A Quantum Analogue of the $\mathcal{Z}$ Algebra

Abstract

We define a natural quantum analogue for the $\mathcal{Z}$ algebra, and which we refer to as the ${\mathcal{Z}}_q$ algebra, by modding out the Heisenberg algebra from the quantum affine algebra $U_q(\widehat{sl(2)})$ with level $k$. We discuss the representation theory of this ${\mathcal{Z}}_q$ algebra. In particular, we exhibit its reduction to a group algebra, and to a tensor product of a group algebra with a quantum Clifford algebra when $k=1$, and $k=2$, and thus, we recover the explicit constructions of $\uq$-standard modules as achieved by Frenkel-Jing and Bernard, respectively. Moreover, for arbitrary nonzero level $k$, we show that the explicit basis for the simplest $\mathcal{Z}$-generalized Verma module as constructed by Lepowsky and primc is also a basis for its corresponding ${\mathcal{Z}}_q$-module, i.e., it is invariant under the q-deformation for generic q. We expect this ${\mathcal{Z}}_q$ algebra (associated with $\uq$ at level $k$), to play the role of a dynamical symmetry in the off-critical $Z_k$ statistical models.

32 pages, LATEX, minor changes

References in corpus (6)

• Quantum Inverse Scattering Method and Correlation Functions
• Diagonalization of the XXZ Hamiltonian by Vertex Operators
• Vertex operators in solvable lattice models
• Notes on highest weight modules of the elliptic algebra ${\mathcal{A}}_{q,p}\left({\widehat{sl}}_2\right)$
• N-point Correlation Functions of the Spin-1 XXZ Model
• On a Bosonic-Parafermionic Realization of $U_q(\widehat{sl(2)})$

Cited by in corpus (2)

• Quantum Z-algebras and representations of quantum affine algebras
• Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations