Quantum vertex representations via finite groups and the McKay correspondence

Details Quantum vertex representations via finite groups and the McKay correspondence Quantum vertex representations via finite groups and the McKay correspondence

1999-07-27

arxiv.org/abs/math/9907175v2

Commun. Math. Phys. 211 (2000), 365--393

Mathematics

Quantum Algebra

34 pages, amslatex, to appear in Comm. Math. Phys

Scientific paper

10.1007/s002200050817

We establish a $q$-analog of our recent work on vertex representations and the McKay correspondence. For each finite group $\Gamma$ we construct a Fock space and associated vertex operators in terms of wreath products of $\Gamma\times \mathbb C^{\times}$ and the symmetric groups. An important special case is obtained when $\Gamma$ is a finite subgroup of $SU_2$, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of $ADE$ type.

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