Chevalley Supergroups (Memoirs of the American Mathematical by R. Fioresi, F. Gavarini

By R. Fioresi, F. Gavarini

"January 2012, quantity 215, quantity 1014 (end of volume)."

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Moreover, as A1 is finitely generated (over A0 ), by assumption, 0 . 6) g0 ≡ f0 mod A1n , g± ≡ f± mod A1n ∀n∈N where, for any A ∈ (salg) , any I ideal of A with πI : A − A I the canonical projection, by x ≡ y mod I we mean that two elements x and y in G(A ) have the same image in G A I via the map G(πI ) . 6) by induction. The case n = 0 is clear (there is no odd part). We divide the induction step in two cases: n even and n odd. 6) be true for n even. In particular, g± ≡ f± mod A1n : then (see the ± n proof of the Claim above) we have ϑ± for all d , hence d ≡ ηd mod A1 ± n+1 n (ϑ± d − ηd ) ∈ A1 ∩ A1 ⊆ A1 for all d by an obvious parity argument.

Next result is a group-theoretical counterpart of the splitting g = g0 ⊕ g1 . It is a super-analogue of the classical Cartan decomposition (for reductive groups). In the differential setting, it was—somewhat differently—first pointed out by Berezin (see [3], Ch. 2, §2). 15. Let A ∈ (salg) . There exist set-theoretic factorizations G(A) = G0 (A) G1 (A) G± (A) = G± 0 (A) G± 1 (A) , G(A) = G1 (A) G0 (A) , ± G± (A) = G± 1 (A) G0 (A) Proof. The proof for G(A) works for G± (A) as well, so we stick to the former.

The proof proceeds by standard arguments, using the commutation rules to expand any unordered monomial in the generators as a Z–linear combination of ordered ones. As in ordinary setting, the Kostant form inside the enveloping superalgebra of g and its PBW–theorem play an absolutely crucial role in the definition and construction of Chevalley supergroups. Again the notion of Kostant superalgebra, and its description via a PBW-like theorem—though not for all types—have appeared previously in the literature in the works [18], [30].

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