October 02, 2025
Wronskian determinants (for coefficients of higher-order differential operators on the affine real line or circle) satisfy the table of Jacobi-type quadratic identities for strong homotopy Lie algebras – i.e.for a particular case of \(L_\infty\)-deformations – for the Lie algebra of vector fields on that one-dimensional affine manifold. We show that the standard realisation of \(\mathfrak{sl}(2)\) by quadratic-coefficient
vector fields is the bottom structure in a sequence of finite-dimensional polynomial algebras \(\Bbbk_N[x]\) with the Wronskians as \(N\)-ary brackets; the structure constants are calculated
explicitly.
Key words: Wronskian determinant, \(N\)-ary bracket, \(L_\infty\)-algebra, strong homotopy Lie algebra, \(\mathfrak{sl}(2)\), Witt algebra,
Vandermonde determinant.
Let us view Lie algebras \(\mathfrak{g}=(V, [{\cdot},{\cdot}])\) as a base class of structures which we seek to generalise in a natural way: (i) the vector space \(V\) (over a chosen field \(\Bbbk=\mathbb{R}\) or \(\mathbb{C}\) of characteristic zero) can be enlarged; (ii) the binary Lie bracket \([{\cdot},{\cdot}]\) can be replaced with \(N\)-ary antisymmetric multi-linear bracket(s) satisfying (collections of) (iii) suitable variants of \(N\)-ary Jacobi-type quadratic identities. In retrospect, the problem and process of such enlargement sheds more light on the nature and properties of the initially taken objects and structures.
Example 1. The Lie algebra \(\mathfrak{sl}(2,\mathbb{C})\) is the main example of a (semi)simple complex Lie algebra; its structure is encoded by the root system \(\mathsf{A}_1\) in \(\mathbb{E}^1\). In the Chevalley basis \(\{e,f,h\}\) for \(\mathfrak{sl}(2)\), the Lie bracket is determined by the relations \[\label{EqRelSL2} [h,e]=2e,\qquad [h,f]=-2f,\qquad [e,f]=h.\tag{1}\] Larger irreducible root systems in higher-dimensional real Euclidean spaces \(\mathbb{E}^r\) exhaustively classify all simple complex Lie algebras of higher ranks \(r\); their list is \(\mathsf{A}_r\) (\(r\geqslant 1\)), \(\mathsf{B}_r\) (\(r\geqslant 2\)), \(\mathsf{C}_r\) (\(r\geqslant 3\)), \(\mathsf{D}_r\) (\(r\geqslant 4\)), and the exceptional five: \(E_6\), \(E_7\), \(E_8\), \(F_4\), and \(G_2\), see [1]; allowing imaginary simple root vectors, we arrive at the Borcherds–Kac–Moody algebras.
Independently, the binary bracket \([{\cdot},{\cdot}]\) can be deformed to a formal series of structures, \[[{\cdot},{\cdot}] \longmapsto \nabla = [{\cdot},{\cdot}] + \nabla_3 +\ldots+\nabla_m+\ldots,\] where \([{\cdot},{\cdot}] \equiv \nabla_2\) and for each \(m\geqslant 2\), the \(m\)-linear term \(\nabla_m\) is totally antisymmetric w.r.t. its \(m\) arguments from the underlying vector space \(V\). The original Lie bracket \(\nabla_2\) in the algebra \(\mathfrak{g}=(V,\nabla_2)\) satisfied the Jacobi identity, \[\tfrac{1}{1!\cdot 2!} \sum_{\tau\in S_3} (-)^\tau\: \nabla_2\bigl( \nabla_2 \bigl( v_{\tau(1)}, v_{\tau(2)} \bigr), v_{\tau(3)} \bigr) = 0,\] for any elements \(v_1,v_2,v_3\in V\). A natural quadratic (w.r.t.the structure \(\nabla\)) Jacobi-type identity for the deformed structure \(\nabla=\nabla_2+\ldots\) is \(\nabla[\nabla]=0\), meaning that for every \(m\)-tuple \(v_1\otimes v_2\otimes\ldots\otimes v_m\) of \(v_j\in V\) the expansion of inner- and outer copy of \(\nabla\) by linearity over \(\Bbbk\) produces the chain of partial (at \(m\geqslant 3\)) identities of the form \[\begin{gather} \sum_{\tau\in S_m} (-)^\tau \Bigl[ \nabla_2\bigl( \nabla_{m-1} \bigl( v_{\tau(1)},\ldots,v_{\tau(m-1)}\bigr),v_{\tau(m)} \bigr) + \ldots {} \\ {}+ \nabla_k\bigl( \nabla_{m+1-k} \bigl( v_{\tau(1)},\ldots,v_{\tau(m+1-k)}\bigr),v_{\tau(m+2-k)},\ldots, v_{\tau(m)} \bigr) + \ldots {} \\ {}+ \nabla_{m-1}\bigl( \nabla_{2} \bigl( v_{\tau(1)},v_{\tau(2)}\bigr), v_{\tau(3)},\ldots, v_{\tau(m)} \bigr) \Bigr] = 0. \end{gather}\] This is the infinite chain (as integer \(m\) starts at \(3\) and increases) of Jacobi identities for the \(L_\infty\)-deformation of the bracket \([{\cdot},{\cdot}]=\nabla_2\) in the Lie algebra \(\mathfrak{g}=(V,[{\cdot},{\cdot}])\), see [2] and [3]. Of particular interest is the case when – in each \(m\)th summand of the above chain of identities – all the quadratic terms vanish separately, i.e.whenever the \(N\)-ary operations \(\nabla_N\) at \(N\geqslant 2\) in the \(L_\infty\)-structure \(\nabla\) are such that \(\nabla_k[\nabla_\ell]=0\) for all \(k,\ell\geqslant 2\). We shall study these strong homotopy deformations of the Lie bracket \([{\cdot},{\cdot}]\), see [4], the tail components \(\nabla_j\) now satisfying the table of identities (at \(k,\ell\geqslant 2\)), \[\sum_{\tau\in S_{\ell,k-1}} (-)^\tau\: \nabla_k\bigl( \nabla_{\ell} \bigl( v_{\tau(1)},\ldots,v_{\tau(\ell)}\bigr),v_{\tau(\ell+1)},\ldots, v_{\tau(k+\ell-1)} \bigr) = 0,\]where the sums are conveniently taken over the sets of \((\ell,k-1)\)-unshuffles \(\tau\in S_{k+\ell-1}\) such that \(\tau(1) <\ldots< \tau(\ell)\) and \(\tau(\ell+1)<\ldots<\tau(k+\ell-1)\); passing from the entire group of permulations \(S_{k+\ell-1}\) to its subset of unshuffles, we divide both sides of the identity \(\nabla_k[\nabla_\ell]=0\) by \((k-1)!\ell!\) occurring from the alternation of arguments \(v_j\) strictly within the totally antisymmetric brackets \(\nabla_k\) and \(\nabla_\ell\), respectively.
Research problem. We are interested in finding a natural source of strong homotopy Lie structures \(\nabla_k\), \(k\geqslant2\), that would deform the Lie algebra \(\mathfrak{sl}(2)\). Secondly, we want to find a class of finite-dimensional vector spaces \(V_N\) such that at each \(N\geqslant 2\), the \(N\)-ary bracket \(\nabla_N\) does restrict to \(V_N\), making it a finite-dimensional Schlessinger–Stasheff Lie algebra \((V_N,\nabla_N)\).
To this end, let us consider the quadratic-coefficient realisation \(\varrho\colon \mathfrak{sl}(2) \to {\mathrm D}_1(\mathbb{R})\) of the Lie algebra \(\mathfrak{sl}(2)\) in the space of vector fields on the line \(\mathbb{R}\) with global affine coordinate \(x\); this standard realisation is given by the formula1 \[\label{EqQuadraticCoeffsSL2} \varrho(e) = 1\cdot \partial/\partial x,\qquad \varrho(h) = -2x\cdot \partial/\partial x,\qquad \varrho(f) = -x^2\cdot \partial/\partial x.\tag{2}\] One readily verifies the standard commutation relations from Eq. 1 ; our choice of sign in the commutator is \(\bigl[\vec{X},\vec{Y}\bigr] = \bigl[\vec{X}(Y) - \vec{Y}(X) \bigr]\cdot \partial/\partial x\) for \(\vec{X} = X(x)\cdot \partial/\partial x\) and \(\vec{Y} = Y(x)\cdot\partial/\partial x\). In the commutator \([{\cdot},{\cdot}]\) on \({\mathrm D}_1(\mathbb{R}^1)\) we recognise the Wronskian determinant of two coefficients: \[\bigl[\vec{X},\vec{Y}\bigr](x) = \det\begin{pmatrix} X & Y \\ X' & Y' \end{pmatrix}(x)\cdot \frac{\partial}{\partial x} = W^{0,1}(X,Y)(x)\cdot \frac{\partial}{\partial x}.\] Of course, the commutator \([{\cdot},{\cdot}]\) of vector fields does satisfy the Jacobi identity, \[\tfrac{1}{2} \sum_{\tau\in S_3} (-)^\tau\: W^{0,1}\bigl( W^{0,1}(X_{\tau(1)},X_{\tau(2)}), X_{\tau(3)} \bigr)\cdot \partial/\partial x=0,\] for any vector fields \(\vec{X}_j = X_j(x)\,\partial/\partial x\) with twice defferentiable coefficients \(X_j(x)\) on \(\mathbb{R}\).
Viewing the Wronskian determinants as (\(N\geqslant 2\))-ary brackets for coefficients of higher-order differential operators on the affine line, we presently describe a class of finite-dimensional Schlessinger–Stasheff Lie algebras — such that this class incorporates the standard realisation \(\varrho\colon \mathfrak{sl}(2) \to {\mathrm D}_1(\mathbb{R})\) with quadratic polynomials in Eq. 2 .
the associative algebra \({\mathrm D}_*(\mathbb{S}^1)\) of differential operators of nonnegative integer orders on the circle \(\mathbb{S}^1\) (or on the other connected one-dimensional affine real manifold \(M^1=\mathbb{R}^1_{\text{aff}}\)). The assumption that allowed coordinate transformations are affine makes well defined the subspaces \({\mathrm D}_p(M^1)\) of differential operators of strict order \(p\).
Definition 1. Take \(N=2p\) and let \(w_1,\ldots,w_N\in{\mathrm D}_p(M^1)\), so that locally we have \(w_j=w_j(x)\cdot\partial_x^p\). By definition, put \[\label{EqAltBrOp} \bigl[w_1,\ldots,w_N\bigr]_N \mathrel{{:}{=}} \mathop{\mathrm{Alt}}\bigl(w_1,\ldots,w_N\bigr) = \sum_{\sigma\in S_N} (-)^\sigma\: w_{\sigma(1)}\circ \ldots \circ w_{\sigma(N)},\tag{3}\] where the r.-h.s.is the alternated associative composition of operators.
In the same way as the commutator of two vector fields is again a vector field, we prove
Theorem 1. The subspace \({\mathrm D}_p(M^1)\) of differential operators \(w_j=w_j(x)\cdot \partial_x^p\) of strict order \(p\in\mathbb{N}\) is closed under the alternated composition \([{\cdot},\ldots,{\cdot}]_{N=2p}\) of twice as many arguments \(w_1,\ldots,w_N\in{\mathrm D}_p(M^1)\). Moreover, the structure constants are explicit: \[\label{EqBrWr} \bigl[w_1(x)\,\partial_x^p,\ldots,w_N(x)\,\partial_x^p\bigr]_N = W^{0,1,\ldots,N-1}(w_1,\ldots,w_N)\cdot \partial_x^p,\qquad{(1)}\] where \(W^{0,1,\ldots,N-1} = \mathbf{1}\wedge \partial_x\wedge\ldots\wedge \partial_x^{N-1}\) is the Wronskian determinant of \(N\) arguments in one independent variable \(x\).
Remark 1. The Wronskian determinant of \(N\) scalar functions itself is not a scalar function: indeed, the Wronskian determinant behaves under a change of base coordinate, \(x=x(y)\) \({}{\rightleftarrows}{}\) \(y=y(x)\), locally on \(M^1\) (see App. 5). Likewise, the coefficients of differential operators of order \(p>0\) do change after a reparametrisation on the base \(M^1\); when this change is affine, the strict order \(p\) is preserved and equality ?? makes sense.
We deduce from Theorem 1 that the alternated composition of \(N=4\) differential operators of order \(p=2\) is again an operator of order two. The same holds for every integer \(N=2p\); this serves an \(N\)-ary generalisation for the commutator of vector fields from \({\mathrm D}_1(M^1)\). Still, given the alternated composition as the bracket for elements of an associative algebra, which quadratic, Jacobi-type identities does this bracket satisfy ?
Proposition 2 ([6], [7]). Let \({{\EuScript A}}\) be an associative algebra and \([{\cdot},\ldots,{\cdot}]_N\in\mathop{\mathrm{Hom}}_\Bbbk\bigl( \bigwedge^N {{\EuScript A}},{{\EuScript A}}\bigr)\) be the alternated composition of \(N\) elements \(a_1,\ldots,a_N\) from \({{\EuScript A}}\) (cf.Eq. 3 ): \[\label{AssocBracketHW} \bigl[a_1,\ldots,a_N\bigr]_N = \sum_{\sigma\in S_N} (-)^\sigma\: a_{\sigma(1)}\circ\ldots\circ a_{\sigma(N)}.\tag{4}\] Suppose also that \(N\) is even. Then the bracket \([{\cdot},\ldots,{\cdot}]_N\) satisfies the quadratic Jacobi-type identity, \[\frac{1}{N!(N-1)!}\: \sum_{\tau\in S_{2N-1}} (-)^\tau\: \bigl[ \bigl[ a_{\tau(1)}, \ldots, a_{\tau(N)} \bigr]_N, a_{\tau(N+1)}, \ldots, a_{\tau(2N-1)} \bigr]_N = 0,\]so that \({{\EuScript A}}\) becomes a Schlessinger–Stasheff Lie algebra.
The proof is by inspecting the coefficient of \(a_1\circ a_2\circ\ldots\circ a_{2N-1}\) in the totally antisymmetric sum over \(S_{2N-1}\ni\tau\); whenever \(N\) is even, the coefficient cancels out.
Corollary 3. For even \(N=2p\in\mathbb{N}\), the Wronskian determinant \(W^{0,1,\ldots,N-1} = \mathbf{1}\wedge \partial_x\wedge\ldots\wedge \partial_x^{N-1}\) over a one-dimensional base \(M^1\) satisfies the \(N\)-ary Jacobi identity \(W^{0,1,\ldots,N-1} \bigl[ W^{0,1,\ldots,N-1} \bigr] = 0\).
In the course of proving Proposition 2 it is readily seen that its idea extends to a not necessarily even number of arguments in either inner- or outer bracket and to a not necessarily coinciding number of arguments in the inner- and outer brackets within the left-hand side of the quadratic Jacobi identity for strong homotopy Lie algebras.
Proposition 4 ([6]). Recall that the subscript \(N\) at the symbol \(\Delta_N\) of bracket 4 denotes its number of arguments: \(\Delta_i\in\mathop{\mathrm{Hom}}_\Bbbk(\bigwedge^N{{\EuScript A}},{{\EuScript A}})\); let \(k\) and \(\ell\) be arbitrary positive integers. Then the following identities hold: \[\begin{align} \Delta_{2k}[\Delta_{2\ell}]&=0,\tag{5}\\ \Delta_{2k+1}[\Delta_{2\ell}]&=\Delta_{2k+2\ell},\tag{6}\\ \Delta_k[\Delta_{2\ell+1}]&=k\cdot\Delta_{2\ell+k}.\tag{7} \end{align}\]
Proof. The proof of 5 repeats literally the proof of Proposition 2. For 6 , we note that the last summand, \[\beta_{2k+1} = (-)^{2\ell \cdot((2k+1)-1)} \Delta_{2k+1}\bigl( \Delta_{2\ell}\bigr(a_{2k+1},\ldots,a_{2k+2\ell}\bigr), a_1,\ldots,a_{2k}\bigr),\] is not compensated. For 7 , the summand \(\alpha=a_1\circ\dots\circ a_{2\ell+k}\) acquires the coefficient \(\sum_{j=1}^k(-1)^{(2\ell+1)(j-1)}\cdot(-1)^{j-1}=k\). This completes the proof. ◻
These properties of totally antisymmetric homomorphisms work immediately for Wronskian determinants of arbitrary and not necessarily coinciding sizes.
Proposition 5 (see [8]). Consider the Wronskian determinants \(W^{0,1,\ldots,N} = \mathbf{1}\wedge \partial_x\wedge\ldots\wedge \partial_x^{N}\) with integral orders of differentiation. Then the strong homotopy Lie algebra Jacobi identities \(W^{0,1,\ldots,k} \bigl[ W^{0,1,\ldots,\ell} \bigr] = 0\) hold for all positive integers \(k,\ell\in\mathbb{N}\), meaning that \[\label{EqSHJacobiKL} \frac{1}{k!(\ell+1)!}\: \sum_{\tau\in S_{k+\ell+1}} (-)^\tau\: W^{0,1,\ldots,k}\bigl( W^{0,1,\ldots,\ell} \bigl( f_{\tau(1)}, \ldots, f_{\tau(\ell+1)} \bigr), f_{\tau(\ell+2)}, \ldots, f_{\tau(k+\ell+1)} \bigr) = 0\tag{8}\] for arbitrary \(f_{1}(x),\ldots,f_{k+\ell+1}(x)\) of one independent variable \(x\).
Proof. Indeed, the inner- and outer Wronskian determinants combined contain \(\tfrac{1}{2} k(k+1) + \tfrac{1}{2} \ell(\ell+1)\) derivatives \(\partial_x\) acting on the arguments of Jacobiator; by construction, the Jacobiator \(W^{0,1,\ldots,k} \bigl[ W^{0,1,\ldots,\ell} \bigr]\) is totally antisymmetric w.r.t.its \(k+\ell+1\) arguments. For this, to let the integral differential orders of all the arguments \(f_j\) be pairwise distinct, at least \(\tfrac{1}{2} (k+\ell+1)(k+\ell+2)\) derivatives are needed. But the actually available number is strictly less, whence the assertion.2 ◻
Remark 2. Although the Wronskian determinant of size \(2\times 2\) does show up in the commutator of vector fields, their differential order \(p=1\) is too low to make noticeable that strong homotopy Jacobi identities 8 are valid for not necessarily equal numbers of arguments in the inner- and outer brackets.
Remark 3. Whenever the number \(N\) of arguments in the Jacobiator \(W^{0,1,\ldots,N-1} \bigl[ W^{0,1,\ldots,N-1} \bigr]\) is not even, we no longer refer to the arguments, depending on the variable \(x\), as coefficients of differential operators of strict (half-)integer order \(p=N/2\). Indeed, there is presently no guarantee that the alternated composition of half-integral order operators would act by integer order differentiations, \(\mathbf{1}\wedge \partial_x\wedge\ldots\wedge \partial_x^{N-1} = W^{0,1,\ldots,N-1}\), on such operators’ coefficients \(f_j(x)\).
quadratic-polynomial realisation of three-dimensional Lie algebra \(\mathfrak{sl}(2)\) can carry the ternary Wronskian bracket \(W^{0,1,2}\) and be closed w.r.t. it. Are there larger, still finite-dimensional Schlessinger–Stasheff \(N\)-ary Lie algebras of polynomials ?
Consider the space \(\Bbbk_N[x]\ni a_j\) of polynomials of degree not greater than \(N\); on this space, the Wronskian determinant is an \(N\)-linear antisymmetric bracket, \[\label{NaryBracket} \bigl[a_1, \ldots, a_N\bigr]_N = W^{0,1,\ldots,N-1}\bigl({a_1},{\ldots},{a_N}\bigr).\tag{9}\] Introduce the basis \(\{a^0_k\}=\{x^k / k! \}\) of monomials in \(\Bbbk_N[x]\), here \(0\leqslant k\leqslant N\); the monomials \(x^k/k!\) are closed w.r.t. derivations — and the Wronskian determinants as well.
Theorem 6. Let \(0\leqslant k\leqslant N\) and bypass the monomial \(x^k/k!\) from our basis in \(\Bbbk_N[x]\). Then the Wronskian determinant of remaining monomials satisfies the identity \[\label{WronskWithConst} W^{0,1,\ldots,N-1}\Bigl({1},{\ldots,\widehat{\frac{x^k}{k!}},\ldots},{\frac{x^N}{N!}}\Bigr)= \frac{x^{N-k}}{(N-k)!}.\qquad{(2)}\] In particular, all the structure constants, whenever nonzero, equal \(\pm 1\) in this \((N+1)\)-dimensional Schlessinger–Stasheff Lie algebra \(\Bbbk_N[x]\) with the Wronskian as \(N\)-ary bracket.
Proof. We have \[\label{WronskDecompose} \mathop{{W}}\left(1,\ldots,\widehat{\frac{x^k}{k!}},\ldots,\frac{x^N}{N!}\right) = \mathop{{W}}\left(1,\ldots,\frac{x^{k-1}}{(k-1)!}\right) \cdot \mathop{{W}}\left(x,\ldots,\frac{x^{N-k}}{(N-k)!}\right),\tag{10}\] where the first factor in the r.h.s.of 10 equals \(1\) and has degree \(0\). Denote by \(W_m\) the second factor, the determinant of the \((N-k)\times(N-k)\) matrix with \(m\equiv N-k\). We claim that \(W_m\) is a monomial: \(\deg W_m=m\); we prove this by induction on \(m\equiv N-k\). For \(m=1\), \(\deg\det(x)=1=m\). Let \(m>1\); the decomposition of \(W_m\) w.r.t.the last row gives \[\label{DecompositionWRTLastRow} W_m=\mathop{{W}}\left(x,{\ldots},\frac{x^m}{m!}\right)= x\cdot\mathop{{W}}\left(x,{\ldots},\frac{x^{m-1}}{(m-1)!}\right) - \mathop{{W}}\left(x,{\ldots},\frac{x^{m-2}}{(m-2)!},\frac{x^m}{m!}\right),\tag{11}\] where the degree of the first Wronskian in r.h.s.of (11 ) is \(m-1\) by the inductive assumption. Again, decompose the second Wronskian in r.h.s.of (11 ) w.r.t.the last row and proceed iteratively by using the induction hypothesis. We obtain the recurrence relation \[\label{RecurrentWronsk} W_m=\sum\limits_{\ell=1}^{m-1} W_{m-\ell}\cdot (-1)^{\ell+1} \frac{x^\ell}{\ell!} - (-1)^m\,\frac{x^m}{m!},\qquad m\geqslant 1,\tag{12}\] whence \(\deg W_m=m\). We see that the initially taken Wronskian (10 ) itself is a monomial of degree \(m=N-k\) with yet unknown coefficient.
Now, we calculate the coefficient \(W_m(x)/x^m\in\Bbbk\) in Wronskian determinant 10 . Consider the generating function \[\label{GenFunction} f(x)\equiv\sum\limits_{m=1}^\infty W_m(x) \qquad \text{such that} \quad W_m(x)=\frac{x^m}{m!}\,\frac{d^mf}{dx^m}(0),\qquad 1\leqslant m\in\mathbb{N}.\tag{13}\] Recall that \(\exp(x)\equiv\sum_{m=0}^\infty x^m/m!\); viewing 13 as the formal sum of equations 12 , we have \[f(x)=f(x)\cdot(\exp(-x)-1)-\exp(-x)+1, \qquad \text{whence}\quad f(x)=\exp(x)-1.\]Hence the required coefficient equals \(1/m!\). The proof is complete. ◻
infinite-dimensional Witt algebra of holomorphic vector fields on \(\mathbb{C}\setminus \{ 0 \}\), defined by the relations \([a_i, a_j]=(j-i)\,a_{i+j}\) for \(i,j\in\mathbb{Z}\), is the Virasoro algebra with zero central charge. We now study its \(L_\infty\)-, yet in fact a strong homotopy deformation by using Wronskians. In the Witt algebra itself, we have the binary bracket (\(N=2\)) of the polynomial coefficient generators \(a_i=x^{i+1}\), where \(x\in\Bbbk\) and \(i\in\mathbb{Z}\).
For \(N\geqslant 2\), the proper choice of index shift in the set of generators is \(a_i=x^{i+N/2}\). We postulate the Wronskian determinant \(W^{0,1,\ldots,N-1}\) be the \(N\)-ary bracket: \[\label{WittGeneral} \bigl[a_{i_1},\ldots,a_{i_N}\bigr]_N=\Omega(i_1,\ldots,i_N)\,a_{i_1+\cdots+i_N};\tag{14}\] the structure constants \(\Omega(i_1,\ldots,i_N)\) are totally antisymmetric w.r.t.their arguments. Let us calculate the function \(\Omega\).
Theorem 7. Let \(\nu_1\), \({\ldots}\), \(\nu_N\in\Bbbk\) be constants and set \(\nu=\sum_{i=1}^N\nu_i\); then we have that\[\label{Vander} W^{0,1,\ldots,N-1}(x^{\nu_1},\ldots,x^{\nu_N})= \prod\limits_{1\leqslant i<j\leqslant N}(\nu_j-\nu_i)\cdot x^{\nu-{N(N-1)}/{2} },\qquad{(3)}\] i.e.the Wronskian determinant of monomials itself is a monomial, and its coefficient is the Vandermonde determinant.
Proof. Consider determinant ?? : \(A=\det\|a_{ij}\,x^{\nu_j-i+1}\|\). From \(j\)th column take the monomial \(x^{\nu_j-N+1}\) out of the determinant: \[A=x^{\nu-N(N-1)}\cdot\det\|a_{ij}\,x^{N-i}\|;\] all rows acquire common degrees in \(x\): \(\deg\)(any element in \(i\)th row) \({}=N-i\). From \(i\)th row take this common factor \(x^{N-i}\) out of the determinant: \[A=x^{\nu-N(N-1)/2}\cdot\det\|a_{ij}\|,\] where the coefficients \(a_{ij}\) originate from the initial derivations in a very special way: for any \(i\) such that \(2\leqslant i\leqslant N\), we have \[a_{1j}=1\qquad\text{and}\qquad a_{ij}=(\nu_j\underline{{}-i+2})\cdot a_{i-1,j} \quad\text{for 1<i\leqslant N.}\] The underlined summand does not depend on \(j\); hence for any \(k=N\),\({\ldots}\),\(2\) the determinant \(\det{}\|a_{ij}\|\) can be split into the sum: \[\begin{gather} \det\|a_{ij}\|=\det\|a'_{kj}=\nu_j\cdot a_{k-1,j};\quad a'_{ij}=a_{ij} \text{ if }i\neq k\| + {}\\ {}+ \det\|a''_{kj}=(2-i)\cdot a_{k-1,j};\quad a''_{ij}=a_{ij} \text{ if }i\neq k\|, \end{gather}\] where the last determinant is vanishing identically. Solving the recurrence relation \(a_{ij}=\nu_j\cdot a_{i-1,j}\), we obtain \[\det\|a_{ij}\|=\det\|\nu_j^{i-1}\|=\prod\limits_{1\leqslant k<\ell\leqslant N} (\nu_\ell-\nu_k).\] This completes the proof. ◻
Remark 4. We have calculated the structure constants in 14 by using a ‘wrong’ basis \(a_i'=x^i\) such that the resulting degree is not \(\sum_{k=1}^N\deg a_k'\). Nevertheless, we use the translation invariance of the Vandermonde determinant, \[\Omega(i_1,\ldots,i_N)=\Omega(i_1+\frac{N}{2},\ldots,i_N+\frac{N}{2}).\] The assertion is established.
author thanks the organisers of 29th international conference on integrable systems &quantum symmetries – ISQS29 (held 7–11 July 2025 at CVUT Prague, CZ) for a warm atmosphere during the meeting. This work has been partially supported by the Bernoulli Institute (Groningen, NL) via project 110135. The author thanks M. Kontsevich and V. Retakh for helpful discussions and advice.
us recall the behaviour of Wronskian determinants w.r.t.coordinate changes \(y=y(x)\).
Theorem 8. Let \(\phi^i(y)\) be smooth functions for \(1\leqslant i\leqslant N\), that is, \(\phi^i\) be a scalar field of conformal weight \(0\) so that \(\phi^i\) is transformed by the rule \(\phi^i(y)\mapsto\phi^i(y(x))\) under a change \(y=y(x)\). Then the transformation law for the Wronskian is \[{\det\left\|\frac{d^j\phi^i}{dx^j}\right\|}_{ \begin{array}{rcl} i&\!=\!&1,\ldots,N\\ j&\!=\!&0,\ldots,N-1\\ \phi^i&\!=\!&\phi^i(y(x)) \end{array} } = {\left(\frac{dy}{dx}\right)}^{\Delta(N)}\, {\det\left\|\frac{d^j\phi^i}{dy^j}\right\|}_{ \begin{array}{rcl} \phi^i&\!=\!&\phi^i(y).\\ y&\!=\!&y(x) \end{array} }\] The conformal weight \(\Delta(N)\) of the Wronskian determinant for \(N\) scalar fields \(\phi^i\), themselves of weight \(0\), is \(\Delta(N)={N(N-1)}/{2}\).
Proof. Consider a function \(\phi^i(y(x))\) and apply the \(j\)th power \(({\mathrm d}/{\mathrm d}x)^j\) of derivative \(\partial_x\) by using the chain rule. The result is \[\frac{d^j\phi^i}{dy^j}\cdot{\left(\frac{dy}{dx}\right)}^j + \text{terms with lower order derivatives }\frac{d^{j'}}{dy^{j'}},\quad j'<j.\] These lower-order terms differ from the leading terms in \(({\mathrm d}/{\mathrm d}x)^{j'}\, \phi^i(y(x))\) with \(0\leqslant j'<j\) by the factors common for all \(i\); those lower-order terms produce no effect since a determinant with coinciding (or proportional) lines equals zero. From \(i\)th row of the Wronskian we extract \((i-1)\)th power of \(dy/dx\), their total number being \(N(N-1)/2\). This is the conformal weight by definition. ◻
arXiv:math.RA/0606241).arXiv:hep-th/9209099).arXiv:math.RA/0202043).arXiv:math.RA/0410185.Independently from our construction of two classes of strong homotopy Lie algebras, one can start from this vector field realisation of \(\mathfrak{sl}(2)\) in \(\Gamma(T\mathbb{R}^1)\) – or of higher-rank semisimple Lie algebras – and study their enlargements to Courant algebra structures on \(\Gamma(TM\oplus T^*M)\) over base manifolds \(M\) (e.g., \(M=\mathbb{S}^1\)): the Lie bracket \([{\cdot},{\cdot}]\) of vector fields is then supplemented with the new rules to commute vector fields with differential \(1\)-forms and similarly, commute two differential \(1\)-form arguments, see [5] and references therein.↩︎
This proof of the claim about Wronskians over one-dimensional base manifolds \(M^1\) does extend to a properly defined class of Wronskian determinants for arguments in \(d\) variables \(x^1,\ldots,x^d\), see [9].↩︎