Spinor representations of Uq( ˆ gl(n)) and quantum boson-fermion correspondence

abstract This is an extension of quantum spinor construction in [DF2]. We define quantum affine Clifford algebras based on the tensor category and the solutions of q-KZ equations, construct quantum spinor representations of U q (ˆ gl(n)) and explain classical and quantum boson-fermion correspondence. I. Introduction. The independent discovery of a q-deformation of universal enveloping algebra of an arbitrary Kac-Moody algebra by Drinfeld [D1] and Jimbo [J1] immediately raised numerous questions about q-deformations of various structures associated to Kac-Moody algebras. A major step in this direction was the work by Lusztig [L], who obtained a q-deformation of the category of highest weight representations of Kac-Moody algebras for generic or formal parameter q. There were a number of successful results on q-deformation of various mathematical structures of finite dimensional and affine Lie algebras [D2], [J2], [FJ], [H], [FR], etc. However each particular problem required its own special insight and in some cases presented formidable difficulties. In [DF2], we propose an invariant approach, which was also stressed in [FRT], where a q-analogue of matrix realization of classical Lie algebras was given. We manage to use such an approach to define quantum Clifford and Weyl algebras using general representation theory of quantum groups. We show that the explicit formulas for quantum Clifford and Weyl algebras match the ones actively studied in physics literature (see e.g. [WZ], [K]). Using those quantum algebras, we construct spinor and oscillator representations of quantum groups of classical types and recover all the relevant formulas for the quantum construction. Uniqueness arguments from representation theory allow us to justify that the quadratic expressions in quantum Clifford and Weyl algebras provide the desired representations. An explicit verification of Serre's relations for quantum groups lead to rather involved formulas (cf. [H]). The key idea consists of reformulating familiar classical constructions entirely in terms of the tensor category of highest weight representations and then, using Lusztig's result on q-deformation of this category to define the corresponding quantum structures. In the quantum case, we often need a qua-sitriangular structure of the tensor category introduced by Drinfeld [D3], since it plays the role of the symmetric structure in the classical case. In particular, we would like to emphasize the central role of the universal Casimir operator implied by the quantum structure. The motivation to develop such an invariant approach for the quantum groups corresponding to the simple finite dimensional algebras is surely to apply it …