Solvable, reductive and quasireductive supergroups

Algebraic supergroups and Harish-Chandra pairs over a commutative ring

“…An important result from the study is the equivalence, shown by Kostant, between the category of Lie supergroups and the category of Harish-Chandra pairs; see [4,Section 7.4], [28]. The corresponding result for algebraic supergroups, that is, the equivalence (1.1) ASG ≈ HCP between the category ASG of algebraic supergroups and the category HCP of Harish-Chandra pairs, was only recently proved by Carmeli and Fioresi [5] when k = C, and then by the first-named author [20] for an arbitrary field of characteristic 2; see [20,12] for applications of the result. As was done for Lie supergroups, Carmeli and Fioresi define a Harish-Chandra pair to be a pair (G, g) of an algebraic group G and a finite-dimensional Lie superalgebra g which satisfy some conditions (see Definition 4.4), and proved that the equivalence (1.1) is given by G → (G ev , Lie(G)) (see the third paragraph above).…”

The Center of Dist (GL(m|n)) in Positive Characteristic

“…If G is connected (or pseudoconnected, in the terminology from [20]; see also §3 of [7]), then the converse statement is also true (analogously to Lemma 5.1 of [7]).…”

“…Proposition 8. 7. For every π, λ ∈ X (r) (T ) we have g π (x λ x 21 x 12 ) = (−1) |π|+1 δ π,λ and g π (x λ ) = 0.…”

BLOCKS FOR THE GENERAL LINEAR SUPERGROUP GL(m|n)

“…We call M cogenerated by an element m ∈ M if the socle of M is irreducible and generated by m. Without loss of generality one can always assume that m is homogeneous. If H is connected, then by Lemma 9.4 of [24] and Proposition 3.4 of [7], for any m ∈ M a supersubmodule generated by m coincides with Dist(H)m. Proof. The proof can be modified from Lemma 9.1 of [22].…”