Group Actions on Central Simple Algebras

Abstract

Let $G$ be a group, $F$ a field, and $A$ a finite-dimensional central simple algebra over $F$ on which $G$ acts by $F$-algebra automorphisms. We study the ideals and subalgebras of $A$ which are preserved by the group action. Let $V$ be the unique simple module of $A$. We show that $V$ is a projective representation of $G$ and $A\cong\text{End}_D(V)$ makes $V$ into a projective representation. We then prove that there is a natural one-to-one correspondence between $G$-invariant $D$-submodules of $V$ and invariant left (and right) ideals of $A$. Under the assumption that $V$ is irreducible, we show that an invariant (unital) subalgebra must be a simply embedded semisimple subalgebra. We introduce induction of $G$-algebras. We show that each invariant subalgebras is induced from a simple $H$-algebra for some subgroup $H$ of finite index and obtain a parametrization of the set of invariant subalgebras in terms of induction data. We then describe invariant central simple subalgebras. For $F$ algebraically closed, we obtain an entirely explicit classification of the invariant subalgebras. Furthermore, we show that the set of invariant subalgebras is finite if $G$ is a finite group. Finally, we consider invariant subalgebras when $V$ is a continuous projective representation of a topological group $G$. We show that if the connected component of the identity acts irreducibly on $V$, then all invariant subalgebras are simple. We then apply our results to obtain a particularly nice solution to the classification problem when $G$ is a compact connected Lie group and $F=\mathbf C$.