Virtual Artin groups

Paolo Bellingeri (Université de Caen Normandie)

Monday 14th March 16:00-17:00 Online + Maths 110B

Abstract

Virtual braid groups were introduced as a braid counterpart of virtual knots. From the combinatorial point of view it is interesting to remark that the virtual braid group VBn admits two surjective homomorphisms onto the symmetric group Sn. The kernels of these two homomorphisms have different meanings and applications: the first one, the virtual pure braid group VPn, coincides with the quasitriangular group QTrn considered by L Bartholdi, B Enriquez, P Etingof, E Rains in relation with Yang–Baxter equations, while the second one, usually denoted KBn, is an Artin group and it turns out to be a powerful tool to study combinatorial properties of VBn.
Starting from the observation that the standard presentation of a virtual braid group mixes the presentations of the corresponding braid group Bn and of the corresponding symmetric group Sn together with the action of the symmetric group on its root system, we define for any Coxeter graph Γ a virtual Artin group VA[Γ] with a presentation that mixes the standard presentations of the Artin group A[Γ] and of the Coxeter group W[Γ] together with the action of W[Γ] on its root system. As in the case of VBn, we will define two surjective homomorphisms from VA[Γ] to W[Γ]: we will provide group presentations for these kernels (completely determined by root systems) and we will show several and general results on virtual Artin groups.
This is a joint work with Luis Paris and Anne-Laure Thiel.

The talk will be preceded by a tea time at 3:45pm. The Zoom link for the seminar is https://uofglasgow.zoom.us/j/98078798957 and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).

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