Last update: 15 October 2012
The affine braid group is the group given by generators and with relations
The generators and can be identified with the diagrams
The pictorial computation
shows that the pairwise commute.
Let be a finite-dimensional complex Lie algebra with a symmetric nondegenerate adinvariant bilinear form, and let be the Drinfel'd-Jimbo quantum group corresponding to The quantum group is a ribbon Hopf algebra with invertible
where (see, [LRa1997, Corollary (2.15)]). For and the map
is a isomorphism. The quasitriangularity of ribbon Hopf algebra provides the braid relation (see, for example, [ORa0401317, (2.12)]),
Let be a finite-dimensional complex Lie algebra with a symmetric nondegenerate bilinear form, let be the corresponding Drinfeld-Jimbo quantum group and let be the center of Let and be Then is a with action given by
with as in (1.24). The action commutes with the on
The relations (1.18) and (1.21) are consequences of the definition of the action of and The relations (1.19) and (1.20) follow from to following computations:
Let be the ribbon element in For a define
(see [Dri1970, Prop. 3.2]). If is a generated by a highest weight vector of weight then
(see [LRa1997, Prop. 2.14] or [Dri1970, Prop. 5.1]). From (1.27) and the relation (1.26) it follows that if and are finite-dimensional irreducible of highest weights and respectively, then acts on the component of the decomposition
By the definition of in (1.23),
so that, by (1.26), the eigenvalues of are functions of the eigenvalues of the Casimir.
This is an excerpt from a paper entitled Affine and degenerate affine BMW algebras: Actions on tensor space written by Zajj Daugherty, Arun Ram and Rahbar Virk.