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Some basic results on super affine groups and super formal groups in arbitrary characteristic ( = 2) are proved in terms of super Hopf algebras, including the fundamental correspondence between subgroups and quotients.

Over an arbitrary field of characteristic = 2, we define the notion of Harish-Chandra pairs, and prove that the category of those pairs is anti-equivalent to the category of algebraic affine supergroup schemes. The result is applied to characterize some classes of affine supergroup schemes such as those which are (a) simply connected, (b) unipotent or (c) linearly reductive in positive characteristic.

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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