Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last update: 15 October 2012
The affine braid group action
The affine braid group is the group given by generators
and with relations
The generators and can be
identified with the diagrams
For define
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
where
with as in (1.24). The
action commutes with the
on
Proof.
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:
and
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.
Notes and References
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.