The affine braid group action

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 15 October 2012

The affine braid group action

The affine braid group Bk is the group given by generators T1,T2,,Tk-1 and Xε1, with relations

TiTj = TjTi, ifji±1, (1.18) TiTi+1 = Ti+1Ti Ti+1, fori=1,2,, k-2, (1.19) Xε1T1 Xε1T1 = T1Xε1 T1Xε1, (1.20) Xε1Ti = TiXε1, fori=2,3,, k-1. (1.21)

The generators Ti and Xε1 can be identified with the diagrams

Ti= i i+1 andXε1= . (1.22)

For i=1,,k define

Xεi= Ti-1Ti-2 T2T1Xε1 T1T2Ti-2 Ti-1= i . (1.23)

The pictorial computation

XεjXεi= i i = i i =XεiXεj

shows that the Xεi pairwise commute.

Let 𝔤 be a finite-dimensional complex Lie algebra with a symmetric nondegenerate adinvariant bilinear form, and let U=Uh𝔤 be the Drinfel'd-Jimbo quantum group corresponding to 𝔤. The quantum group U is a ribbon Hopf algebra with invertible -matrix

=R1R2 inUU, and ribbon elementv=e-hρ u,

where u=S(R2)R1 (see, [LRa1997, Corollary (2.15)]). For U-modules M and N, the map

RMN: MN NM mn R2nR1 m M N N M (1.24)

is a U-module isomorphism. The quasitriangularity of ribbon Hopf algebra provides the braid relation (see, for example, [ORa0401317, (2.12)]),

M N P P N M = M N P P N M ( RMNidP ) ( idN RMP ) ( RNPidM ) = ( idM RNP ) ( RMP idN ) ( idP RMN ) .

Let 𝔤 be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form, let U=Uh𝔤 be the corresponding Drinfeld-Jimbo quantum group and let C=Z(U) be the center of Uh𝔤. Let M and V be U-modules. Then MVk is a CBk-module with action given by

Φ: CBk EndU ( MVk ) Ti Ri, Xε1 R02, z zM, 1ik-1, (1.25)

where zM= idVk,

Ri=idM idV(i-1) RVV idV(k-i-1) and R020 ( RMV RVM ) idV(k-1),

with RMV as in (1.24). The CBk action commutes with the U-action on MVk.


The relations (1.18) and (1.21) are consequences of the definition of the action of Ti and Xε1. The relations (1.19) and (1.20) follow from to following computations:

Ri Ri+1 Ri = = = Ri+1 Ri Ri+1


R02 R1 R02 R1 = = = = = R1 R02 R1 R02

Let v=e-hρu be the ribbon element in U=Uh𝔤. For a Uh𝔤-module M define

CM:MM mvm so that CMN= ( RMN RNM ) -1 (CMCN) (1.26)

(see [Dri1970, Prop. 3.2]). If M is a Uh𝔤-module generated by a highest weight vector vλ+ of weight λ, then

CM= q - λ,λ+2ρ idM,whereq= eh/2 (1.27)

(see [LRa1997, Prop. 2.14] or [Dri1970, Prop. 5.1]). From (1.27) and the relation (1.26) it follows that if M=L(μ) and N=L(ν) are finite-dimensional irreducible Uh𝔤-modules of highest weights μ and ν respectively, then RMN RNM acts on the L(λ)-isotypic component L(λ) cμνλ of the decomposition

L(μ)L(ν)= λ L(λ) cμνλ by the scalar q λ,λ+ 2ρ - μ,μ+ 2ρ - ν,ν+ 2ρ . (1.28)

By the definition of Xεi in (1.23),

Φ (Xεi) = R M V(i-1) ,V R V,M V(i-1) = , (1.29)

so that, by (1.26), the eigenvalues of Φ(Xεi) 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.

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