Embeddings of fields into simple algebras: generalizations and applications

TL;DR

This paper provides a numerical criterion for the existence of algebra homomorphisms between semi-simple algebras over any field, characterizes when the set of such homomorphisms modulo conjugation is finite, and offers explicit formulas for counting these orbits.

Contribution

It generalizes the understanding of homomorphisms between semi-simple algebras and provides explicit criteria and formulas, extending previous results to arbitrary ground fields.

Findings

Numerical criterion for non-empty Hom set

Conditions for finiteness of orbit set

Explicit formula for orbit set cardinality

Abstract

For two semi-simple algebras $A$ and $B$ over an arbitrary ground field $F$, we give a numerical criterion when $\Hom_F(A,B)$, the set of $F$-algebra homomorphisms between them, is non-empty. We also determine when the orbit set $B^\times \backslash \Hom_\falg(A,B)$ is finite and give an explicit formula for its cardinality. A few applications of main results are given.