University of Sydney Algebra Seminar

Azat Gainutdinov (Universitat Hamburg)

Friday 31 July, 2-3pm, Place: Carslaw 373

Small quantum groups: modular group representations and CFT

The small quantum group is a finite-dimensional quotient of the Kac-De Concini specialisation of the quantum algebra \(U_q\mathfrak{g}\) at roots of unity, where \(\mathfrak{g}\) is a finite-dimensional simple Lie algebra. This quotient is known to be a ``factorisable'' Hopf algebra and has an interesting connection to Logarithmic Conformal Field Theory based on a particular non-rational VOA: the Lyubashenko's modular group action on the small quantum group's centre is equivalent to the one on the conformal blocks. There is also a certain equivalence of categories and a match for fusion rules but the equivalence can not be extended to braided tensor categories, as the small quantum group has no R-matrix. I will present an explicit coassociator and a new R-matrix such that the small quantum group becomes a quasi-triangular quasi-Hopf algebra. This allows us to prove the braided tensor categories equivalence. I will also discuss related problems on the invariant theory for this small quantum group in the rank-1 case.