Centers of centralizers of nilpotent elements in Lie superalgebras 𝔰𝔩(m|n) or 𝔬𝔰𝔭(m|2n)

Let [Formula: see text] be the simple algebraic supergroup [Formula: see text] or [Formula: see text] over [Formula: see text]. Let [Formula: see text] and let [Formula: see text], where [Formula: see text] is considered as a superalgebra concentrated in even degree. Suppose [Formula: see text] is nilpotent. We describe the centralizer [Formula: see text] of [Formula: see text] in [Formula: see text] and its center [Formula: see text]. In particular, we give bases for [Formula: see text], [Formula: see text] and [Formula: see text]. We also determine the labeled Dynkin diagram [Formula: see text] with respect to [Formula: see text] and subsequently describe the relation between [Formula: see text] and [Formula: see text].

On Lie Superalgebras of Algebraic Supergroups

Let G be an algebraic supergroup over a field k, whose pure-even group scheme G ev is a reduced algebraic group scheme, i.e., [Formula: see text] is reduced. In this paper, we prove that Lie (G) can be identified with the Lie superalgebra of admissible left-invariant derivations of k[G].

ON ALGEBRAIC SUPERGROUPS AND QUANTUM DEFORMATIONS

We give the definitions of affine algebraic supervariety and affine algebraic supergroup through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroups GL(m|n) and SL(m|n) and give explicitly the Hopf algebra structure of the algebra representing the functors of points. In the end we also give the quantization of GL(m|n) together with its coaction on suitable quantum spaces according to Manin's philosophy.

SUPERGROUPS, QUANTUM SUPERGROUPS AND THEIR HOMOGENEOUS SPACES

We give a more algebraic definition of algebraic supergroup via its Hopf algebra. The Hopf algebra of the supergroup SL (m|n) is given together with its quantization. The example of supercoadjoint orbits is examined at the end.