Bimodule Structure of Central Simple Algebras

TL;DR

This paper establishes a semiring isomorphism linking bimodules of central simple algebras to bisets of Galois groups, offering a combinatorial perspective on their structure and growth behavior.

Contribution

It introduces a novel semiring isomorphism between bimodules of central simple algebras and Galois bisets, enabling combinatorial analysis of algebraic growth patterns.

Findings

Semiring isomorphism between $K$-$K$-bimodules and Galois bisets

Combinatorial interpretation of $ ext{dim}_K((KaK)^i)$ growth

Application to Kummer sets and algebra structure analysis

Abstract

For a maximal separable subfield $K$ of a central simple algebra $A$, we provide a semiring isomorphism between $K$-$K$-bimodules $A$ and $H$-$H$ bisets of $G = \Gal(L/F)$, where $F = \operatorname{Z}(A)$, $L$ is the Galois closure of $K/F$, and $H = \Gal(L/K)$. This leads to a combinatorial interpretation of the growth of $\dim_K((KaK)^i)$, for fixed $a \in A$, especially in terms of Kummer sets.