G-Frobenius algebras, twisting, and the Drinfeld double

This thesis concerns the relationship between G-Frobenius algebras (G-FAs) and Dω( k[G]), the twisted Drinfeld double of a finite group G. One of the main results of this thesis is that a Frobenius object in the category D(k[G]) - Mod encodes most of the axioms of a G-FA. More precisely, a G-FA is simply a Frobenius object in D(k[G])-Mod which satisfies two additional constraints. In addition to this, we extend the notion of a G-FA to the case of finite groupoids. With regard to the problem of twisting G-FAs, this generalization provides a convenient conceptual framework for twisting certain kinds of G -FAs. Lastly, we show that a certain subalgebra of D ω(k[G]) can be identified with a specific twist of the G-FA arising from the stringy cohomology of the global quotient (I(pt,G), G). We also show that, in a sense, the aforementioned subalgebra is the best approximation of Dω( k[G]) by a G-FA