Double-loop algebras and the Fock space

Details Double-loop algebras and the Fock space Double-loop algebras and the Fock space

1996-12-30

arxiv.org/abs/q-alg/9612035v3

Mathematics

Quantum Algebra

final version, accepted for publication in Invent. Math

Scientific paper

The fermionic Fock space admits two different actions of the quantized enveloping algebra of $\hat\sln$> The first one is a q-deformation of the well-known level-one representation of the affine Lie algebra and the second one is a new level-zero action arising from solvable lattices models. In this paper we prove that these two constructions can be glued together to get a representation of a new object: the toridal quantum group. This representation involves some difference operatorators. It is due to the fact that the specialization at q=1 of the toroidal algebra is isomorphic to the enveloping algebra of the universal central extension of a current algebra over a quantum torus.

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