Virtual Artin groups

Paolo Bellingeri (Université de Caen Normandie)

Monday 14th March, 2022 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 VB_n admits two surjective homomorphisms onto the symmetric group S_n. The kernels of these two homomorphisms have different meanings and applications: the first one, the virtual pure braid group VP_n, coincides with the quasitriangular group QTr_n considered by L Bartholdi, B Enriquez, P Etingof, E Rains in relation with Yang–Baxter equations, while the second one, usually denoted KB_n, is an Artin group and it turns out to be a powerful tool to study combinatorial properties of VB_n.
Starting from the observation that the standard presentation of a virtual braid group mixes the presentations of the corresponding braid group B_n and of the corresponding symmetric group S_n 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 VB_n, 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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