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

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**Example text**

5), so we are done. 24. This theorem asserts that Chevalley supergroup functors actually provide algebraic supergroups. This is quite remarkable, as some of these supergroups had not yet been explicitly constructed before. In fact, giving the 46 5. , showing that there is a superscheme whose functor of points is the given one—can be very hard. For example using the procedure described above, it is possible to construct the algebraic supergroups corresponding to all of the exceptional classical Lie superalgebras F (4), G(3) and D(2, 1; a)—for a ∈ Z —and to the strange Lie superalgebras.

Step (1) is non-trivial. ” i Then we apply any one of steps (2), (3) or (4) to M . Step (2) gives M = z M , with ht M for some z ∈ Z and some monomial M Step (3) yields ht M , (possibly zero). M = z M∨ , with ht M∨ = ht M , fac M∨ fac M ∨ for some z ∈ Z and some monomial M . Finally, step (4) instead gives M = M + k ht M zk Mk , with ht Mk and ht M = ht M , inv M ∀ k, inv M where zk ∈ Z (for all k ), and M and the Mk ’s are monomials. 16); and then we iterate. 17), this process will stop after ﬁnitely many iterations.

Vμ may have a non-trivial weight + Moreover, let us denote by component is Vμ itself, as NΔ− 1 ∩ NΔ1 = {0} . vμ inside Vμ+γ + . d 44 5. vμ ∀ d = 1, . . vμ ∀ d = 1, . . , N+ d = d In fact, this is certainly true for γd± simple, and one can check directly case by case that any odd root γd± can never be the sum of three or more odd roots all positive or negative like γd± itself. d. ♦ Let’s now go on with the proof of the theorem. By deﬁnition of G0 (A) , both g0 and f0 are products of ﬁnitely many factors of type xα (tα ) and hi (si ) for some tα ∈ A0 , si ∈ U (A0 ) —with α ∈ Δ0 , i = 1, .