By J. Baker, C. Cleaver, J. Diestel, G. Bennett, S.Y. Chang, D.E. Marshall, J.A. Cima, W. Davis, W.J. Davis, W.B. Johnson, J.B. Garnett, J. Johnson, J. Wolfe, H.E. Lacey, D.R. Lewis, A.L. Matheson, P. Orno, J.W. Roberts

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E. all products (without repetitions) of factors (n ) i and Xγ —with α ∈ Δ0 , i ∈ 1, . . , , γ ∈ Δ1 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.

CHEVALLEY SUPERGROUPS where the factors in the product are taken in any order (as they do commute). (b) Let γ, δ ∈ Δ1 , A ∈ (salg) , ϑ, η ∈ A1 . 3) xγ (ϑ) , xδ (η) = xγ+δ −cγ,δ ϑ η = 1−cγ,δ ϑ η Xγ+δ ∈ G0 (A) if δ = −γ ; otherwise, for δ = −γ , we have xγ (ϑ) , x−γ (η) = 1− ϑ η Hγ = hγ 1− ϑ η ∈ G0 (A) (c) Let α, β ∈ Δ , A ∈ (salg) , t ∈ U (A0 ) , u ∈ A0 ×A1 = A . Then hα (t) xβ (u) hα (t)−1 = xβ tβ(Hα ) u ∈ Gp(β) (A) where p(β) := s , by deﬁnition, if and only if β ∈ Δs . Proof. The result follows directly from the classical results in [29], pg.

CHAPTER 5 Chevalley supergroups This chapter is the core of our work: it presents the construction of “Chevalley supergroups” and gives their main and fundamental properties. Our construction goes through some main steps we are going to outline here. The ﬁrst step demands to ﬁx a speciﬁc g–module, which has to be faithful, semisimple, ﬁnite-dimensional and rational. Then any such g–module V , contains a Z–integral form M which is KZ (g)– stable; such an M will then be called an “admissible lattice” of V .