2025/12/22 by Elitza Hristova, Hristova, Elitza
Computer Science · Mathematics · #Advanced Topics in Algebra #Associative algebra #Associative property #Codimension #Field (mathematics) #Finite Group Theory Research #Integer (computer science) #Lie algebra #Matrix Theory and Algorithms #Nilpotent #Tensor product #Vector space #math.RA
paper · pdf · doi:10.48550/arxiv.2512.19610
openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28
Let G and H be Lie nilpotent associative algebras over a field K such that in addition H satisfies the identity [x1, x2] ⋯ [x2k-1, x2k]=0 for some k ≥ 2. In this paper, extending results of Deryabina and Krasilnikov, we show that the tensor product G ⊗ H is again a Lie nilpotent associative algebra. Moreover, we give a lower and an upper bound on the minimal value of q for which [x1, …, xq+1] = 0 is an identity for G⊗ H. In the case when H satisfies the identities [x1, x2, x3] = 0 and [x1, x2][x3, x4] = 0 and charK ≠ 3, we determine a better upper bound for q, which in many cases is equal to the minimal index of Lie nilpotency for G⊗ H. As a corollary, we reprove a result of Drensky saying that any product of Grassmann algebras of the form E⊗ Ei1⊗ ⋯ ⊗ Eis or Ej1 ⊗ Ej2 ⊗ ⋯ ⊗ Ejt, where E denotes the Grassmann algebra over a countable dimensional vector space and Er denotes the Grassmann algebra over an r-dimensional vector space, satisfies an identity of the form [x1, …, xq+1] = 0. We also provide several particular cases in which the minimal value of q can be explicitly computed. As an application, we consider a field of characteristic zero, the variety \mathfrakNp of Lie nilpotent associative algebras of index at most p and the corresponding relatively free algebras of finite rank, Fn(\mathfrakNp). We exhibit many explicit irreducible Sn-modules in the Sn-module decomposition of the space of proper multilinear polynomials of degree n in Fn(\mathfrakNp) for any p. This gives a lower bound for the dimensions of the spaces of multilinear and proper multilinear polynomials of degree n in Fn(\mathfrakNp).