Tensor products of Lie nilpotent associative algebras and applications to codimension sequences

TL;DR

This paper investigates the Lie nilpotency of tensor products of associative algebras satisfying specific identities, providing explicit bounds, reproofs of known results, and applications to Grassmann algebras and polynomial identities.

Contribution

It extends previous results on Lie nilpotency of tensor products, explicitly determines bounds for certain identities, and applies these findings to Grassmann algebras and polynomial identities.

Findings

Tensor product of certain Lie nilpotent algebras is again Lie nilpotent.

Explicit bounds for the nilpotency index in specific cases.

Reproof of a result on identities satisfied by products of Grassmann algebras.

Abstract

Let $G$ and $H$ be unital associative algebras over a field $K$, such that $G$ satisfies the identity $[x_1, \dots, x_p] = 0$ for some integer $p \geq 3$ and $H$ satisfies the identities $[x_1, x_2, x_3] = 0$ and $[x_1, x_2] \cdots [x_{2k-1}, x_{2k}]=0$ for some $k \geq 2$. In this paper, extending results of Deryabina and Krasilnikov, we show that the tensor product $G \otimes H$ is again a Lie nilpotent associative algebra, i.e., it satisfies $[x_1, \dots, x_{q}] = 0$ for some $q \geq p$. We also determine an explicit value of $q$ in the case $k = 2$, i.e., when $H$ satisfies the identity $[x_1, x_2][x_3, x_4] = 0$. As a corollary, we reprove a result of Drensky saying that any product of Grassmann algebras of the form $E\otimes E_{i_1}\otimes \cdots \otimes E_{i_s}$ or $E_{j_1} \otimes E_{j_2} \otimes \cdots \otimes E_{j_t}$, where $E$ denotes the Grassmann algebra over a countable dimensional vector space and $E_r$ denotes the Grasmann algebra over an $r$-dimensional vector space, satisfies an identity of the form $[x_1, \dots, x_q] = 0$ for some integer $q \geq 3$. In addition, we show that for products of the form $E\otimes E_{i_1}\otimes \cdots \otimes E_{i_s}$ the minimal value of $q$ is always and odd integer. We also provide several particular cases in which a value of $q$ can be explicitly computed. As an application, we consider a field of characteristic zero, the variety $\mathfrak{N}_p$ of Lie nilpotent associative algebras of index at most $p$ and the corresponding relatively free algebras of finite rank, $F_n(\mathfrak{N}_p)$. We exhibit many explicit irreducible $S_n$-modules in the $S_n$-module decomposition of the space of proper multilinear polynomials in $F_n(\mathfrak{N}_p)$ for any $p$. This gives a lower bound for the dimensions of the spaces of multilinear and proper multilinear polynomials in $F_n(\mathfrak{N}_p)$.