Gr\"obner-Shirshov bases and embeddings of algebras

TL;DR

This paper uses Gr"obner-Shirshov bases to demonstrate embedding properties of various algebraic structures, including associative and Lie algebras, into simpler or two-generated algebras, extending classical embedding results.

Contribution

It introduces new embedding theorems for classes of algebras using Gr"obner-Shirshov bases, including embeddings into simple or two-generated algebras, and provides alternative proofs for known theorems.

Findings

Every countably generated algebra in specified classes can be embedded into a simple or two-generated algebra.

Countably generated modules over free associative algebras can be embedded into cyclic modules.

Classical embedding theorems for groups, algebras, semigroups, and Lie algebras are reproved with new methods.

Abstract

In this paper, by using Gr\"obner-Shirshov bases, we show that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associative $\Omega$-algebras, associative $\lambda$-differential algebras. We show that in the following classes, each countably generated algebra over a countable field $k$ can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative $\Omega$-algebras, associative $\lambda$-differential algebras. Also we prove that any countably generated module over a free associative algebra $k< X>$ can be embedded into a cyclic $k< X>$-module, where $|X|>1$. We give another proofs of the well known theorems: each countably generated group (resp. associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (resp. associative algebra, semigroup, Lie algebra).