By Minoru Wakimoto

The illustration conception of affine lie algebras has been built in shut reference to numerous components of arithmetic and mathematical physics within the final twenty years. There are 3 worthy works on it, written through Victor G Kac. This quantity starts with a survey and assessment of the fabric taken care of in Kac's books. specifically, modular invariance and conformal invariance are defined in additional aspect. The publication then is going extra, facing a number of the contemporary subject matters regarding the illustration conception of affine lie algebras. when you consider that those issues are very important not just in themselves but additionally of their software to a couple components of arithmetic and mathematical physics, the booklet expounds them with examples and certain calculations.

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Integrable 1) A(X) : Representations chL(2n-l,n-l) A °° 3=1 - e2naoqini)(l X(l 2)A2 2(2) - e-2na0g4n(j-l))| . • (i) chi(2n-l,n-l) A V - °° < ^ n ) I I { ( ! - enaiq2nj)(l X(l (ii) - e2naoq2nj)(l - e-^i^nb-i)) - e -2na 0 g 2n(j-l)O chL(4n - 1,ra- 1) X(l - e 4 n a °^ 4 " J ')(l - e -4n« 0 g 4n(j-l)) j . In particular when n = 1, one has the following: oo chL(Ao) = eA° J I ( l + e ° V ' ) ( l + e ~ a ( Y " 1 ) (iii) 3=1 chL(3A0) = '[[Ul + qj){l + ea°qj)(l (iv) e-a°qj-1) + 3=1 *- x ( l + e2aoq2j)(l + e-2aoq2(i-x))\ .

P. 52). 2 2. Integrable Representations Specialized Characters In this section we assume that A = (aij)i,j=i,.. ,n is a symmetrizable generalized Cartan matrix and put S:=[( Sl ,---, Sn )e(z>or ; ] [ > > o ) , which is naturally in ono-one correspondence with the set f n *• i = l n x i=l J of non-zero dominant co-integral forms by n t=0 Namely p^ is an element in h* satisfying (pi,*) = Si. 25) Given A G h*, we define s\ G C " by sx := « a i , A > , - " , (a„, A)). 24). For s — (si, • • • , sn) G S, we consider a homomorphism F^ : C[[e-a\---,e-*»}]3e-a< ,—> q3< E C[[q}}, of associative algebras, called the "specialization of type s ".

We simply write rj := raj and fj :=r&j. This identification of Weyl groups enables us to calculate specialized characters of integrable g(A)-modules in terms of the root system of g(A), since the numerator of the character formula is described by the Weyl group. 1, one can compute the numerator of the character formula for A\ -modules by the root system of A\ '. For this sake, it is necessary to identify the Cartan subalgebra of Q(A) with that of Q{A) in a consistent way with the identification of Weyl groups.