lecture notes2

X 3 in any characteristic let v v f be a nondegenerate

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Unformatted text preview: ations, exp adx = exp (λx + ρ−x ) = exp λx exp ρ−x But clearly, exp λx = λexp x and exp ρ−x = ρexp(−x) . So, exp adx (y ) = (exp x)y (exp(−x)) = (exp x)y (exp(x)−1 ) ￿ 8 MATH 223A NOTES 2011 LIE ALGEBRAS 2.3. Exercises. What about derivations in characteristic p? (1) Show that δ p is a derivation. (2) For the Lie algebra of an associative algebra over a field of characteristic p show that adp = adxp . x (3) (in any characteristic) Let ϕ : V × V → F be a nondegenerate skew-symmetric bilinear pairing. Then the Heisenberg algebra of f is given by L = V ⊕ F with [(v, a)(w, b)] = (0, f (v, w))....
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