Integrable and Weyl modules for quantum affine \(\operatorname {sl}_2\). (English) Zbl 1034.17008

Chari, Vyjayanthi; Pressley, Andrew

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].

Reviewer: Volodymyr Mazorchuk (Uppsala)

MSC:

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

Keywords:

quantum group; affine algebra; integrable module; Weyl module