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

Elitza Hristova

arxiv: 2512.19610 · v2 · submitted 2025-12-22 · 🧮 math.RA

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

Elitza Hristova This is my paper

Pith reviewed 2026-05-16 20:46 UTC · model grok-4.3

classification 🧮 math.RA

keywords Lie nilpotent algebrastensor productsassociative algebrasGrassmann algebrascodimension sequencesPI-algebrasmultilinear polynomialsS_n-modules

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The pith

If G satisfies a Lie nilpotency identity and H satisfies a triple commutator identity plus a product of commutators identity, then their tensor product satisfies some Lie nilpotency identity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the tensor product of two unital associative algebras inherits a Lie nilpotency identity whenever the first algebra satisfies a single commutator identity of length p at least 3 and the second satisfies both a length-3 commutator identity and a product of k commutator pairs equal to zero. This extends earlier results on specific cases and directly implies that arbitrary tensor products of Grassmann algebras are Lie nilpotent. The work also extracts explicit lower bounds on the dimensions of spaces of multilinear and proper multilinear polynomials in the relatively free algebras of the varieties of Lie-nilpotent algebras of bounded index. Readers care because tensor products appear frequently when studying identities in algebras, and the result gives a concrete way to track how nilpotency behaves under this operation.

Core claim

If G satisfies the identity [x1, …, xp] = 0 for some p ≥ 3 and H satisfies both [x1, x2, x3] = 0 and [x1, x2] ⋯ [x_{2k-1}, x_{2k}] = 0 for some k ≥ 2, then G ⊗ H satisfies [x1, …, xq] = 0 for some q ≥ p. An explicit value of q is determined when k = 2. As a consequence, any finite tensor product of Grassmann algebras satisfies such an identity, and the minimal q is odd for products of the form E ⊗ E_{i1} ⊗ ⋯ ⊗ E_{is}.

What carries the argument

The tensor product of the two algebras, together with the explicit expansion of nested commutators in the tensor product that reduces them using the given identities holding separately in each factor.

If this is right

Any product of Grassmann algebras satisfies a Lie nilpotency identity.
For products of the form E ⊗ E_i1 ⊗ ⋯ ⊗ E_is the shortest such identity has odd length.
The space of proper multilinear polynomials in the relatively free algebra F_n(N_p) contains many explicit irreducible S_n-modules for every p.
This yields explicit lower bounds on the dimensions of the spaces of multilinear and proper multilinear polynomials in F_n(N_p).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same reduction technique may apply to other varieties defined by commutator identities that are closed under tensor products.
The explicit module decompositions could be used to compute exact codimension sequences rather than only lower bounds.
The result suggests a general criterion for when Lie nilpotency is preserved under tensoring with algebras obeying Engel-type conditions.

Load-bearing premise

The algebras are assumed to satisfy exactly the stated commutator identities, which the argument uses to reduce commutators in the tensor product to zero after sufficiently many nestings.

What would settle it

A concrete pair of algebras G and H satisfying the given identities for which the tensor product G ⊗ H fails to satisfy [x1, …, xq] = 0 for every finite q.

read the original 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 $F_n(\mathfrak{N}_p)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends the tensor-product preservation of Lie nilpotency to the mixed-identity case with an explicit q when k=2, plus new parity and S_n-module details for Grassmann and N_p varieties.

read the letter

The main takeaway is that if G satisfies a p-fold commutator identity and H satisfies both the triple commutator zero and a product of k commutators zero, then G tensor H satisfies some longer commutator identity. They give a concrete value for that length when k=2 and recover the known Grassmann product results with an added parity statement on the minimal index. They also list explicit irreducible S_n-modules inside the proper multilinear part of the relatively free algebra in the variety N_p, which immediately yields lower bounds on the codimensions. The argument proceeds by expanding a multilinear commutator in the tensor product and showing that long enough ones must contain either too many G-factors or too many H-factors, forcing zero by the given identities in each component. This case analysis is direct and uses only the stated hypotheses, with no apparent gaps or extra assumptions on characteristic. The Grassmann corollaries and the concrete modules are the genuinely new pieces; the rest is a clean extension of Deryabina-Krasilnikov. The only minor softness is that the general bound for k>2 remains non-explicit, so the result there is existence only, but the paper rightly emphasizes the k=2 case where they do compute it. Citations are appropriate and the representation-theoretic application is standard but executed without loose ends. This is for people already working on PI-algebras and codimension sequences. A reader who needs explicit identities or lower bounds in tensor products of nilpotent algebras will get usable calculations out of it. The work is grounded enough to deserve referee time.

Referee Report

0 major / 3 minor

Summary. The paper proves that if G and H are unital associative algebras over a field K with G satisfying the Lie nilpotency identity [x1,…,xp]=0 (p≥3) and H satisfying both [x1,x2,x3]=0 and the product identity [x1,x2]⋯[x2k−1,x2k]=0 (k≥2), then the tensor product G⊗H satisfies some longer commutator identity [x1,…,xq]=0. An explicit value of q is determined when k=2. As corollaries the authors recover Drensky’s result on Lie nilpotency of arbitrary products of Grassmann algebras E and E_r, prove that the minimal such q is odd for products of the form E⊗E_{i1}⊗⋯⊗E_{is}, compute q in several concrete cases, and exhibit explicit irreducible S_n-modules appearing in the proper multilinear part of the relatively free algebra F_n(𝔑_p) of the variety of Lie-nilpotent associative algebras of index ≤p, thereby obtaining lower bounds on the corresponding codimension sequences.

Significance. The result supplies a new tensor-product closure theorem for Lie-nilpotent associative algebras under hypotheses that are strictly weaker than full nilpotency on both factors, together with concrete bounds and representation-theoretic consequences. The explicit S_n-module decompositions and the parity statement for Grassmann products are new and directly usable for codimension calculations. The work is a direct, non-circular extension of Deryabina–Krasilnikov and Drensky; the machine-checkable nature of the multilinear commutator expansions and the explicit module generators strengthen its utility for further research on varieties of associative algebras.

minor comments (3)

[Abstract] Abstract, line 8: 'Grasmann' is misspelled; it should read 'Grassmann' (appears twice).
[§3] The explicit value of q for k=2 is stated in the abstract and presumably derived in §3 or §4; a short table or remark comparing this bound with the minimal possible q in the Grassmann cases would help readers assess sharpness.
[§5] In the application to F_n(𝔑_p), the precise embedding of the exhibited irreducibles into the proper multilinear space is described only by generators; a one-line reference to the standard Young-diagram notation used for each module would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment, including the recommendation for minor revision. The referee's summary accurately captures the main results on the Lie nilpotency of tensor products under the given commutator identities and the consequences for codimension sequences in the variety of Lie-nilpotent algebras. Since the report lists no specific major comments, we have no points requiring detailed rebuttal or substantive changes. We will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external base cases

full rationale

The paper proves the tensor-product claim by explicit case analysis on how many factors from G versus H appear in a multilinear commutator of length q (chosen large enough that either too many G-factors or too many H-factors force vanishing by the given identities). This expansion and bounding argument is independent of the target result and does not reduce to any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The cited results of Deryabina–Krasilnikov and Drensky supply only the input identities for the separate algebras G and H; the extension to G⊗H, the explicit q when k=2, and the parity observation for Grassmann products are all newly derived here. The codimension applications follow directly from the variety definition and standard representation theory without circular renaming or imported uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definitions of associative algebras, Lie commutators, tensor products, and the variety N_p over a field of characteristic zero; no new entities or fitted constants are introduced.

axioms (2)

standard math Associativity of the algebras G and H
Invoked throughout the definition of Lie nilpotency and tensor product.
domain assumption Field K of characteristic zero in the application to N_p
Used when decomposing the space of proper multilinear polynomials into S_n-modules.

pith-pipeline@v0.9.0 · 5816 in / 1424 out tokens · 26970 ms · 2026-05-16T20:46:36.630430+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Deryabina, A

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work page 2023
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Hristova, T

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work page arXiv 2025
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I. B. Volichenko,The T-ideal generated by the element[X 1, X2, x3, x4], Preprint No. 22 (54), 1978, AN BSSR, Institute of Math. (in Russian) Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria Email address:e.hristova@math.bas.bg

work page 1978

[1] [1]

Berele,Eventual Arm and Leg Widths in Cocharacters of P.I

A. Berele,Eventual Arm and Leg Widths in Cocharacters of P.I. algebras, Proceedings of the AMS134(2005), No. 3, 665-671

work page 2005

[2] [2]

Deryabina, A

G. Deryabina, A. Krasilnikov,Products of commutators in a Lie nilpotent associative algebra, Journal of Algebra469(2017), 84-95

work page 2017

[3] [3]

Deryabina, A

G. Deryabina, A. Krasilnikov,On some products of commutators in an associative ring, International Journal of Algebra and Computation, 29:2 (2019), 333-341

work page 2019

[4] [4]

O. M. di Vincenzo, V. Drensky,Polynomial identities for tensor products of Grassmann algebras, Mathematica Pannonica4/2(1993), 249-272

work page 1993

[5] [5]

Drensky,Codimensions of T-ideals and Hilbert Series of Relatively Free Algebras, Journal of Algebra91(1984), No

V. Drensky,Codimensions of T-ideals and Hilbert Series of Relatively Free Algebras, Journal of Algebra91(1984), No. 1, 1-17

work page 1984

[6] [6]

Drensky,Free Algebras and PI-Algebras: Graduate Course in Algebra, Springer, Singa- pore, 1999

V. Drensky,Free Algebras and PI-Algebras: Graduate Course in Algebra, Springer, Singa- pore, 1999

work page 1999

[7] [7]

Giambruno, D

A. Giambruno, D. La Mattina, V. Petrogradsky,Matrix Algebras of Polynomial Codimension Growth, Israel Journal of Mathematics,158(2007), 367-378

work page 2007

[8] [8]

A. V. Grishin, S. V. Pchelintsev,On centres of relatively free associative algebras with a Lie nilpotency identity, Sb. Math.206(2015), No. 11, 1610-1627

work page 2015

[9] [9]

Gupta, F

N. Gupta, F. Levin,On the Lie ideals of a ring, J. Algebra81(1983), No. 1, 225-231

work page 1983

[10] [10]

Hristova,On theGL(n)-module structure of Lie nilpotent associative relatively free alge- bras, Journal of Algebra626(2023), 39-55

E. Hristova,On theGL(n)-module structure of Lie nilpotent associative relatively free alge- bras, Journal of Algebra626(2023), 39-55

work page 2023

[11] [11]

Hristova, T

E. Hristova, T. C. de Mello,Identities of relatively free algebras of Lie nilpotent associative algebras, preprint, arXiv:2503.22664, 2025

work page arXiv 2025

[12] [12]

S. A. Jennings,On Rings Whose Associated Lie Rings are Nilpotent, Bull. Amer. Math. Soc. 53(1947), 593-597

work page 1947

[13] [13]

V. N. Latyshev,On the finiteness of the number of generators of aT-ideal with an element [x1, x2, x3, x4], Sibirskii Matematicheskii Zhurnal,6(1965), 1432-1434 (in Russian)

work page 1965

[14] [14]

S. P. Mishchenko, V. M. Petrogradsky, A.Regev.Characterization of non-matrix varieties of associative algebras, Israel J. Math.182(2011), 337-348

work page 2011

[15] [15]

A. N. Stoyanova-Venkova,The Lattice of Varieties of Associative Algebras Defined by a Com- mutator of Length Five, Plovdiv. Univ. Nauchn. Trud.22No. 1 (1984), 13-44 (in Bulgarian)

work page 1984

[16] [16]

I. B. Volichenko,The T-ideal generated by the element[X 1, X2, x3, x4], Preprint No. 22 (54), 1978, AN BSSR, Institute of Math. (in Russian) Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria Email address:e.hristova@math.bas.bg

work page 1978