# Graphical Calculus for the Double Affine Q-Dependent Braid Group

Abstract : We define a double affine $Q$-dependent braid group. This group is constructed by appending to the braid group a set of operators $Q_i$, before extending it to an affine $Q$-dependent braid group. We show specifically that the elliptic braid group and the double affine Hecke algebra (DAHA) can be obtained as quotient groups. Complementing this we present a pictorial representation of the double affine $Q$-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation we can fully describe any DAHA. Specifically, we graphically describe the parameter $q$ upon which this algebra is dependent and show that in this particular representation $q$ corresponds to a twist in the ribbon.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal-cea.archives-ouvertes.fr/cea-01059518
Contributor : Emmanuelle de Laborderie <>
Submitted on : Monday, September 1, 2014 - 11:05:05 AM
Last modification on : Monday, February 10, 2020 - 6:13:39 PM

### Identifiers

• HAL Id : cea-01059518, version 1
• ARXIV : 1307.4227

### Citation

Glen Burella, Paul Watts, Vincent Pasquier, Jiri Vala. Graphical Calculus for the Double Affine Q-Dependent Braid Group. 2013. ⟨cea-01059518⟩

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