Braid group invariance of quantum tori and applications

Abstract: A well-known fact in classical Lie theory is that the Weyl group W associated with a simple, complex, finite-dimensional Lie algebra g acts by isometries on the dual of the Cartan subalgebra. In particular, the symmetric bilinear form defined on the weight lattice is invariant under the W-action.

In this talk, I will present an analogue of this invariance phenomenon for skew-symmetric bicharacters in the context of quantum affine algebras. Specifically, we will consider the groups of fundamental and pre-fundamental l-weights (which can be viewed as affine analogues of the fundamental weights of g). By results of Chari and Frenkel–Hernandez, the braid group B associated with g acts on these groups of l-weights. We then introduce the skew-symmetric bicharacters on the groups of l-weights defined by Nakajima and Hernandez, and extended by Bittmann and myself. I will show that these bicharacters are invariant under the B-action on l-weight groups, and present an application concerning quantum QQ-systems in the theory of shifted quantum groups.

This talk will take place in hybrid mode at the Institut Henri Poincaré.

K-theory and homologyquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Paris algebra seminar

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