Structure of Zariski-closed algebras

TL;DR

This paper investigates the structure of Zariski-closed algebras, extending finite-dimensional algebra theory with new structural results, explicit descriptions, and the construction of generic algebras, especially over finite fields.

Contribution

It provides a version of Wedderburn's theorem for Zariski-closed algebras and introduces explicit descriptions via representations and gluing techniques.

Findings

Established a Wedderburn-type theorem for Zariski-closed algebras

Developed explicit descriptions using representations and gluing

Constructed generic Zariski-closed algebras, including infinite-dimensional cases

Abstract

The objective of this paper is to describe the structure of Zariski closed algebras, which provide a useful generalization to finite dimensional algebras in the study of representable algebras over finite fields. Our results include a version of Wedderburn's principal theorem, as well as a more explicit description using representations, in terms of "gluing" in Wedderburn components. Finally, we construct "generic" Zariski closed algebras, whose description is considerably more complicated than the description of generic algebra of finite dimensional algebras. Special attention is given to infinite dimensional algebras over finite fields.