Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models

TL;DR

This paper constructs a quantum group at a root of unity corresponding to (1,p) Virasoro logarithmic models, demonstrating that its tensor products match the fusion rules of these models, thus linking algebraic and conformal field theory structures.

Contribution

It introduces a Kazhdan--Lusztig-dual quantum group as a Lusztig limit of quantum sl(2) at roots of unity and shows its tensor products reproduce the fusion rules of (1,p) Virasoro logarithmic models.

Findings

Tensor products of quantum group representations match fusion rules.

The quantum group is a Hopf algebra at the Lusztig limit.

Fusion rules are realized algebraically via tensor products.

Abstract

We introduce a Kazhdan--Lusztig-dual quantum group for (1,p) Virasoro logarithmic minimal models as the Lusztig limit of the quantum sl(2) at pth root of unity and show that this limit is a Hopf algebra. We calculate tensor products of irreducible and projective representations of the quantum group and show that these tensor products coincide with the fusion of irreducible and logarithmic modules in the (1,p) Virasoro logarithmic minimal models.