34 research outputs found
Universal Verma modules and the Misra-Miwa Fock space
The Misra-Miwa -deformed Fock space is a representation of the quantized
affine algebra of type A. It has a standard basis indexed by partitions and the
non-zero matrix entries of the action of the Chevalley generators with respect
to this basis are powers of . Partitions also index the polynomial Weyl
modules for the quantum group as tends to infinity. We explain
how the powers of which appear in the Misra-Miwa Fock space also appear
naturally in the context of Weyl modules. The main tool we use is the
Shapovalov determinant for a universal Verma moduleComment: 15 pages. v2: Minor corrections and clarifications; 4 new reference
Fock space representations of quantum affine algebras and generalized Lascoux-Leclerc-Thibon algorithm
We construct the Fock space representations of classical quantum affine
algebras using combinatorics of Young walls. We also show that the crystal
graphs of the Fock space representations can be realized as the abstract
crystal consisting of proper Young walls. Finally, we give a generalized
version of Lascoux-Leclerc-Thibon algorithm for computing the global bases of
the basic representations of classical quantum affine algebras.Comment: 70 page
Diagonalization of the XXZ Hamiltonian by Vertex Operators
We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the
thermodynamic limit, where the model becomes invariant under the action of
affine U_q( sl(2) ).
Our method is based on the representation theory of quantum affine algebras,
the related vertex operators and KZ equation, and thereby bypasses the usual
process of starting from a finite lattice, taking the thermodynamic limit and
filling the Dirac sea. From recent results on the algebraic structure of the
corner transfer matrix of the model, we obtain the vacuum vector of the
Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex
operators, which act as particle creation operators in the space of
eigenvectors.
We check the agreement of our results with those obtained using the Bethe
Ansatz in a number of cases, and with others obtained in the scaling limit ---
the -invariant Thirring model.Comment: 65 page