Families of Dirac operators and quantum affine groups

Twisted K-theory classes over compact Lie groups can be realized as families of Fredholm operators using the representation theory of loop groups. In this talk I want to show how to deform the Fredholm family, in the sense of quantum groups. The family of Dirac type operators is parametrized by vectors in the adjoint module for a quantum affine algebra and transform covariantly under a (central extension of) the algebra.

💡 Research Summary

The paper investigates a quantum‑group deformation of the well‑known construction that realizes twisted K‑theory classes on a compact Lie group G as families of Fredholm operators. In the classical setting, one uses the representation theory of the loop group LG (or equivalently the affine Kac‑Moody algebra (\widehat{\mathfrak g})) to build, for each point g∈G, a Dirac‑type operator D_g acting on a suitable Hilbert space of spinors. The family {D_g} is continuous, its index reproduces the twisted K‑theory class, and the operators transform covariantly under the central extension of (\widehat{\mathfrak g}).

The author’s goal is to replace the affine Kac‑Moody algebra by its Drinfeld‑Jimbo quantum analogue (U_q(\widehat{\mathfrak g})) and to construct a corresponding “quantum” family of Dirac operators. The key steps are:

1. Quantum affine algebra and the adjoint module.
   The quantum affine algebra (U_q(\widehat{\mathfrak g})) is generated by (E_i, F_i, K_i^{\pm1}) (i runs over the simple roots) together with a central element C. It carries a Hopf‑algebra structure (coproduct, antipode, counit) and a universal R‑matrix. The adjoint representation (\mathfrak{ad}_q) is the q‑deformation of the classical adjoint module; its underlying vector space has the same dimension as (\mathfrak g) but the action is twisted by the R‑matrix, and its weight spaces are measured by q‑integers (