Yuri Bazlov (University of Manchester)
Groups generated by reflections are ubiquitous in mathematics, and there are strong connections from reflection groups to geometry and topology. We are interested in analogues of reflection groups in noncommutative geometry. More specifically, we are looking for deformations of reflection groups which would act on noncommutative spaces and still satisfy the defining property of reflection groups - that their invariants are polynomial.
In this talk I will focus on the n-dimensional quantum plane, which is one of the simplest kinds of a noncommutative space. It can be obtained from the usual affine space by applying Moyal-type quantisation, expressed in the language of quantum groups as a Drinfeld twist. If the parameter q is a root of unity, this method can quantise certain finite reflection groups. Moreover, if q = -1, some of the quantum groups thus obtained are ordinary finite groups, which explains a recent construction due to Bazlov-Berenstein and Kirkman-Kuzmanovich-Zhang.