dorsal/arxiv
View SchemaA Quantum Analogue of the ${\cal Z}$ Algebra
| Authors | A. Hamid Bougourzi, Luc Vinet |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9504009 |
| URL | https://arxiv.org/abs/q-alg/9504009 |
| DOI | 10.1063/1.531581 |
Abstract
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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"abstract": "We define a natural quantum analogue for the ${\\cal Z}$ algebra, and which we\nrefer to as the ${\\cal Z}_q$ algebra, by modding out the Heisenberg algebra\nfrom the quantum affine algebra $U_q(\\hat{sl(2)})$ with level $k$. We discuss\nthe representation theory of this ${\\cal Z}_q$ algebra. In particular, we\nexhibit its reduction to a group algebra, and to a tensor product of a group\nalgebra with a quantum Clifford algebra when $k=1$, and $k=2$, and thus, we\nrecover the explicit constructions of $\\uq$-standard modules as achieved by\nFrenkel-Jing and Bernard, respectively. Moreover, for arbitrary nonzero level\n$k$, we show that the explicit basis for the simplest ${\\cal Z}$-generalized\nVerma module as constructed by Lepowsky and primc is also a basis for its\ncorresponding ${\\cal Z}_q$-module, i.e., it is invariant under the\nq-deformation for generic q. We expect this ${\\cal Z}_q$ algebra (associated\nwith $\\uq$ at level $k$), to play the role of a dynamical symmetry in the\noff-critical $ Z_k$ statistical models.",
"arxiv_id": "q-alg/9504009",
"authors": [
"A. Hamid Bougourzi",
"Luc Vinet"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1063/1.531581",
"title": "A Quantum Analogue of the ${\\cal Z}$ Algebra",
"url": "https://arxiv.org/abs/q-alg/9504009"
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