Maschke functors, semisimple . . . Applications

Maschke functors, semisimple functors and separable functors of the second kind. Abstract We introduce separable functors of the second kind (or H-separable functors) and H-Maschke functors. H-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be H-separable or H-Maschke, in terms of generalized (co)Casimir elements (integrals, in the case of Hopf algebras), are given. An H-separable functor is always H-Maschke, but the converse holds in particular situations. A special role will be played by Frobenius functors and their relations to H-separability. One of the fundamental results in classical representation theory is Maschke's Theorem, stating that a finite group algebra kG over a field k is semisimple if and only if the characteristic of k does not divide the order of G. Several generalizations of this result have appeared in the literature. To illustrate that there is a subtle difference, let us look more carefully at one of the earliest generalization, where the ground field k is replaced by a commutative ring k. Algebras over a commutative ring are rarely semisimple, and one arrives at the following result: a finite group algebra kG over a field k is separable if and only if the characteristic of k does not divide the order of G. The interesting thing is that, over a field k, a separable finite dimensional algebra is semisimple, but not conversely: it suffices to look at a purely inseparable field extension. A consequence of the two versions of Maschke's Theorem is then that, for a finite group algebra (and, more generally, for a finite dimensional Hopf algebra) over a field, separability and semisimplicity are equivalent. An elegant categorical definition of separability has been proposed by Nˇastˇasescu et al. in [19]. A functor F is called separable if and only if the natural transformation F induced by F is split by a natural transformation P. It is a proper generalization of the notion of separable algebra, in the sense that a k-algebra A is separable if and only if the restriction of scalars functor F : M A → M k is separable [19, Prop. 1.3]. Moreover, a separable functor F between two abelian categories satisfies the following version of Maschke's Theorem: an exact sequence, that becomes split after applying F is itself split. If we apply this property to the restriction of scalars functor in the case of an algebra A over …