Matrix wreath products of algebras and embedding theorems

TL;DR

This paper introduces a new matrix wreath product construction for algebras and uses it to prove embedding theorems, including constructing finitely generated nil algebras with arbitrary Gelfand-Kirillov dimension over countable fields.

Contribution

It presents a novel matrix wreath product construction for algebras and applies it to establish embedding theorems and construct nil algebras with specified Gelfand-Kirillov dimensions.

Findings

Established embedding theorems for Jacobson radical, nil, and primitive algebras.

Constructed finitely generated nil algebras with Gelfand-Kirillov dimension ≥ 8.

Introduced a new algebraic construction similar to wreath products of groups.

Abstract

We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. In \S\ref{Section6}, we construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimension $\geq 8$ over a countable field which answers a question from \cite{8}.