Free Algebras in Mal'cev-Type Subvarieties of Associative Algebras

TL;DR

The paper addresses free objects in two Mal'cev-type subvarieties of the associative algebra variety, focusing on the second-type $ $mathcal{A}s_2$ and the third-type $ $mathcal{A}s_3$. It constructs explicit bases for the relatively free algebras $ $mathcal{A}s_2 langle X angle$ and $ $mathcal{A}s_3 langle X angle$ using rewriting systems, and analyzes dimensions $ $dim $ $mathcal{A}s_2(n)$ and $ $dim $ $mathcal{A}s_3(n)$. It further examines symmetric polynomials in $ $mathcal{A}s_2$ and proves they are generated by a specific $S_n$-invariant family $p_n$, linking operad theory to classical algebra through Koszul duality. Collectively, the work provides explicit bases, combinatorial interpretations, and a structural bridge between Mal'cev-type subvarieties, operads, and classical algebraic structures such as Lie and Jordan algebras.

Abstract

In this paper, we study free algebras in subvarieties of the variety of associative algebras singled out by Mal'cev's classification. For each subvariety, we construct the bases for the corresponding free algebras and describe the space of symmetric polynomials they contain.

Theorems & Definitions (16)

• Lemma 1
• proof
• Theorem 1
• proof
• Example 1
• Theorem 2
• proof
• Definition 1
• Lemma 2
• Lemma 3