Universität Paderborn, FB 17 Mathematik, AG von zur Gathen

Article:

Unique Tensor Factorization of Loop-Resistant Algebras
over a Field of Finite Characteristic

Journal of Algebra 251 (2002) 2, 509-528.

This article transfers the main result of my dissertation to a great extent to finite characteristics. 

Abstract

Tensor product decomposition of algebras is known to be non-unique in many cases. But, as will be shown here, an additively indecomposable, finite-dimensional C-algebra A has an essentially unique tensor factorization

A = A1 tensor ... tensor Ar

into non-trivial, tensor-indecomposable factors Ai. Thus the semiring of isomorphism classes of finite-dimensional C-algebras is a polynomial semiring N[X]. Moreover, the field C of complex numbers can be replaced by an arbitrary (not necessarily algebraically closed) field of characteristic zero if we restrict ourselves to split algebras.
Here, we show that the above result still holds in finite characteristic if we only consider loop-resistant algebras.

Michael Nüsken,