434 research outputs found
Perfect Crystals for U_q(D_4^{(3)})
A perfect crystal of any level is constructed for the Kirillov-Reshetikhin
module of corresponding to the middle vertex of the Dynkin
diagram. The actions of Kashiwara operators are given explicitly. It is also
shown that this family of perfect crystals is coherent. A uniqueness problem
solved in this paper can be applied to other quantum affine algebras.Comment: 27 page
Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras
In this paper, we prove Khovanov-Lauda's cyclotomic categorification
conjecture for all symmetrizable Kac-Moody algebras. Let be the
quantum group associated with a symmetrizable Cartan datum and let
be the irreducible highest weight -module with a dominant integral
highest weight . We prove that the cyclotomic Khovanov-Lauda-Rouquier
algebra gives a categorification of .Comment: Typoes correcte
Young tableaux and crystal for finite simple Lie algebras
We study the crystal base of the negative part of a quantum group. An
explicit realization of the crystal is given in terms of Young tableaux for
types , , , , and . Connection between our realization
and a previous realization of Cliff is also given
On dual canonical bases
The dual basis of the canonical basis of the modified quantized enveloping
algebra is studied, in particular for type . The construction of a basis for
the coordinate algebra of the quantum matrices is appropriate for
the study the multiplicative property. It is shown that this basis is invariant
under multiplication by certain quantum minors including the quantum
determinant. Then a basis of quantum SL(n) is obtained by setting the quantum
determinant to one. This basis turns out to be equivalent to the dual canonical
basis
Fusion of the -Vertex Operators and its Application to Solvable Vertex Models
We diagonalize the transfer matrix of the inhomogeneous vertex models of the
6-vertex type in the anti-ferroelectric regime intoducing new types of q-vertex
operators. The special cases of those models were used to diagonalize the s-d
exchange model\cite{W,A,FW1}. New vertex operators are constructed from the
level one vertex operators by the fusion procedure and have the description by
bosons. In order to clarify the particle structure we estabish new isomorphisms
of crystals. The results are very simple and figure out representation
theoretically the ground state degenerations.Comment: 35 page
The Kazhdan-Lusztig conjecture for W-algebras
The main result in this paper is the character formula for arbitrary
irreducible highest weight modules of W algebras. The key ingredient is the
functor provided by quantum Hamiltonian reduction, that constructs the W
algebras from affine Kac-Moody algebras and in a similar fashion W modules from
KM modules. Assuming certain properties of this functor, the W characters are
subsequently derived from the Kazhdan-Lusztig conjecture for KM algebras. The
result can be formulated in terms of a double coset of the Weyl group of the KM
algebra: the Hasse diagrams give the embedding diagrams of the Verma modules
and the Kazhdan-Lusztig polynomials give the multiplicities in the characters.Comment: uuencoded file, 29 pages latex, 5 figure
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