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Noncommutative reflections
Groups generated by reflections are ubiquitous in mathematics and play a major role in modern representation theory. We introduce their noncommutative-geometric generalisation via the notion of a reflection of a noncommutative space (graded associative algebra) A. If S is a set of reflections of A, then S-twisted derivations of A, S, and A may generate an algebra with triangular decomposition, which serves as a noncommutative analogue of a nil Hecke algebra. We work out the case where A is an n-dimensional quantum plane in more detail. We show that A can be obtained as a Drinfeld twist of an ordinary polynomial algebra and that the triangular decomposition property holds for the odd nil Hecke algebra. This yields a family of "twisted" Coxeter groups and explains two recent constructions: one due to Berenstein and Bazlov and independently due to Kirkman, Kuzmanovich and Zhang, the other due to Ellis, Khovanov and Lauda.
- Speaker
- Yuri Bazlov (Manchester)