Level-0 action of \(U_q(\widehat{\mathfrak sl}_n)\) on the \(q\)-deformed Fock spaces
The additional symmetry structure of the quantum affine algebra \(U_q(\widehat{sl}_n)\) manifested by a level-0 action is studied by the authors. Starting with a \(q\)-deformed Fock space graded by the degree operator of the level-1 action of \(U_q(\widehat{sl}_n)\), a level-0 action of \(U_q' (\widehat{sl}_n)\) is defined in such a way that the homogeneous components are invariant. Identifying homogeneous components with projective limits of certain sequences of finite dimensional representation spaces of \(U_q'(\widehat{sl}_n)\), the level-0 action is defined in the components. Moreover the decomposition of the \(q\)-deformed Fock space as a level-0 \(U_q'(\widehat{sl}_n)\) module is investigated. The paper is organized as follows: Section 1 deals with the background of the problem and some results are stated. Section 2 contains necessary informations on affine Hecke algebras and \(q\)-wedge products. In section 3 the finite \(q\)-wedges are decomposed by a certain level-0 action. In section 4 the level-0 action in the Fock space is defined and the authors describe a family of Hamiltonians commuting with the level-0 action. The decomposition of the Fock space with respect to the defined action is given in section 5 by constructing a base of the Fock space in terms of non-symmetric Macdonald polynomials. There is an appendix summarizing two realizations of the algebra \(U_q'(\widehat{sl}_n)\) and results on the calculation of Drinfeld polynomials.
scientific article