# 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.