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Modular transformation of twisted characters of admissible representations and fusion algebras associated to nonsymmetric transformation matrices. (English) Zbl 0970.17014
Kobayashi, Toshiyuki (ed.) et al., Analysis on homogeneous spaces and representation theory of Lie groups. Based on activities of the RIMS Project Research ’97, Okayama-Kyoto, Japan, during July and August 1997. Tokyo: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 26, 325-353 (2000).
The author extends the orbifolds theory on modular transformations of twisted characters and on the connection of the fusion algebra of integrable representations of affine Lie algebras with that of finite groups, developed by V. G. Kac and I. T. Todorov [Commun. Math. Phys. 190, 57-111 (1997; Zbl 0904.17021)], to the characters of principal admissible representations of affine Lie algebras. Already for the simplest case of $$\hat{sl}(2,{\mathbb C})$$ the appearance of non-symmetric transformation matrices and associated fusion algebras is observed. To handle them the author extends G. Lusztig’s theory of “fusion datum” with a non-symmetric transformation matrix [see Duke Math. J. 73, 227-241 (1994; Zbl 0815.20031] to a more general situation.
For the entire collection see [Zbl 0941.00016].

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations