# Manchester Geometry Seminar 2011/2012

**Thursday 29 September 2011. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm*
##
Reflection Groups on Noncommutative Spaces

Yuri Bazlov (University of Manchester)

`yuri.bazlov@manchester.ac.uk`

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.