By Georgia Benkart

Quantity 197, quantity 920 (second of five numbers).

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Xpm ). In the special case that m = 1, the resulting Lie algebra W (1; 1) is called the p-dimensional Witt algebra or simply the Witt algebra. The JacobsonWitt algebras may be viewed as “thickenings” of it by addition of variables. By identifying cosets with their representatives, we may assume that the elements xa := xa11 · · · xamm with a = (a1 , . . , am ) and 0 ≤ ai < p for all i determine a basis for F[x1 , . . , xm ]/ xp1 , . . , xpm . Then the derivations xa Di , as a ranges over such m-tuples and i = 1, .

P − 1. If η is not a 0 multiple of α, this is impossible, since root strings in classical Lie algebras have length at most 4. If η is a multiple of α, then all multiples kα for k = 1, . . 5 of Chapter 2 for more details). 63 and the assumption q = 1, χ(x) = 0 for all x ∈ gα0 and all α > 0; the g0 -module g−1 has a common eigenvector for ad b+ By symmetrical arguments, χ(x) = 0 for all x ∈ g−α and all α > 0; 0 the g0 -module g−1 has a common eigenvector for ad b− . 12. 63, there is a linear functional χ on the sum g0 of the nonabelian ideals such that (ad x)p − ad x[p] = χ(x)p id holds on g−1 for all x ∈ g0 .

We want to prove that for some λ, there exists a nonzero v ∈ gλ−1 such that [x, v] = 0 for all x ∈ gα0 , α > 0. 1) and will be a common eigenvector for ad b+ . 68. 63 and the hypothesis that the depth q = 1, the g0 -module g−1 contains a common eigenvector for b+ = t ⊕ α>0 gα0 (and by symmetry, for b− = t ⊕ α>0 g−α 0 ). Proof. 64)) (ad x)p − ad x[p] = χ(x)p id holds on g−1 for all x ∈ g0 . If x ∈ gα0 is a root vector of g0 , then x[p] = 0, and (ad x)p = χ(x)p id on g−1 . 47). As the space of those eigenvectors is invariant under ad t, there will be a common eigenvector for ad b+ in g−1 .

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