Quantum toroidal algebras, quantum affine algebras, and their representation theory by Laurie, D

Subjects: “…Representations of quantum groups…”

Langlands duality for representations and quantum groups at a root of unity by McGerty, K

“…We give a representation-theoretic interpretation of the Langlands character duality of Frenkel and Hernandez, and show that the "Langlands branching multiplicities" for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules.…”

Hall algebras and the Quantum Frobenius by McGerty, K

“…Lusztig has constructed a Frobenius morphism for quantum groups at an $\ell$-th root of unity, which gives an integral lift of the Frobenius map on universal enveloping algebras in positive characteristic. …”

The Kronecker quiver and bases of quantum affine sl_2 by McGerty, K

“…We compare various bases of the affine quantum group $\mathbfU^ (\hat\mathfrak{sl}_2)$ in the context of the Kronecker quiver, and relate them to the Drinfeld presentation.…”

On the geometric realization of the inner product and canonical basis for quantum affine $\mathfrak{sl}_n$ by McGerty, K

“…We give a geometric interpretation of the inner product on the modified quantum group of $\hat{\mathfrak{sl}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.…”

Cells in quantum affine sl_n by McGerty, K

“…Using the geometric construction of the quantum group due to Lusztig and Ginzburg--Vasserot, we describe explicitly the two-sided cells, the number of left cells in a two--sided cell, and the asymptotic algebra, verifying conjectures of Lusztig.…”