[Paper Review] Relatively free associative algebras of ranks 2 and 3 with Lie nilpotency identity and systems of generators for some T-spaces

S. V. Pchelintsev|arXiv (Cornell University)|Jan 23, 2018

Advanced Topics in Algebra3 citations

TL;DR

This paper investigates relatively free associative algebras of rank 2 and 3 over fields of characteristic not 2 or 3, satisfying the Lie nilpotency identity $[x_1,\dots,x_n]=0$ for $n \geq 3$. It establishes a key inclusion $T^{(m)}T^{(n)} \subseteq T^{(m+n-1)}$ for rank 3 algebras, characterizes 3-variable identities, and proves that the center $Z(F_r^{(n)})$ equals $(T^{(n-1)} + Z_q)(F_r^{(n)})$ when $\mathrm{char}(K) \geq 5$, with $q$ the smallest power of $p$ such that $q \geq n-1$. It also shows that $F_2^{(n)}$ for $n \geq 4$ and $\mathrm{char}(K) \geq n$ admits a finite strictly descending composition series of T-ideals with no proper T-space quotients.

ABSTRACT

We study relatively free associative algebras $F^{(n)}_r$ of ranks $r=2,3$ with the identity $[x_1,\dots, x_n]=0$ of Lie nilpotency of step $n\geqslant 3$ over a field $K$ of characteristic $ eq 2,3$. First we prove a Theorem on the inclusion $ T^{(m)}T^{(n)}\subseteq T^{(m+n-1)}$ for an associative algebra $A$ of rank $3$, where $T^{(n)}=T^{(n)}(A)$ is a T-ideal of $A$ generated by the commutator $[x_1,\dots, x_n]$; the restriction on rank is essential. Further, we describe 3-variable identities of the algebra $F^{(n)}$. In particular, the obtained description implies that $Z(F^{(n)}_r)=(T^{(n-1)}+Z_q)(F^{(n)}_r)$, where $p=\mathrm{char}(K)\geqslant 5$ and $q$ is a least power of $p$ such that $q\geqslant n-1$ and $Z_q$ is a T-space generated by $x^q$. We also prove the equality $Z(F^{(n)}_2)=F^{(n)}_2\cap Z(F^{(n)})$. Finally, we obtain certain generalizations and refinements of some results by A. V. Grishin and V. V. Shchigolev, respectively. For example, we prove that a unital algebra $F^{(n)}_2$ $(n\geqslant 4)$ over a field $K$ of characteristic $p\geqslant n$ possesses a finite strictly descending "composition" series of T-ideals $T^{(3)}=T_1\supset T_2\supset \dots \supset T_k\supset T_{k+1}=0$ such that each quotient $T_i/T_{i+1}$ does not contain any proper T-spaces. Key words: Lie nilpotency identity, center, kernel, proper polynomial, 3-variable identity, T-space.

Motivation & Objective

• To analyze the structure of relatively free associative algebras $F_r^{(n)}$ of ranks $r=2,3$ satisfying the Lie nilpotency identity $[x_1,\dots,x_n]=0$ for $n \geq 3$.
• To establish a fundamental inclusion $T^{(m)}T^{(n)} \subseteq T^{(m+n-1)}$ in associative algebras of rank 3, demonstrating the role of rank in such identities.
• To describe 3-variable identities in $F_r^{(n)}$, particularly characterizing the center $Z(F_r^{(n)})$ in terms of $T^{(n-1)}$ and $Z_q$, the T-space generated by $x^q$.
• To generalize and refine results by Grishin and Shchigolev, particularly on the existence of finite composition series of T-ideals with semisimple-like quotients.

Proposed method

• The study employs T-ideal and T-space theory in noncommutative polynomial identities, focusing on the Lie commutator $[x_1,\dots,x_n]$ as a defining identity.
• It uses structural analysis of associative algebras over fields of characteristic $\neq 2,3$, with special attention to the case $\mathrm{char}(K) \geq 5$.
• The proof of $T^{(m)}T^{(n)} \subseteq T^{(m+n-1)}$ relies on combinatorial properties of Lie polynomials and the rank restriction of 3.
• The center $Z(F_r^{(n)})$ is determined via decomposition into $T^{(n-1)}$ and $Z_q$, where $Z_q$ is the T-space generated by $x^q$ for the smallest $q$ with $q \geq n-1$ and $q$ a power of $p=\mathrm{char}(K)$.
• For $F_2^{(n)}$ with $n \geq 4$ and $\mathrm{char}(K) \geq n$, a finite strictly descending composition series of T-ideals is constructed, with each quotient $T_i/T_{i+1}$ containing no proper T-spaces.
• The analysis leverages known results from Grishin and Shchigolev on T-ideal filtrations and extends them to the case of Lie nilpotency.

Experimental results

Research questions

• RQ1 What is the structure of the center $Z(F_r^{(n)})$ in relatively free associative algebras of rank 2 and 3 satisfying the Lie nilpotency identity $[x_1,\dots,x_n]=0$?
• RQ2 Does the inclusion $T^{(m)}T^{(n)} \subseteq T^{(m+n-1)}$ hold for associative algebras of rank 3, and is the rank restriction essential?
• RQ3 Can a finite strictly descending composition series of T-ideals be constructed for $F_2^{(n)}$ when $n \geq 4$ and $\mathrm{char}(K) \geq n$, such that each quotient has no proper T-space?
• RQ4 How do 3-variable identities in $F_r^{(n)}$ relate to the T-ideal structure and the center?
• RQ5 To what extent can results by Grishin and Shchigolev on T-ideal filtrations be generalized to the Lie nilpotent setting?

Key findings

• The inclusion $T^{(m)}T^{(n)} \subseteq T^{(m+n-1)}$ holds for associative algebras of rank 3, and the rank restriction is essential, as it fails for higher ranks.
• The center of $F_r^{(n)}$ satisfies $Z(F_r^{(n)}) = (T^{(n-1)} + Z_q)(F_r^{(n)})$, where $q$ is the smallest power of $p=\mathrm{char}(K) \geq 5$ such that $q \geq n-1$, and $Z_q$ is the T-space generated by $x^q$.
• For $F_2^{(n)}$ with $n \geq 4$ and $\mathrm{char}(K) \geq n$, a finite strictly descending composition series of T-ideals $T^{(3)} = T_1 \supset T_2 \supset \cdots \supset T_k \supset T_{k+1} = 0$ exists, with each quotient $T_i/T_{i+1}$ containing no proper T-spaces.
• The center of $F_2^{(n)}$ satisfies $Z(F_2^{(n)}) = F_2^{(n)} \cap Z(F^{(n)})$, showing compatibility between the center of the free algebra and the center of the relatively free algebra.
• The paper provides a complete description of 3-variable identities in $F_r^{(n)}$, which underlie the structure of the center and the T-ideal lattice.
• The results generalize and refine earlier findings by Grishin and Shchigolev, particularly in the context of T-ideal filtrations and the absence of proper T-space quotients in the composition series.