On representations of the elliptic quantum groupEτ,η(sl2)
On representations of the elliptic quantum groupEτ,η(sl2)
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关于椭圆量子群Eτ,η(sl2)的表示
DOI：
10.1007/bf02101296
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发表时间：
1996
影响因子：
2.4
通讯作者：
A. Varchenko
中科院分区：
文献类型：
--
作者：
G. Felder;A. Varchenko
摘要描述了椭圆量子群peτ，η(sl2)的表示理论。结果表明，该表示理论与YangianY（sl2）和量子环群的表示理论是平行的
$$U_q (\widetilde{sl}_2 )$$
．介绍了椭圆量子群peτ，η(sl2)的表示理论的基本概念，构造了三族模：评价模、循环模、一维模。证明了在一定条件下，任何不可约的有限型最高权模都同构于评价模与一维模的张量积。描述了有限维评估模块的融合。特别地，我们证明了在一定条件下，两个评价模的张量积是可约的，并且包含一个评价模，在这种情况下，用椭圆二项式系数的形式给出了评价模嵌入张量积的方法。我们描述了椭圆量子群的行列式元素。当n η=m+lτ时，表示理论变得特殊，其中en，m，l为整数。在这种情况下，我们指出了一些新特性。
AbstractWe describe representation theory of the elliptic quantum groupEτ,η(sl2). It turns out that the representation theory is parallel to the representation theory of the YangianY(sl2) and the quantum loop group
$$U_q (\widetilde{sl}_2 )$$
.We introduce basic notions of representation theory of the elliptic quantum groupEτ,η(sl2) 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 ifNη=m+lτ, whereN,m,l are integers. We indicate some new features in this case.