Chevalley Supergroups by R. Fioresi, F. Gavarini

By R. Fioresi, F. Gavarini

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

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This implies that A1n = {0} and (n) A1 = 0 for n larger than the number of odd generators of A1 . Let us consider two consecutive factors gi gi+1 in g . If they are already ordered, we are done. Otherwise, there are four possibilities: — (1) gi = xαi (ti ) , gi+1 = hαi (ui ) , or gi = xγi (ϑi ) , gi+1 = hαi (ui ) . 13(c). In particular we replace a pair of unordered factors with a new pair of ordered factors. Even more, this shows that any factor of type hαi(ui ) can be flushed to the left of our product so to give a new product of the same nature, but with all factors of type hαi (ui ) on the left-hand side.

1 and nα , of type Xα α , H ni ni ∈ N—taken in the right order with respect to . Proof. Let us call “monomial” any product in any possible order, possibly (n ) i ’s—with zi ∈ Z—and several with repetitions, of several Xα α ’s, several His−z i mγ ’s, mγ ∈ N. For any such monomial M , we define three numbers, namely: — its “height” ht(M) , namely the sum of all nα ’s and mγ ’s occurring in M (so it does not depend on the binomial coefficients in the Hi ’s); — its “factor number” fac(M) , defined to be the total number of factors (n ) i (namely Xα α , Hin−z or Xγ ) within M itself; i — its “inversion number” inv(M) , which is the sum of all inversions of the order among the indices of factors in M when read from left to right.

INDEPENDENCE OF CHEVALLEY AND KOSTANT SUPERALGEBRAS 47 G1 (A) are also normalized by G0 (A) . 15 we conclude that G± 1 (A) and G1 (A) are normal in G(A) . 23. The statement for G clearly follows as well. 4. Independence of Chevalley and Kostant superalgebras Next question is the following: what is the role played by the representation V ? Moreover, we would like to show that our construction is independent of the choice of an admissible Z–lattice M in a fixed g–module (over K ). Let G and G be two Chevalley supergroups obtained by the same g , possibly with a different choice of the representation.

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