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Varieties of bicommutative algebras with identity of degree three

This paper provides a complete classification of varieties of bicommutative algebras over a field of characteristic zero that satisfy a polynomial identity of degree three and establishes a necessary and sufficient condition for such varieties to possess a distributive lattice of subvarieties.

Original authors: Vesselin Drensky, Bekzat Zhakhayev

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Vesselin Drensky, Bekzat Zhakhayev

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, infinite library of mathematical structures called algebras. Most of these structures follow strict rules, like the familiar rule that $2 \times 3$ is the same as $3 \times 2$ (commutativity). But in this paper, the authors are exploring a very specific, quirky corner of this library: Bicommutative Algebras.

Think of a bicommutative algebra as a game with two special rules:

Right-Commutativity: If you have three players, $A$ , $B$ , and $C$ , the order in which $B$ and $C$ interact after $A$ has acted doesn't matter. It's like saying, "If I hand a gift to Bob, it doesn't matter if Bob gives it to Charlie or Charlie gives it to Bob first; the result is the same."
Left-Commutativity: Similarly, if $B$ and $C$ act before $A$ , their order also doesn't matter.

The authors are asking a very specific question: What happens if we add a third rule to this game? Specifically, what if we force the game to follow a rule involving three variables (a "degree three" identity)?

Here is the breakdown of their findings, translated into everyday concepts:

1. The "Rulebook" and the "Symmetry Group"

In mathematics, every algebra has a "rulebook" (identities) that defines it. The authors use a tool called Representation Theory (which sounds scary but is just a way of counting patterns) to analyze these rulebooks.

Imagine the variables in your algebra as actors on a stage. The "Symmetric Group" is like a director who can swap the actors around in any order. The authors discovered that for bicommutative algebras, the "actors" can be grouped into specific "troupes" (mathematical modules).

The Big Discovery: When they looked at rules involving three variables, they found there are only two main types of "troupes" that can appear.
- Troupe A: A rule that looks like $x(xx)$ or $(xx)x$ (repeating the same variable).
- Troupe B: A rule that looks like $x(xy - yx)$ (involving a difference or "commutator").

2. The Two Main Families

The paper essentially splits all possible bicommutative algebras that follow a 3-variable rule into two main families, based on which "troupe" they belong to.

Family 1: The "Repetition" Family

This family follows rules like "If you do $x$ twice, then do $x$ again, it equals doing it in a different order."

The Twist: Depending on the specific numbers (coefficients) in the rule, the behavior of the algebra changes drastically.
- If the numbers are "generic" (random and non-zero), the algebra becomes very rigid. It stops producing new, complex patterns after degree 3. It's like a machine that runs out of fuel after three steps.
- If the numbers hit specific "sweet spots" (like being zero or equal to each other), the algebra becomes more flexible, allowing for longer chains of operations.
The Graph Metaphor: The authors draw a map (a graph) to show how these rules connect. If you have a rule at step 3, does it force a rule at step 4? Sometimes yes, sometimes no. They mapped out every possible path on this graph for this family.

Family 2: The "Difference" Family

This family follows rules involving the difference between $xy$ and $yx$ (how much the order matters).

The Twist: Similar to the first family, the specific numbers in the rule determine the fate of the algebra.
- In most cases, the algebra is "rich" with patterns.
- In specific cases (like when the numbers cancel each other out), the algebra becomes "poor," meaning it stops generating new patterns very quickly.
The Graph Metaphor: They also mapped this family. They found that some rules act like a dead end (stopping the game), while others act like a bridge to infinite complexity.

3. The "Distributive Lattice" (The Organized Library)

The paper concludes with a question about organization.
Imagine all the possible sub-varieties (sub-groups) of these algebras as books in a library.

A Distributive Lattice is a very special, perfectly organized library where the rules of combining sections are simple and predictable. It's like a family tree where you can clearly see who is related to whom without any confusing loops.
The Condition: The authors proved that a bicommutative algebra has this "perfectly organized" structure if and only if it follows both types of rules (the Repetition rule AND the Difference rule) simultaneously.
The Analogy: If you only follow one type of rule, your library is messy and chaotic. But if you follow both specific types of 3-variable rules, the library snaps into perfect order.

Summary of the "So What?"

The paper doesn't claim this will cure diseases or build better bridges. Instead, it solves a pure math puzzle:

Classification: It completely lists every possible type of bicommutative algebra that follows a 3-variable rule.
Structure: It draws the "family trees" (graphs) showing how these rules relate to one another.
Order: It identifies the exact condition needed to make the collection of these algebras perfectly organized (distributive).

In short, the authors took a chaotic, infinite set of mathematical possibilities, found the two main "flavors" of rules that govern them, and showed exactly how those rules shape the structure of the mathematical world they inhabit.

Technical Summary: Varieties of Bicommutative Algebras with Identity of Degree Three

Problem Statement
The paper addresses the classification of varieties of bicommutative algebras over a field $K$ of characteristic zero that satisfy a polynomial identity of degree three. A bicommutative algebra is a non-associative algebra satisfying the right-commutative identity $(x_1x_2)x_3 = (x_1x_3)x_2$ and the left-commutative identity $x_1(x_2x_3) = x_2(x_1x_3)$ . While the structure of the free bicommutative algebra $F(B)$ and its numerical invariants (Hilbert series, codimension sequences) were previously established by Dzhumadil'daev, Ismailov, and Tulenbaev, a complete description of the subvarieties defined by identities of degree three remained an open problem. Additionally, the paper seeks to establish necessary and sufficient conditions for a variety of bicommutative algebras to possess a distributive lattice of subvarieties.

Methodology
The authors employ the representation theory of the symmetric group $S_n$ and the general linear group $GL_d(K)$ to analyze the multilinear components $P_n(B)$ of the free bicommutative algebra.

Representation Theory Framework: The multilinear component $P_n(B)$ is decomposed into a direct sum of irreducible $S_n$ -modules $M(\lambda)$ , where $\lambda$ is a partition of $n$ . The multiplicity $m(\lambda)$ of these modules determines the cocharacter sequence $\chi_n(B)$ .
Graph Theory Approach: The authors describe the lattice of subvarieties using oriented graphs. Vertices represent irreducible $S_n$ -submodules, and directed edges indicate that identities in one module are consequences of identities in another. Subvarieties correspond to subgraphs of this structure.
Algebraic Isomorphism: The analysis is conducted within a specific algebra $G_d$ , which is isomorphic to the relatively free algebra $F_d(B)$ . $G_d$ is constructed using a polynomial algebra in $2d$ variables with a specific multiplication rule that simplifies the handling of consequences of identities.
Highest Weight Vectors: The authors utilize highest weight vectors to generate irreducible submodules. They explicitly compute the consequences of degree 3 identities (specifically linearizations of degree 3 polynomials) to determine which higher-degree identities are generated, thereby characterizing the cocharacter sequences of the resulting subvarieties.

Key Contributions and Results

Classification of Degree 3 Identities:
The paper identifies that any degree 3 identity in a bicommutative algebra is a linear combination of two fundamental types:
- Type 1 (Right/Left Symmetric): $f^{(3)}(x) = \alpha_1 x(xx) + \alpha_2 (xx)x = 0$ .
- Type 2 (Commutator-based): $f^{(2,1)}(x, y) = \beta_1 x(xy - yx) + \beta_2 (xy - yx)x = 0$ .
The authors provide a complete description of the varieties $U_\alpha$ (defined by Type 1) and $V_\beta$ (defined by Type 2) for all possible coefficients $(\alpha_1, \alpha_2)$ and $(\beta_1, \beta_2)$ .
Cocharacter Sequences and Module Structure:
For each case of the coefficients (e.g., $\alpha_1\alpha_2(\alpha_1+\alpha_2)(\alpha_1-\alpha_2) \neq 0$ , $\alpha_2=0$ , etc.), the authors determine the exact decomposition of the $S_n$ -module $P_n(U_\alpha)$ and $P_n(V_\beta)$ .
- They calculate the multiplicities $m(\lambda)$ for partitions $\lambda$ of $n$ .
- They identify specific cases where the multiplicities exceed 1 (specifically $m(2,1)=2$ or $m(4)=2$ ) and cases where the cocharacter sequence vanishes for $n \geq 4$ or $n \geq 5$ .
- The results are summarized in propositions detailing the generators of the irreducible $GL_d(K)$ -submodules for each specific variety.
Lattice of Subvarieties:
The paper constructs the directed graphs $\Gamma(U_\alpha)$ and $\Gamma(V_\beta)$ for the various cases. These graphs explicitly map the inclusion relationships between subvarieties. The authors demonstrate how the structure of these graphs changes based on the parameters of the defining identities, providing a visual and algebraic representation of the subvariety lattice.
Distributive Lattice Condition:
The paper establishes a necessary and sufficient condition for the lattice of subvarieties of a bicommutative algebra variety $W$ to be distributive.
- Theorem: The lattice $L(W)$ is distributive if and only if all multiplicities $m(\lambda)$ in the cocharacter sequence $\chi_n(W)$ satisfy $m(\lambda) \leq 1$ .
- Application: This condition is equivalent to $W$ satisfying both a non-trivial identity of Type 1 and a non-trivial identity of Type 2. If a variety satisfies only one type of identity, it may contain multiplicities greater than 1 (specifically $m(2,1)=2$ ), resulting in a non-distributive lattice.

Significance
The paper provides a complete structural description of the subvarieties of bicommutative algebras defined by identities of degree three. By translating the problem into the language of graph theory and representation theory, the authors offer a systematic method for analyzing the lattice of subvarieties. The primary theoretical contribution is the characterization of the distributive property of the subvariety lattice, linking it directly to the multiplicities in the cocharacter sequence and the simultaneous satisfaction of specific polynomial identities. This work extends previous studies on the free bicommutative algebra and the subvarieties generated by low-dimensional algebras, filling a gap in the classification of varieties satisfying low-degree identities.