Group Actions on Central Simple Algebras
نویسنده
چکیده
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 = EndD(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 = C.
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تاریخ انتشار 2000