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

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11. ,N± }k (−1)(2) η ±d Xγ ± d where ϑ± := ϑd1 · · · ϑdk , Xγ ± := Xγ ± · · · Xγ ± , d d1 dk η ± := ηd1 · · · ηdk k for every k > 1 and every k–tuple d := (d1 , . . , dk ) ∈ {1, . . , N± } . ,N+ }k d d where the Φγ ± ’s are suitable monomials (in the ϑi ’s and ηj ’s) of degree k with a d coeﬃcient ±1 , and Φγ + = ±Φγ − . 9 to deﬁne G(A) , all the identities above actually hold inside End Vk (A) . We proceed now to prove the following, intermediate result: Claim: Let g± , f± ∈ G±,< (A) be such that g− g+ = f− f+ .

We shall prove the ﬁrst mentioned bijection. 17 gives G(A) = G0 (A) G< 1 (A) , so the product map from G0 (A) × G1 (A) to G(A) is onto; but in particular, we can choose an ordering on Δ1 for which −,< < Δ− Δ+ (A) G+,< (A) , so we are done for surjectivity. 3. CHEVALLEY SUPERGROUPS AS ALGEBRAIC SUPERGROUPS 43 To prove that the product map is also injective amounts to showing that for any g ∈ G(A) , the factorization g = g0 g− g+ with g0 ∈ G0 (A) and g± ∈ G±,< (A) is 1 unique. In other words, if we have g = g0 g− g+ = f0 f− f+ , g0 , f0 ∈ G0 (A) , g± , f± ∈ G±,< (A) 1 we must show that g0 = f0 and g± = f± .

That is to say M = μ∈h∗ Proof. The proof is the same as in the classical case. Without loss of generality, we can assume that V be irreducible of highest weight. v ; then M spans V over K , and it is clearly KZ (g)–stable because KZ (g) is a subalgebra of U (g) . The proof that M is actually a Z–lattice of V and that M splits into M = ⊕μ M ∩ Vμ is detailed in [29], §2, Corollary 1, and it applies here with just a few minor changes. 4. M ⊆ M . If V is faithful, then gV = hV ⊕α∈Δ Z Xα , hV := H ∈ h μ(H) ∈ Z , ∀ μ ∈ Λ where Λ is the set of all weights of V .