Action Of Uq Research Articles

Descriptions of crystal [formula omitted] for E6 and E7 types via tableaux and Kashiwara embedding

Let Uq(g) be the quantized universal enveloping algebra for a Lie algebra g, and let Vq(λ) be the irreducible highest weight module for Uq(g). The crystal base B(λ) is a colored directed graph that captures the structure of Vq(λ) and the action of Uq(g) on Vq(λ) in a rudimentary manner. Likewise, the crystal base B(∞) holds the bare structure of the negative part Uq−(g).In this work, we describe realizations of the crystal B(λ) via two separate approaches for the cases when the base Lie algebra g is of E6 and E7 types. Our first description relies on the fact that B(λ) appears as a connected component within the much larger crystal B(∞)⊗{rλ}, where {rλ} is a certain single-element crystal. Choosing to represent elements of B(∞) with marginally large tableaux, we identify those elements belonging to the mentioned connected component. Our second description of B(λ) is a translation of the first description into one involving the Kashiwara embedding, which is an embedding of B(∞) into a tensor product of a series of much simpler crystals.

The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere

Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the orthogonal quantum 4-sphere S^4_q. These representations are the constituents of a spectral triple on this sphere with a Dirac operator which is isospectral to the canonical one on the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an `instanton' projection. We also introduce a real structure which satisfies all required properties modulo smoothing operators.

U q [sl (2| ∧ 1)] vertex operators, screen currents, and correlation functions at an arbitrary level

Bosonized q-vertex operators related to the four-dimensional evaluation modules of the quantum affine superalgebra Uq[sl(2|∧1)] are constructed for arbitrary level k=α, where α≠0,−1 is a complex parameter appearing in the four-dimensional evaluation representations. They are intertwiners among the level-α highest weight Fock–Wakimoto modules. Screen currents which commute with the action of Uq[sl(2|∧1)] up to total differences are presented. Integral formulas for N-point functions of type I and type II q-vertex operators are proposed.

Commutation relations of vertex operators related with the spin representation of Uq(Dn

A new scheme to study the XXZ spin chain in massive regime is introduced by Davies et al in [1]. In this scheme, which is called symmetry approach, the space of states is identified with some Uq(A\_ ) -module. Under this identification, the XXZ Hamiltonian can be constructed on this module as an operator which commutes with the action of Uq(A '). Furthermore, the row transfer matrix and creation (annihilation) operators can be constructed by using vertex operators. In a similar way to [1], the higher spin chain [2] and the models related to the vector representation ofUq(A^) [3], Uq(B^ U q ( D ( n } ) [5] are studied. In this paper, we consider the model related with the spin representation of Uq(Dn ). In physics the model that we consider here is explained as follows. It is a one dimensional quantum spin chain model constructed from the spin representation of Uq(Dn )• The Hamiltonian acts on the space of the infinite tensor product