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

This paper establishes that the tensor product of specific Lie nilpotent associative algebras remains Lie nilpotent, provides explicit bounds for the nilpotency index in certain cases, and applies these results to analyze the codimension sequences and $S_n$ -module decompositions of relatively free algebras in the variety of Lie nilpotent associative algebras.

Original authors: Elitza Hristova

Published 2026-04-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Elitza Hristova

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, chaotic library of mathematical objects called Algebras. These aren't books you read; they are structures where you can add and multiply things, but unlike normal numbers, the order in which you multiply matters. If you multiply $A \times B$ , it might not be the same as $B \times A$ . This difference is called a commutator (think of it as the "friction" or "chaos" created when you swap the order).

Some of these algebras are very orderly. If you keep swapping the order of multiplication enough times, the chaos eventually dies out and becomes zero. Mathematicians call these Lie Nilpotent Algebras. The "index" ( $p$ ) is like a "chaos meter": it tells you how many times you need to swap things around before the result becomes zero.

The Main Story: Mixing Two Orderly Systems

The author, Elitza Hristova, is asking a simple but tricky question: What happens if you mix two of these orderly systems together?

Imagine you have two different types of "orderly boxes":

Box G: A box where the chaos dies out after $p$ swaps.
Box H: A box that is very special. It has two rules:
- It becomes zero after just 3 swaps.
- It has a special "double-commutator" rule (a specific pattern of swaps) that also forces things to zero.

Hristova proves that if you take the Tensor Product of these two boxes (which is like gluing them together side-by-side to create a giant, combined box), the new giant box is also orderly! It will eventually become zero after some number of swaps ( $q$ ).

The Analogy:
Think of Box G as a room where people get tired and stop arguing after 5 minutes. Think of Box H as a room where people stop arguing after 3 minutes, but they also have a special "silence rule" that kicks in if they argue in pairs. When you combine these two rooms, the new, bigger room will still eventually go silent. Hristova's job was to figure out exactly how long it takes for the new room to go silent ( $q$ ) based on how long it took the original rooms ( $p$ ).

The "Grassmann" Connection

A big part of the paper deals with Grassmann Algebras. These are mathematical structures built from "anti-commuting" variables (like $x \times y = -y \times x$ ). They are the "gold standard" of orderly algebras in this field.

Hristova shows that if you take a chain of these Grassmann algebras and glue them together (e.g., $E \otimes E_2 \otimes E_3 \dots$ ), the resulting chain is still orderly.

The Surprise: She discovered a pattern in the "chaos meter" ( $q$ ). If you have a chain starting with the infinite Grassmann algebra ( $E$ ), the number of swaps needed to silence the system is always an odd number. It's like a rhythm that only beats on the 1st, 3rd, 5th, etc., beat.

The Second Act: Counting the Patterns

The second half of the paper shifts from "how long until silence?" to "what does the silence look like?"

Mathematicians love to break complex shapes into simple, irreducible building blocks (like LEGO bricks). In this context, the "building blocks" are called $S_n$ -modules (related to how you can rearrange $n$ items).

Hristova looks at the "space of all possible polynomials" in these orderly algebras. She asks: Which specific LEGO bricks (patterns of rearrangement) appear in this space?

The Discovery: She identifies many specific patterns (partitions) that are guaranteed to appear.
The Result: By finding these patterns, she can calculate a lower bound for the size of the space.
- Analogy: Imagine you are trying to estimate the size of a hidden treasure chest. You can't open it, but you know for a fact that inside there are at least 5 gold coins of type A, 3 of type B, and 2 of type C. Even if you don't know the total, you now know the chest is at least that big. Hristova found the "coins" (the specific patterns) and proved the chest is bigger than previously thought.

Why Does This Matter?

Predictability: It tells us that combining these specific types of mathematical structures preserves their "orderly" nature. We don't have to fear that mixing them creates infinite chaos.
Precision: She didn't just say "it's orderly"; she gave exact formulas for how orderly it is (the value of $q$ ).
New Tools: By identifying the specific patterns (the LEGO bricks) inside these algebras, she gives future mathematicians a better map to navigate these complex structures. This helps in understanding the "growth" of these algebras—how big they get as you add more variables.

Summary in One Sentence

Elitza Hristova proved that gluing together specific types of "orderly" mathematical systems creates a new system that is also orderly, calculated exactly how much order remains, and discovered a hidden rhythm (odd numbers) and specific building blocks that define the structure of these combined systems.

1. Problem Statement

The paper addresses two interconnected problems in the theory of associative algebras with polynomial identities (PI-algebras):

Lie Nilpotency of Tensor Products: Determining conditions under which the tensor product $G \otimes H$ of two Lie nilpotent associative algebras remains Lie nilpotent. Specifically, if $G$ satisfies the identity $[x_1, \dots, x_p] = 0$ and $H$ satisfies specific commutator identities, what is the index $q$ such that $G \otimes H$ satisfies $[x_1, \dots, x_q] = 0$ ?
Codimension Sequences: Analyzing the $S_n$ -module structure of the relatively free algebra $F_n(\mathcal{N}_p)$ in the variety $\mathcal{N}_p$ of Lie nilpotent associative algebras of index at most $p$ . The goal is to identify explicit irreducible $S_n$ -modules appearing in the decomposition of the space of proper multilinear polynomials and to derive lower bounds for the codimension sequences $c_n(\mathcal{N}_p)$ and proper codimension sequences $\gamma_n(\mathcal{N}_p)$ .

2. Methodology

The author employs a combination of structural algebraic techniques and representation theory:

Commutator Calculus and Ideal Theory: The paper utilizes the structure of T-ideals generated by commutators ( $I_p$ ). It relies on known results regarding products of commutators (Theorems of Latyshev, Gupta-Levin, and Deryabina-Krasilnikov) to analyze how commutators behave under tensor products.
Inductive Construction of Commutators: In Section 2, the author defines specific types of elements (Type I, II, III) within the tensor product $G \otimes H$ and uses induction to track how long commutators $c_k = [g_1 \otimes h_1, \dots, g_k \otimes h_k]$ evolve. This involves analyzing the interaction between the commutator structure of $G$ and the "product of commutators" structure of $H$ .
Grassmann Algebras as Test Cases: The author extensively uses Grassmann (exterior) algebras ( $E$ and $E_r$ ) as concrete examples. These algebras are known to be Lie nilpotent and satisfy specific identities, serving as building blocks to test the general theorems.
Representation Theory of Symmetric Groups: In Section 3, the paper shifts to characteristic 0. It utilizes the theory of $S_n$ -modules, Young diagrams, and hook formulas. The author constructs specific "proper" polynomials (linear combinations of products of pure commutators) and demonstrates that their linearizations generate specific irreducible $S_n$ -modules $M(\lambda)$ .
Evaluation on Tensor Products: To prove that certain polynomials are not identities (and thus generate non-zero modules), the author constructs explicit evaluations of these polynomials on tensor products of Grassmann algebras (e.g., $E \otimes E_2 \otimes \dots \otimes E_2$ ) and shows the result is non-zero.

3. Key Contributions and Results

A. Tensor Product Theorems (Section 2)

Generalization of Deryabina-Krasilnikov: The paper proves that if $G$ satisfies $[x_1, \dots, x_p] = 0$ and $H$ satisfies $[x_1, x_2, x_3] = 0$ and $[x_1, x_2]\cdots[x_{2k-1}, x_{2k}] = 0$ , then $G \otimes H$ is Lie nilpotent.
Explicit Index Determination: For the specific case where $H$ $H$ satisfies $[x_1, x_2][x_3, x_4] = 0$ $[x_{1}, x_{2}] [x_{3}, x_{4}] = 0$ (i.e., $k=2$ $k = 2$ ) and $\text{char } K \neq 3$ $char K \neq = 3$ :
- If $p$ is even, $G \otimes H$ satisfies $[x_1, \dots, x_{p+1}] = 0$ .
- If $p$ is odd, $G \otimes H$ satisfies $[x_1, \dots, x_{p+2}] = 0$ .
Parity of Minimal Index: A significant result (Proposition 2.15) shows that for algebras of the form $F = E \otimes E_{i_1} \otimes \dots \otimes E_{i_s}$ , the minimal integer $q$ such that $[x_1, \dots, x_q] = 0$ is always an odd integer.
Specific Bounds for Grassmann Products: The paper provides explicit values for $q$ for various tensor products of Grassmann algebras (e.g., $E_{2k} \otimes E_{2k}$ satisfies $[x_1, \dots, x_{2k+2}] = 0$ but not $2k+1$ ).

B. $S_n$ -Module Structure and Codimensions (Section 3)

Identification of Irreducible Modules: The author identifies explicit classes of partitions $\lambda \vdash n$ $λ ⊢ n$ such that the irreducible $S_n$ $S_{n}$ -module $M(\lambda)$ $M (λ)$ appears with non-zero multiplicity in the decomposition of $\Gamma_n(\mathcal{N}_p)$ $Γ_{n} (N_{p})$ (the space of proper multilinear polynomials).
- Case $p=2k$ or $p=2k+1$ : The paper lists partitions including $(p-1, 1)$ , $(p-1, p-1)$ , and families of the form $(a+2, 2b, 1^c)$ and $(l+3, l+1, 1^{n-2l-4})$ .
- Construction: These modules are generated by linearizations of specific polynomials constructed from standard polynomials and commutators (denoted $g_i^{(j)}$ ).
Lower Bounds for Codimensions:
- Using the identified modules, the author derives lower bounds for the proper codimension sequence $\gamma_n(\mathcal{N}_p)$ .
- For $p=2k$ or $p=2k+1$ with $k \ge 3$ , $\gamma_n(\mathcal{N}_p)$ is bounded below by a polynomial $p_k(n)$ of degree $2k-2$ .
- The leading coefficient of this polynomial involves the Catalan number $C_k$ . Specifically, the leading coefficient is $\frac{2^{2k-3} + C_k}{(2k-2)!}$ (for $k \ge 4$ ).
- Using the relation between codimensions and proper codimensions (Proposition 3.1), lower bounds for the full codimension sequence $c_n(\mathcal{N}_p)$ are also established. These bounds grow as $2^n \cdot n^{2k-2}$ .

4. Significance

Extension of Known Results: The paper extends previous work by Drensky, Deryabina, and Krasilnikov, providing a unified framework for understanding the Lie nilpotency of tensor products of Grassmann algebras and other PI-algebras.
Resolution of Parity Question: The proof that the minimal Lie nilpotency index for products involving the infinite Grassmann algebra $E$ must be odd is a non-trivial structural insight.
Advancement in Codimension Theory: By explicitly identifying the $S_n$ -modules involved, the paper moves beyond asymptotic estimates to provide concrete lower bounds with precise leading coefficients. This refines the understanding of the growth rate of the codimension sequences for Lie nilpotent varieties.
Connection to Conjectures: The results support (and provide evidence for) Drensky's conjecture that the variety $\mathcal{N}_p$ is generated by finite sets of Grassmann algebra products, and the author's own conjecture regarding the asymptotic equivalence of $\mathcal{N}_{2k}$ and $\mathcal{N}_{2k+1}$ .

In summary, Hristova's work bridges the gap between the structural properties of tensor products of Lie nilpotent algebras and the quantitative representation-theoretic analysis of their codimension sequences, offering both explicit identities and refined asymptotic bounds.