Grassmann Real Forms of Osp(2r,s, L C) Lie Supergroups

We introduce the concept of the Grassmann real form of an arbitrary matrix supergroup and of its Lie superalgebra. This new notion includes as special cases the ordinary real forms (considered and classified by Serganova) but it is more general. Indeed, the Grassmann real forms are characterized by certain Lie superal-gebra automorphisms of order four, while the ordinary real forms by those of them who are moreover of the order two. As an example, we consider the classical series OSp(2r, s, l C). We classify all Grassmann real forms on the superalgebra and supergroup level. It turns out, that for every values of parameters r and s there is precisely one compact Grassmann real form whose characteristic automorphism does not have the order two.