In this article we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If G is an affine supergroup and H is its normal supersubgroup, then we prove that a dur K-sheafG/H is again affine supergroup. Additionally, if G is algebraic, then a K-sheafG /H is also algebraic supergroup and it coincides withG/H. In particular, any normal supersubgroup of an affine supergroup is faithfully exact.

It is shown that the category of rational supermodules over a general linear supergroup is a highest weight category. More exactly, we construct superanalogs of the theory of modules with good filtration and of the dual theory of modules with Weyl's. Using these, we show that indecomposable injective supermodules have good filtration of a certain kind.

To generalize some fundamental results on group schemes to the super context, we study the quotient sheaf G/H of an algebraic supergroup G by its closed supersubgroup H, in arbitrary characteristic = 2. Our main theorem states that G/H is a Noetherian superscheme. This together with derived results give positive answers to interesting questions posed by J. Brundan.We have the braided tensor category J J YD of Yetter-Drinfeld modules with left J-action and left J-coaction. To be more precise, an object, say V , in J J YD is a left J-module and left J-comodule, whose structures we denote bythese are required to satisfyThe category J J YD has the same tensor product just as the category of left J-(co)modules, and has the braiding given bywhere V, W ∈ J J YD; see [15], Section 10.6. Suppose that J is the group algebra KZ 2 of the group Z 2 . Regard each object V ∈ SMod K as a left J-module by letting the generator of Z 2 act on homogeneous elements v ∈ V by (6.2). Then, V turns into an object in J J YD. We can thus embed SMod K into J J YD as a braided tensor full subcategory. Therefore, results on J J YD can apply to SMod K .

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We define new concept of mixed representations of quivers which naturally arose from the adjoint action of orthogonal or symplectic groups on several matrices. We find generating invariants of mixed representations of quivers and prepare necessary facts to describe defining relations between them.