# Infinite-Dimensional Lie Algebras by Victor G. Kac

By Victor G. Kac

This is often the 3rd, considerably revised variation of this significant monograph. The publication is worried with Kac-Moody algebras, a specific classification of infinite-dimensional Lie algebras, and their representations. it truly is in keeping with classes given over a few years at MIT and in Paris, and is satisfactorily self-contained and particular for use for graduate classes. each one bankruptcy starts off with a motivating dialogue and ends with a set of routines, with tricks to the more difficult difficulties.

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A, a^) < 0). Ch. 3 Integrable Representations and the Weyl Group b) If A is a weight ofV, multiplicity. 6. Let V be an integrable g'(A)-module. 6, with f) replaced by f)', still hold. Furthermore, the local nilpotency of e,- and /, on V guarantees that V is /i,-diagonalizable, and hence Jj'-diagonalizable provided that n < oo. 8. 7. Now we introduce the important notion of the Weyl group of a Kac-Moody algebra &(A). For each i = 1 , . . (A) = A - ( A , a ) > < > A G Ij*. It is clear that r, is a reflection since its fixed point set is TJ = {A G f)1(A,c*y) = O}, and r,(a t ) = -<*<.

It follows from b), c) and e) that the bilinear form (. | . ) is symmetric. If d) fails, then, by c), the form (. | . ) is degenerate. Let i = Ker(. | . ) . It is an ideal and by b), we have iPI t) = 0, which contradicts the definition of 9(A). 3. Suppose that A = (a,j) is a symmetrizable generalized Cartan matrix. Jsl where ef- are positive rational numbers and (6,-j) is a symmetric rational matrix. Such a decomposition always exists. 1) is equivalent to a system of homogeneous linear equations and inequalities over Q with unknowns e,"1 and 6^: By definition, it has a solution over C.

1) (exp a)6(exp - a ) = (exp(ad a))(6). 2). 8. Let IT be an integrable representation of space V. For i = 1 , . . (Vx) = Vri(x) b)rfeAutQ(A) Proof. Let v G Vx. (v)) = (\,h)rf(v) if (a,-,h) = 0. (r*(v)) = —(\,aY)r*(v). 2) in the 2-dimensional natural representation of ^ ( C ) . 2) is clear. 1) applied to the adjoint representation. • Ch. 8. Let (K, TT) be an integrable g(A)-module whose kernel lies in f). 6, the action of the subalgebra 0(,) (i = l , . . , n ) on V can be integrated to a representation 7rt- : SX2(C) —> GL(V).