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Integrable and Weyl modules for quantum affine $$\operatorname {sl}_2$$. (English) Zbl 1034.17008
Pressley, Andrew (ed.), Quantum groups and Lie theory. Lectures given at the Durham symposium on quantum groups, Durham, UK, July 19–29, 1999. Cambridge: Cambridge University Press (ISBN 0-521-01040-3/pbk). Lond. Math. Soc. Lect. Note Ser. 290, 48-62 (2001).
The authors study some maximal finite-dimensional quotients $$W_q(\pi)$$ of integrable modules $$W_q(m)$$ generated by extremal vectors of weight $$m$$ over the quantum affine sl$$_2$$. They show that all $$W_q(\pi)$$ have a classical limit and use this to prove that their dimensions are $$2^m$$. It is also shown that all simple finite-dimensional modules arise as simple quotients of $$W_q(\pi)$$. The integrable modules $$W_q(m)$$ are realized as the spaces of invariants of an action of the Hecke algebra $$\mathcal{H}_m$$ on the tensor product $$(V\otimes \mathbb{C}(q)[t,t^{-1}])^{\otimes m}$$, where $$V$$ is a two-dimensional vector space over $$\mathbb{C}(q)$$. In the last section the authors formulate some conjectures on how the results of the present paper can be extended to the general case.
For the entire collection see [Zbl 0980.00028].

MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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