Zhukavets N.M., Shestakov I.P.'s A base of the free alternative superalgebra on one odd PDF

By Zhukavets N.M., Shestakov I.P.

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12). 15. THEOREM. 1 is nilpotent; (ii) for all x E o. ad x is nilpotent. (i) ~ (ii). 5 (ii). (ii)::::;. (i). Assume that ad xis nilpotent for all xE g. Let g' be a solvable Lie subalgebra of maximal dimension of g; suppose that g' "F g. Consider the adjoint representation of g, and let !! be its restriction to g'. Then g' is stable under g(g'). Let q be the quotient representation of Q in o/g', and (/1 the canonical mapping of 9 onto g/g'. 12). Then yE! g' and [g',y] c g'. Consequently, o' + ky is a Lie subalgebra of g, and [g' + ky, g' + ky] c g', so that g' + ky is solvable.

A) We assume that f! 1 and its proof. - 1(Q(x,)f(y,) + ··· + g(xn)f(yn)). (Xt)f(yj) = 0. I,) (b) We assume that g is simple. Let a= Kerg. Then g = axb, with a and b semi-simple. From (a) there exists v E V such that f(x) = g(x)v for all x E b. Since g(a) = 0, /([a,a]) = 0 by assumption (i). 9), hence f(x) = 0 = g(x)v for all x E a. l,§6] SEMI-SIMPUCITY OF REPRESEl'~TATIONS 23 (c) We consider the general case by induction on dim V. If !! is not simple, let W be a vector subspace of V which is stable under !!

Since g, b and b' have the same rank, the nilspaces of ad9x, ad6x and ad 6,x are equal to a Cartan subalgebra of g. 19. PROPOSITION. Let (g,l)) be a split semi-simple Lie algebra. Let w be an element of the Weyl group operating In {). There exists an elementary = w. automorphism (J of 0 such that n, It is sufficient to consider the case where ex E R exists such that w = s_,. Let X..

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A base of the free alternative superalgebra on one odd generator by Zhukavets N.M., Shestakov I.P.


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