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This paper determines all the possible endomorphism algebras for polarizable  Q -Hodge structures with Hodge numbers ( n ,0,…,0, n ). This generalizes the classification of the possible endomorphism algebras of abelian varieties by Albert and Shimura. As with abelian varieties, the most interesting feature of the classification is that in certain cases, every Hodge structure on which a given algebra acts must have extra endomorphisms.

Hodge Structures of Type ( n , 0, …, 0, n )