Wednesday, July 17, 2013

1307.4227 (Glen Burella et al.)

Graphical Calculus for the Double Affine Q-Dependent Braid Group    [PDF]

Glen Burella, Paul Watts, Vincent Pasquier, Jiri Vala
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.
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