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Graphical Calculus for the Double Affine Q-Dependent Braid Group

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.

preprint2013arXivOpen access
Glen BurellaPaul WattsVincent PasquierJiri Vala
math-phmath.MP
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Glen BurellaPaul WattsVincent PasquierJiri Vala

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