Combinatorial Foundation of Homology and Homotopy by Hans-Joachim Baues

By Hans-Joachim Baues

A new combinatorial origin of the 2 recommendations, in accordance with a attention of deep and classical result of homotopy concept, and an axiomatic characterization of the assumptions less than which leads to this box carry. contains a number of specific examples and purposes in quite a few fields of topology and algebra.

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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.

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