theory of shifted quantum loop algebras from [33]. For any \({\mathbf{r}}\in {\mathbb{Z}^I}\), define a shifted quantum loop algebra \[\label{eqn:shifted32algebra} \mathbf{U}^{\mathbf{r}}= {\mathbb{K}}\Big \langle e_{i,d}, f_{i,d}, \varphi^+_{i,d'}, \varphi^-_{i,d'}\Big \rangle_{i \in I, d \in {\mathbb{Z}}, d' \geq 0} \Big/ \Big(\text{\eqref{eqn:rel32quantum321}-\eqref{eqn:rel32quantum328} and modified \eqref{eqn:rel32quantum329}} \Big)\tag{267}\] where the appropriate modification of 89 is to replace \(\varphi^-_j(w)\) by \(w^{-r_j} \varphi^-_j(w)\) in the right-hand side. Clearly, we have \(\mathbf{U}^{\boldsymbol{0}} = \mathbf{U}\). It was shown in [32] that for any loop weight \({\boldsymbol{\psi}}\) with \({\mathbf{r}}= \boldsymbol{ord }{\boldsymbol{\psi}}\), one can construct a simple module \[\label{eqn:shifted32module} \mathbf{U}^{\mathbf{r}}\curvearrowright L^{\text{sh}}({\boldsymbol{\psi}}) = {\mathcal{S}}^- \Big/ J^{\text{sh}}({\boldsymbol{\psi}})\tag{268}\] where \[\label{eqn:j32shifted} J^{\text{sh}}({\boldsymbol{\psi}})= \left\{F \in {\mathcal{S}}^- \;\Big| \;\langle E \psi, F_1 * S(F_2) \rangle = 0 , \;\forall E \in {\mathcal{S}}^+ \right \}\tag{269}\] Explicitly, the vanishing condition in the formula above reads \[\begin{gather} \label{eqn:psi32pairing32shifted} \left\langle E(z_{i1},\dots,z_{in_i})_{i \in I} \prod_{i \in I} \prod_{a=1}^{n_i} \psi_{i}(z_{ia}), \right. \\ \left. \sum_{{\boldsymbol{0}} \leq {\boldsymbol{m}}\leq {\boldsymbol{n}}} F_1(z_{i1},\dots,z_{im_i})_{i \in I} * S(F_2(z_{i,m_i+1},\dots,z_{in_i})_{i \in I}) \right\rangle = 0 \end{gather}\tag{270}\] and the pairing is calculated by expanding in the range \[\label{eqn:expansion} z_{i1},\dots,z_{i m_i} \sim 0 \qquad \text{and} \qquad z_{i,m_i+1},\dots,z_{in_i} \sim \infty\tag{271}\] This is the reason why 270 does not vanish identically, despite the fact that \(F_1 * S(F_2) = 0\) in any topological Hopf algebra due to the properties of the antipode.

In 268 , elements of \(L^{\text{sh}}({\boldsymbol{\psi}})\) are given by \(F |\varnothing\rangle\) for various \(F \in {\mathcal{S}}^-\). The action is given as follows: \({\mathcal{S}}^-\) acts by left multiplication, while \({\mathcal{S}}^+\) acts by the formula \[\label{eqn:action32verma322} E \cdot F |\varnothing\rangle= F_2|\varnothing\rangle\cdot \langle E_1,F_1\rangle \langle E_2 \psi, S(F_3) \rangle\tag{272}\] for all \(E \in {\mathcal{S}}^+\) and \(F \in {\mathcal{S}}^-\) (viewed as halves of the shifted algebra \(\mathbf{U}^{\mathbf{r}}\)), see [32].

4.10 Regular\(^{\neq 0}\) loop weights↩︎

A rational loop weight \({\boldsymbol{\psi}}\) is called regular\(^{\neq 0}\) if \[\boldsymbol{ord }{\boldsymbol{\psi}}= {\boldsymbol{0}}\] i.e. each constituent rational function \(\psi_i(z)\) is regular and non-zero at \(z = 0\) (beside being regular and non-zero at \(z = \infty\), as all loop weights are). The following result generalizes a well-known feature of simple modules of quantum affine algebras.

As a consequence of Proposition 24, we will denote the simple modules 273 as \[\label{eqn:simple32module32without32slope} \mathbf{U}\curvearrowright L({\boldsymbol{\psi}})\tag{274}\] for any regular\(^{\neq 0}\) loop weight \({\boldsymbol{\psi}}\), without any reference to the defining slope \({\boldsymbol{p}}\in {\mathbb{R}^I}\).

4.11 \(R\)-matrices↩︎

Consider now two modules \(\mathbf{U}\curvearrowright V,W\) in category \({\mathcal{O}}\), for example simple modules associated to regular\(^{\neq 0}\) loop weights, as per Proposition 24. The various coproducts \(\Delta_{{\boldsymbol{p}}}\) give rise to a host of module structures \[\label{eqn:structures} \mathbf{U}\curvearrowright V \otimes_{{\boldsymbol{p}}} W\tag{275}\] on the tensor product \(V \otimes W\); the fact that the completion \(\stackrel{\mathsf{H}}{\otimes}\) in which \(\Delta_{{\boldsymbol{p}}}\) takes values acts by finite sums on any element of \(V \otimes W\) is an immediate consequence of the fact that \(V,W\) are in category \({\mathcal{O}}\), as we saw in Proposition 16. As \({\boldsymbol{p}}\) varies, the module structures 275 are all related by the evaluation of the universal \(R\)-matrices of Subsection 3.7 in \(V\otimes W\). For instance, we may consider 173 \[_{{\boldsymbol{p}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} \in \mathbf{U}\;\bar{\otimes} \;\mathbf{U}\quad \leadsto \quad _{{\boldsymbol{p}}^2}R_{{\boldsymbol{p}}^1} \in \text{End}(V) \otimes \text{End}(W)\] for any \({\boldsymbol{p}}^1, {\boldsymbol{p}}^2\) in \({\mathbb{R}^I}\), and then 175 implies that we have a \(\mathbf{U}\)-intertwiner \[\label{eqn:isomorphism321} _{{\boldsymbol{p}}^2}R_{{\boldsymbol{p}}^1} : V \otimes_{{\boldsymbol{p}}^1} W \rightarrow V \otimes_{{\boldsymbol{p}}^2} W\tag{276}\] To do the same for the universal \(R\)-matrix of 174 , we need to modify the construction as is usually done in the theory of integrable systems for quantum affine algebras. The reason is that while elements of the completion 159 act by finite sums on \(V \otimes W\) for any \(V,W\) in category \({\mathcal{O}}\), the universal \(R\)-matrix 174 actually has infinitely many terms in any horizontal degree, corresponding to arbitrarily high vertical degrees in the first tensor factor. Therefore, we consider \[\mathbf{U}\curvearrowright V^u = V \quad \text{with the action of any }X \in \mathbf{U}\text{ renormalized by }u^{\text{vdeg }X}\] Therefore \[\begin{gather} _{\bar{{\boldsymbol{p}}}^2}{\mathcal{R}}_{{\boldsymbol{p}}^1} = \sum_{k} A_k \otimes B_k \in \mathbf{U}\;\widehat{\otimes} \;\mathbf{U}\quad \leadsto \\ \leadsto \quad _{\bar{{\boldsymbol{p}}}^2}R_{{\boldsymbol{p}}^1}(u) = \sum_k A_k u^{\text{vdeg }A_k} \otimes B_k \in \text{End}(V) \otimes \text{End}(W)((u)) \end{gather}\] With this modification, formula 176 implies that we have a \(\mathbf{U}\)-intertwiner \[\label{eqn:isomorphism322} _{\bar{{\boldsymbol{p}}}^2}R_{{\boldsymbol{p}}^1}(u) : V^u \otimes_{{\boldsymbol{p}}^1} W \rightarrow V^u \otimes^{\text{op}}_{{\boldsymbol{p}}^2} W((u))\tag{277}\] When \(V = L({\boldsymbol{\psi}})\) and \(W = L({\boldsymbol{\psi}}')\) are the simple modules 273 associated to arbitrary regular\(^{\neq 0}\) loop weights \({\boldsymbol{\psi}}\) and \({\boldsymbol{\psi}}'\), it is customary to renormalize the above intertwiner so that the matrix coefficient of the highest weight vector is 1: \[\label{eqn:reduced32r-matrix} _{\bar{{\boldsymbol{p}}}^2}R'_{{\boldsymbol{p}}^1}(u) = \frac{_{\bar{{\boldsymbol{p}}}^2}R_{{\boldsymbol{p}}^1}(u)}{\langle \varnothing \otimes \varnothing | _{\bar{{\boldsymbol{p}}}^2}R_{{\boldsymbol{p}}^1}(u) | \varnothing \otimes \varnothing \rangle}\tag{278}\] The following generalizes a classic result of Drinfeld ([1], 0?) for finite type \({\mathfrak{g}}\).

We note that the theorem above also requires the following technical condition, which holds automatically if one replaces the ground field \({\mathbb{K}}\) by its algebraic closure: the zeta functions that define \(\mathbf{U}\) are fully factored, in the sense of 100 , and the rational loop weights \({\boldsymbol{\psi}},{\boldsymbol{\psi}}'\) are also fully factored as follows: \[\label{eqn:factored32loop32weight} \psi_i(z) = \ell_i\frac{(z - s_{i|1})\dots (z-s_{i|\flat_i})}{(z - t_{i|1})\dots (z-t_{i|\flat_i})} \quad \text{and} \quad \psi'_i(z) = \ell_i'\frac{(z - s'_{i|1})\dots (z-s'_{i|\flat'_i})}{(z - t'_{i|1})\dots (z-t'_{i|\flat'_i})}\tag{279}\] for various \(\ell_i,\ell_i',s_{i|\bullet},t_{i|\bullet},s'_{i|\bullet},t'_{i|\bullet} \in {\mathbb{K}}^*\).

One could also define \(u\)-twisted versions of the intertwiner 276 , but it would actually be Laurent polynomial in \(u\): \[\label{eqn:isomorphism32132bis} _{{\boldsymbol{p}}^2}R_{{\boldsymbol{p}}^1} : V^u \otimes_{{\boldsymbol{p}}^1} W \rightarrow V^u \otimes_{{\boldsymbol{p}}^2} W[u^{\pm 1}]\tag{288}\] There is a host of interesting problems that can be asked about the \(R\)-matrices above, such as the Bethe ansatz, calculation of transfer matrices and XXZ-type Hamiltonians, see [12] and numerous other works. The infrastructure we have set up in the present paper shows that it is reasonable to ask these questions in the generality of quantum loop algebras for all \((I,{\mathbb{K}},\zeta_{ij}(x))\), which goes beyond the quantum affine or toroidal algebras that have been studied so far ([9], [10]).

References↩︎

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1. Strictly speaking, 95 96 follow from 52 and 97 98 follow from 79 only if we assume that \(\kappa_i^+\) and \(\kappa_i^-\) are invertible. We will tacitly tolerate this slight abuse in the logical flow of our constructions, and henceforth assume that 95 98 hold as stated.↩︎

2. We also make the convention that \[\begin{align} &{\mathcal{S}}^+_{[{\boldsymbol{p}},\boldsymbol{\infty}]} := {\mathcal{S}}^+_{\geq {\boldsymbol{p}}} \otimes {\mathcal{B}}^{\geq}_{\boldsymbol{\infty}} \tag{120} \\ &{\mathcal{S}}^-_{[{\boldsymbol{p}},\boldsymbol{\infty}]} := {\mathcal{S}}^-_{\geq {\boldsymbol{p}}} \otimes {\mathcal{B}}^{\leq}_{\boldsymbol{\infty}} \tag{121} \end{align}\] i.e. we include the finite Cartan elements \(\kappa_i^\pm\) in the algebras \({\mathcal{S}}^\pm_{[{\boldsymbol{p}},\boldsymbol{\infty}]}\), alongside \({\mathcal{B}}_{\boldsymbol{\infty}}^{\pm} = {\mathbb{K}}[p_{i,\pm d}]\).↩︎

3. A little care must be taken with the formulas above if \(E = E'\kappa^+\) and \(F \in F' \kappa^-\) where \[E' \in {\mathcal{S}}^+ \otimes {\mathcal{B}}_{\boldsymbol{\infty}}^+ \quad \text{and} \quad F' \in {\mathcal{S}}^- \otimes {\mathcal{B}}_{\boldsymbol{\infty}}^-\] and \(\kappa^\pm\) denote various products of finite Cartan elements \(\kappa^\pm_i\). In this case, we set \[[E]_{\geq {\boldsymbol{p}}} = [E']_{\geq {\boldsymbol{p}}} \kappa^+, \qquad [F]_{<{\boldsymbol{p}}} = [F']_{<{\boldsymbol{p}}}, \qquad [F]_{\geq {\boldsymbol{p}}} = [F']_{\geq {\boldsymbol{p}}} \kappa^-, \qquad [E]_{<{\boldsymbol{p}}} = [E']_{<{\boldsymbol{p}}}\] with \([E']_{\geq {\boldsymbol{p}}}, [F']_{<{\boldsymbol{p}}}, [F']_{\geq {\boldsymbol{p}}}, [E']_{<{\boldsymbol{p}}}\) uniquely determined by 127 130 .↩︎

4. See the discussion of Subsection 3.9 for certain technicalities necessary to make this precise.↩︎

5. We tacitly modify the pairing of \(\mathbf{U}\) by dividing the RHS of ?? ?? by the constant \[\left(q_{i_1}^{-1} - q_{i_1} \right) \dots \left( q_{i_n}^{-1} - q_{i_n}\right)\] where \(q_i = q^{d_i}\), in order to match the existing conventions for quantum affine algebras. This has a trivial effect on our algebras, which can be countered by rescaling either the \(e\) or \(f\) generators.↩︎

6. A crucial detail which ensures the preceding argument works is that the variable \(z_{m-1}\) has the smallest index among all variables involved, and so it can only afford a diagonal going out of it to the right; this is in tune with the fact that the variable \(z_{m-1}\) is causing the bottom line in the picture to be shifted between \(0\) and \(|d_{\iota(A_r) \iota(A_s)}|\) units to the right of the top line.↩︎

7. The decomposition above exists on general grounds if \({\mathbb{K}}\) is algebraically closed, but otherwise we restrict attention only to those \(V\)’s for which such a decomposition exists.↩︎

8. While the sum \(F' \otimes F''\) is infinite for any given \(F\), we note that the map ?? is well-defined because all but finitely many of the \(F'\) that appear have vertical degree bounded below by any arbitrarily large number, and thus vanish in \(L^{{\boldsymbol{p}}}(z^{-{\mathbf{r}}})\).↩︎

9. Note that we must use the following straightforward consequence of 56 and 64 : \[\label{eqn:consequence321} \langle E'', S(\varphi F'') \rangle = \varepsilon(\varphi) \langle E'',S(F'')\rangle\tag{248}\] for any \(E'' \in {\mathcal{S}}^+\), \(F'' \in {\mathcal{S}}^-\) and \(\varphi= \varphi^-_{i_1,d_1}\varphi^-_{i_2,d_2}\dots\). We also note that \(\varphi\) is on opposite sides of \(F''\) in the RHS of 245 and in the LHS of 248 . This is not an issue, since commuting \(\varphi\) past \(F''\) happens at the cost of multiplying \(F' \otimes F''\) by a power series in \(\{z_{ia}/z_{jb}\}_{i,j \in I, a \leq m_i, b > m_j}\) with non-zero constant term, which does not change whether \(F' \otimes F'' \in J^{{\boldsymbol{p}}}(z^{-{\mathbf{r}}}) \otimes {\mathcal{S}}^- + {\mathcal{S}}^-_{<{\boldsymbol{p}}} \otimes J^{\neq 0}({\boldsymbol{\psi}})\).↩︎

10. Note that we must use the following straightforward consequence of 57 and 65 : \[\label{eqn:consequence322} \langle \varphi E', S(F') \rangle = \varepsilon(\varphi) \langle E',S(F')\rangle\tag{254}\] for any \(E' \in {\mathcal{S}}^+\), \(F' \in {\mathcal{S}}^-\) and \(\varphi= \varphi^+_{i_1,d_1}\varphi^+_{i_2,d_2}\dots\).↩︎

11. Note that term in the expression below involving the product of \((t_{i|\bullet}^d - s_{i|\bullet}^d)\) and \(({t'_{j|\bullet}}^{-d} - {s'_{j|\bullet}}^{-d})\) is missing because we renormalized the \(R\)-matrix in the right-hand side of 278 .↩︎