Hermitian Morita equivalences between maximal orders in central simple algebras

Abstract

Let R be a Dedekind domain with quotient field $K$. That every maximal order in a finite dimensional central simple $K$-algebra $A$, (the algebra of nxn matrices over $D$), where $D$ is separable over $K$, is Morita equivalent to every maximal order in $D$ is a well known linear result. Hahn defined the notion of Hermitian Morita equivalence (HME) for algebras with anti-structure, generalizing previous work by Frohlich and McEvett. The question this paper investigates is the hermitian analogue of the above linear result. Specifically, when are maximal orders with anti-structure in $A$, HME to maximal orders with anti-structure in $D$ in the sense of Hahn? Two sets of necessary and sufficient conditions are obtained with an application which provides the hermitian analogue under some conditions.