Structure of Zariski-closed algebras

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.