There are various versions of nil algebras in non-associative and multilinear settings. These notions are closely connected with problems concerning polynomial automorphisms.

Let \(A\) be a vector space and \(\mu: A^d \to A\) be a \(d\)-linear map, so that the pair \((A,\mu)\) forms a (multilinear) algebra (in the sense of general algebra). The algebra \(A\) is called \(n\)-Engel (or simply Engel) if for each element \(x\in A\) the linear operator of multiplication by \(x\) is \(n\)-nilpotent, that is, the \(n\)-th Engel identity \[\mathop{\mathrm{Ad}}_x^n(y) \equiv 0\] holds in \(A\), where \(\mathop{\mathrm{Ad}}^1_x (y) = \mathop{\mathrm{Ad}}_x (y) = \mu (x, \dots, x, y), \mathop{\mathrm{Ad}}^{k+1}_x (y) = \mu (x, \dots, x, \mathop{\mathrm{Ad}}^{k}_x (y))\).

Gerstenhaber connected the Engel property with another kind of nilpotence. We call \(A\) Gerstenhaber nil (of nilindex \(n\)) if for each element \(x\in A\), all multilinear multiple compositions of at least \(n\) copies of \(\mu\) applied to \(x\) are zero. In other words, \(A\) is Gerstenhaber nil if each subalgebra generated by a single element is nilpotent of bounded degree.

From now on, assume that the ground field \({\Bbbk}\) has zero characteristic and the algebra \(A\) is commutative in the sense that the operation \(\mu\) is symmetric, i. e., for every \(\sigma \in S_d\) we have \(\mu(x_1, \dots , x_d ) = \mu(x_{\sigma 1}, \dots , x_{\sigma d})\). The following theorem was proved by Gerstenhaber [1] for binary algebras and generalized to the general case \(d\ge 2\) by Umirbaev [2].

In the binary case Gerstenhaber proved the following estimate: if \(A\) is Gerstenhaber \(p\)-nil, then it is \(n\)-Engel for \(n = 2p-3\) [1]. For a generalization of this estimate to the case of \(d\)-linear algebras in a more general setting, see Theorem 5 below.

Another version of nilpotence appears in an equivalent formulation of the Jacobian problem. Let \(T^{mult}_q (x_1, \dots , x_q)\) be the sum of all multilinear multiple compositions of \(\mu\) with \(q\) arguments, and let \(T_q(x) = \frac{1}{q!}T_q^{mult}(x,x, \dots, x)\). In other words, \[T_1(x) = xand T_q (x) = \sum_{i_1 +\dots + i_d = q}\mu(T_{i_1}, \dots, T_{i_d})for q>1.\]

Let us call an algebra \(A\) Yagzhev nil (of nilindex \(p\)) if the identities \(T_q(x) \equiv 0\) hold in \(A\) for all \(q\ge p\). Such algebras are called Yagzhev, or weakly nilpotent in [3], and weakly nil in [4]. Note that Yagzhev nil algebras need not be Gerstenhaber nil. Indeed, Gorni and Zampieri [5] have shown that there exists a 4-dimensional 3-linear algebra which is Yagzhev nil and Engel but not Gerstenhaber nil (their example is induced by a polynomial automorphism due to van den Essen). However, in the important case of binary algebras we do not know whether these two nil properties are equivalent? At least, a straightforward calculation with linearized identities shows that binary Yagzhev nil algebras of nilindex \(4\) are Gerstenhaber nil of nilindex \(6\).

Yagzhev nil algebras appear in his reformulation of the famous Jacobian conjecture, see [2], [3] and references therein. Recall that the conjecture states that a complex polynomial endomorphism with constant Jacobian determinant has a polynomial inverse. It is equivalent to the following statement ([6]; see also [7]):

The conjecture is known to be true for \(d=2\) [8], and it is sufficient to prove it for \(d=3\) ([6], [7]).

It follows from the results by Yagzhev [9] (based on an earlier work by Drużkowski and Rusek [10]) that the Jacobian conjecture for homogeneous mapping of degree \(d\) for polynomials of \(n\) variables is equivalent to the following:

In the notation of Conjecture 2, here \(\mu\) is the complete linearization of \(H\) so that \(H(X) =\mu(X, \dots, X)\).

The easy reverse implication of the Jacobian conjecture (stating that the Jacobian determinant of a polynomial automorphism is a nonzero constant) is well known as the Jacobian theorem. Being reformulated in the Yagzhev terms, it is a refinement of the Gerstenhaber–Umirbaev theorem for finite-dimensional algebras.

The aim of this note is to prove the same result without the assumption that \(A\) is finite-dimensional. We regard this as an infinite-dimensional version of the Jacobian theorem.

Note that the assumption of the second implication can be replaced by the following: the identities \(T_q(x) \equiv 0\) for all \(p \le q \le d(p-1)+1\) (cf. [4]).

Dotsenko has asked whether, in the binary case \(d=2\), the identity \(T_q(x) \equiv 0\) implies the \(n\)-Engel identity for some \(n\) [4]? Theorem 5 gives a partial answer to this question. Indeed, if the identities \(T_q(x) \equiv 0\) hold for all \(p \le q \le 2p-1\), the algebra is \(n\)–Engel for \(n = 2p-3\). Moreover, one can show that if the identities \(T_4(x) \equiv T_5(x) \equiv 0\) hold in a binary algebra \(A\), then it is Gerstenhaber nil of nilindex 6 (hence, Yagzhev nil of nilindex 4); therefore, it is 5-Engel (cf. [4]).

Proof of Theorem 5. Let \(q_0\) be the minimal number such that the identities \(T_q(x) =0\) hold in \(A\) for all \(q\ge q_0\). Since by definition \(T_q(x) = 0\) for \(q \notin (d-1){\mathbb{Z}}+1\), we have \(q_0 = (d-1)N+2\), where the integer \(N\) satisfies \(N \le \left[ \frac{p-2}{d-1}\right]\).

We may assume that all relations of \(A\) are consequences of the identities \(T_q(x) \equiv 0\) for all \(q\ge p\), so that \(A\) is a free algebra of some variety of multilinear algebras. In particular, we assume that \(A\) is graded. Moreover, we will assume that the set of free generators of \(A\) consists of at least two elements.

The operator \(g:A\to A\) defined by \(g(x) = x - \mu (x, \dots ,x)\) is invertible; the inverse is given by \(\gamma(y) := g^{-1}(y) = \sum_{j\ge 1} T_j(y)\) [10] (where \(T_j(y)=0\) for \(j\ge q_0\)). We obtain the identities \[\gamma (g(x)) = \sum_{j\ge 1} T_j(g(x)) = xandg(\gamma(y)) = \gamma(y)-\mu (\gamma(y), \dots ,\gamma(y)) = y.\] Replacing \(x\) by \(x+z\), \(y\) by \(y+t\), and collecting all terms which are linear in \(z\) and \(t\), we obtain partial linearizations of the above identities \[\mathrm{d}\gamma (g(x), \mathrm{d}g(x,z)) = z and\mathrm{d}g (\gamma(y), \mathrm{d}\gamma(y,t)) = t,\] where \[\mathrm{d}\gamma(y,t) = \sum_{n= 1}^{q_0-1} \frac{1}{(j-1)!}T_j^{mult}(y,\dots, y,t)and\mathrm{d}g(x,z) = z - d \mathop{\mathrm{Ad}}_x(z)\] are the corresponding partial linearizations of \(\gamma\) and \(g\). So, the linear operator \(\mathrm{d}g_x: z \mapsto \mathrm{d}g(x,z)\) (the “Jacobian”) is invertible; the inverse is given by the operator \(t \mapsto \mathrm{d}\gamma(g(x),t)\).

On the other hand, the linear operator \(\mathrm{d}g_x\) can be extended to the completion \(\widehat A\) of the graded algebra \(A\) by the same formula \(\widehat{\mathrm{d}g_x}: z\mapsto z - d \mu(x, \dots, x,z)\). It admits an inverse \(\widehat{\mathrm{d}g_x}^{-1}: t \mapsto t+ \sum_{i\ge 1} d^i \mathop{\mathrm{Ad}}_x^i(t)\). Since the restriction of the operator \(\widehat{\mathrm{d}g_x}\) to the subset \(A\subset \widehat A\) is a bijection \(\mathrm{d}g_x:A\to A\), we have \(\widehat{\mathrm{d}g_x}^{-1}(A)\subset A\). So, the higher homogeneous components \(d^i \mathop{\mathrm{Ad}}_x^i(t)\) with \(i>>0\) of the element \(\widehat{\mathrm{d}g_x}^{-1}(t) \in \widehat A\) vanish for \(x, t\in A\). It follows that for each pair of generators \(x\) and \(t\) of \(A\) we have \(\mathop{\mathrm{Ad}}_x^i(t) =0\) for some \(i\). Since \(A\) is a free algebra of some variety, this equality is an identity in \(A\).

Let \(n\) be the smallest number such that the Engel identity \(\mathop{\mathrm{Ad}}_x^n(t) =0\) holds. We have an equality \[\widehat{\mathrm{d}g_x}^{-1}(t) = {\mathrm{d}g_x}^{-1}(t)\] in \(A\) for the generators \(x,t\) of \(A\). Since the algebra is graded, this equality implies the equality of corresponding homogeneous components. On the left-hand side \(\widehat{\mathrm{d}g_x}^{-1}(t) = t+ \sum_{i\ge 1} d^i \mathop{\mathrm{Ad}}_x^i(t)\), the highest nonzero component is the one with \(i=n-1\); its degree is \((d-1)(n-1)+1\). On the right-hand side \[{\mathrm{d}g_x}^{-1}(t) = \mathrm{d}\gamma(g(x),t) = \sum_{j= 1}^{q_0-1} \frac{1}{(j-1)!}T_j^{mult}(g(x),\dots, g(x),t),\] the degrees of the nonzero homogeneous components do not exceed the number \(d(q_0-2) +1 = d(d-1)N+1\). Therefore, we obtain the inequality \[(d-1)(n-1)+1 \le d(d-1)N+1.\] Thus, \(n\le dN +1 \le d \left[ \frac{p-2}{d-1}\right]+1\). to 7ptto 7ptheight 7pt width 7pt
The converse of Theorem 5 (whether an arbitrary Engel algebra is Yagzhev nil?) is a challenging problem which generalizes the Jacobian conjecture to the case of infinite number of variables. For binary algebras, it is stated in [3] as the Generalized Jacobian conjecture for quadratic mappings. This last conjecture holds for power-associative algebras and for those satisfying the identity \((x^2)^2\equiv 0\) [11] as well as for 3-Engel algebras [12]. Note that in all these cases the algebras turn out to be not only Yagzhev nil but also Gerstenhaber nil.

Acknowledgments

I am grateful to Vladimir Dotsenko and Fouad Zitan for stimulating discussions. This work was supported by the Russian Science Foundation, grant RSF 24-21-00341.

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