On representations of the elliptic quantum group E τ,η( sl 2)
Abstract
We describe representation theory of the elliptic quantum group E τ,η( sl 2). It turns out that the representation theory is parallel to the representation theory of the Yangian Y( sl 2) and the quantum loop groupU_q (widetilde{sl}_2 ). We introduce basic notions of representation theory of the elliptic quantum group E τ,η( sl 2) and construct three families of modules: evaluation modules, cyclic modules, one-dimensional modules. We show that under certain conditions any irreducible highest weight module of finite type is isomorphic to a tensor product of evaluation modules and a one-dimensional module. We describe fusion of finite dimensional evaluation modules. In particular, we show that under certain conditions the tensor product of two evaluation modules becomes reducible and contains an evaluation module, in this case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients. We describe the determinant element of the elliptic quantum group. Representation theory becomes special if Nη=m+lτ, where N,m,l are integers. We indicate some new features in this case.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- December 1996
- DOI:
- 10.1007/BF02101296
- arXiv:
- arXiv:q-alg/9601003
- Bibcode:
- 1996CMaPh.181..741F
- Keywords:
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- Mathematics - Quantum Algebra
- E-Print:
- 21 pages, amstex. An explicit formula for the general R matrix is given in this revised version