Tensor products, characters, and blocks of finite-dimensional representations of quantum affine algebras at roots of unity

Details Tensor products, characters, and blocks of finite-dimensional representations of quantum affine algebras at roots of unity Tensor products, characters, and blocks of finite-dimensional representations of quantum affine algebras at roots of unity

2009-09-11

arxiv.org/abs/0909.2198v3

Int. Math. Res. Notices (2011), no. 18, 4147-4199

Mathematics

Quantum Algebra

revised version; to appear in IMRN

Scientific paper

10.1093/imrn/RNQ250

We establish several results concerning tensor products, q-characters, and the block decomposition of the category of finite-dimensional representations of quantum affine algebras in the root of unity setting. In the generic case, a Weyl module is isomorphic to a tensor product of fundamental representations and this isomorphism was essential for establishing the block decomposition theorem. This is no longer true in the root of unity setting. We overcome the lack of such a tool by utilizing results on specialization of modules. Furthermore, we establish a sufficient condition for a Weyl module to be a tensor product of fundamental representations and prove that this condition is also necessary when the underlying simple Lie algebra is sl(2). We also study the braid group invariance of q-characters of fundamental representations.

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