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

Details A Quantum Analogue of the ${\cal Z}$ Algebra A Quantum Analogue of the ${\cal Z}$ Algebra

1995-04-19

arxiv.org/abs/q-alg/9504009v2

Mathematics

Quantum Algebra

32 pages, LATEX, minor changes

Scientific paper

10.1063/1.531581

We define a natural quantum analogue for the ${\cal Z}$ algebra, and which we refer to as the ${\cal Z}_q$ algebra, by modding out the Heisenberg algebra from the quantum affine algebra $U_q(\hat{sl(2)})$ with level $k$. We discuss the representation theory of this ${\cal Z}_q$ 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 ${\cal Z}$-generalized Verma module as constructed by Lepowsky and primc is also a basis for its corresponding ${\cal Z}_q$-module, i.e., it is invariant under the q-deformation for generic q. We expect this ${\cal Z}_q$ algebra (associated with $\uq$ at level $k$), to play the role of a dynamical symmetry in the off-critical $ Z_k$ statistical models.

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