Affine Braids, Markov Traces and the Category 𝒪

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

Last update: 22 December 2013

Affine Braid Group Representations and the Functors Fλ

There are three common ways of depicting affine braids [Cri1997], [GLa1997], [Jon1994]:

(a) As braids in a (slightly thickened) cylinder,
(b) As braids in a (slightly thickened) annulus,
(c) As braids with a flagpole.
See Figure 1. The multiplication is by placing one cylinder on top of an- other, placing one annulus inside another, or placing one flagpole braid on top of another. Figure 1. These are equivalent formulations: an annulus can be made into a cylinder by turning up the edges, and a cylindrical braid can be made into a flagpole braid by putting a flagpole down the middle of the cylinder and pushing the pole over to the left so that the strings begin and end to its right.

The affine braid group is the group k formed by the affine braids with k strands. The affine braid group k can be presented by generators T1,T2,,Tk-1 and Xε1 Ti= i i+1 andXε1= (3.1) with relations (3.2a) TiTj=TjTi, if|i,j|>1, (3.2b) TiTi+1Ti=Ti+1TiTi+1, for1ik-2, (3.2c) Xε1T1Xε1T1 =T1Xε1T1Xε1, (3.2d) Xε1Ti=Ti Xε1,for2 ik-1. Define Xεi=Ti-1 Ti-2T2T1 Xε1T1T2 Ti-1,1ik. (3.3) By drawing pictures of the corresponding affine braids it is easy to check that the Xεi all commute with each other and so X=Xεi|1ik is an abelian subgroup of k. Let Lk be the free abelian group generated by ε1,,εk. Then L={λ1ε1++λkεk|λi} andX={Xλ|λL}, (3.4) where Xλ=(Xε1)λ1(Xε2)λ2(Xεk)λk, for λL.

The k Module MVk

Let Uh𝔤 be the Drinfeld-Jimbo quantum group associated to a finite dimensional complex semisimple Lie algebra 𝔤. Let M be a Uh𝔤-module in the category 𝒪 and let V be a finite dimensional Uh𝔤 module. Define Ři, 1ik-1, and Ř02 in EndUh𝔤(MVk) by Ři=idM idV(i-1) ŘVV idV(k-i-1) andŘ02= (ŘMVŘVM) idV(k-1).

The following proposition is well known (see [Suz2000, Prop. B.2], [LRa1977, Prop. 2.19], or [Res1990]).

The map defined by Φ: k EndUh𝔤(MVk) Ti Ři, 1ik-1, Xε1 Ř02, makes MVk into a right k module.


It is necessary to show that

(a) ŘiŘj=ŘjŘi, if |i-j|>1,
(b) Ř02Ři=ŘiŘ02, i>2,
(c) ŘiŘi+1Ři= Ři+1ŘiŘi+1, 1ik-2,
(d) Ř02Ř1Ř02Ř1= Ř1Ř02Ř1Ř02.
The relations (a) and (b) follow immediately from the definitions of Ři and Ř02 and (c) is a particular case of the braid relation (2.12). The relation (d) is also a consequence of the braid relation: Ř02Ř1 Ř02Ř1 = (ŘMVŘVMid) (idŘVV) (ŘMVŘVMid) (idŘVV) = (ŘMVid) (idŘMV) (ŘVVid) (idŘVM) (ŘVMid) (idŘVV) = (idŘVV) (ŘMVid) (idŘMV) (ŘVVid) (idŘVM) (ŘVMid) = (idŘVV) (ŘMV ŘVMid) (idŘVV) (ŘMV ŘVMid) = Ř1Ř02 Ř1Ř02, or equivalently, Ř02Ř1Ř02Ř1= = = = =Ř1Ř02Ř1Ř02.

A k module N is calibrated if the abelian group X defined in (3.4) acts semisimply on N, i.e. if N has a basis of simultaneous eigenvectors for the action of Xε1,,Xεk.

If M and V are finite dimensional Uh𝔤 modules then the k module MVk defined in Proposition 3.1 is calibrated.


Let P+ be the set of dominant integral weights. Since M and V are finite dimensional the Uh𝔤-module MVi is semisimple for every 1ik and MVi= λP+ (MVi)[λ] λP+ L(λ)mλ, where mλ0 and (MVi)[λ]L(λ)mλ. Given a basis of MV(i-1) which respects the decomposition MV(i-1)=μ(MV(i-1))[μ] one can construct a basis of MVi which respects the decomposition MVi= (MV(i-1)) V=λ,μ,ν ((MV(i-1))[μ]V[ν])[λ]. Since ((MV(i-1))[μ]V[ν])[λ](MVi)[λ] this new basis respects the decomposition MVi=λ(MVi)[λ]. This procedure produces, inductively, a basis B of MVk which respects the decompositions MVk= (MVi) V(k-i)= λ(MVi)[λ] V(k-i), for all 0ik. The central element e-hρu in Uh𝔤 acts on (MVi)[λ] by the constant q-λ,λ+2ρ. From (2.10), (2.11) and (2.14) it follows that Xεi acts on MVk by Ři-1Ř1 Ř02Ř1 Ři-1 = ŘMV(i-1),V ŘV,MV(i-1) idV(k-i) = (CMV(i-1)CV) CMVi-1 idV(k-i) = λ,μ,ν qλ,λ+2ρ-μ,μ+2ρ-ν,ν+2ρ PμνλidV(k-i) where Pμνλ:MidViMidVi is the projection onto ((MV(i-1))[μ]V[ν])[λ]. Thus Xεi acts diagonally on the basis B.

Define an anti-involution on k by θ(Ti)=Ti andθ(Xλ) =Xλ, for 1ik-1 and λL. A contravariant form on a k module N is a symmetric bilinear form ,:N×N such that bn1,n2= n1,θ(b)n2 forn1,n2N, bk. Suppose M is a Uh𝔤-module in the category 𝒪 and V is a finite dimensional Uh𝔤 module. Let ,M and ,V be Uh𝔤-contravariant forms on M and V respectively. By (2.16), (v1v2) ŘVV,v1 v2 = v1v2, (v1v2) ŘVV , for v1,v2,v1,v2V, and (mv)ŘMV ŘVM,mv = (mv)ŘMV, (mv)ŘMV = mv,(mv) ŘMVŘVM , for m,mM, v,vV. Thus it follows that the form , on MVk given by mv1vk, mv1 vk = m,mM v1,v1V v2,v2V vk,vkV, (3.5) for m,mM, vi,viV is a k contravariant form on the k module MVk.

The Functor Fλ

Fix a finite dimensional Uh𝔤 module V and an integrally dominant weight λ in 𝔥*. Let 𝒪k be the category of finite dimensional k modules and define a functor Fλ: 𝒪 𝒪k M HomUh𝔤(M(λ),MVk) . (3.5) Since an element of HomUh𝔤(M(λ),MVk) is determined by the image of a generating highest weight vector of M(λ), the space Fλ(M) can be identified with the vector space of highest weight vectors of weight λ in MVk. If λ is integrally dominant the highest possible weight of (MVk)[λ] is λ. Thus, viewing Fλ(M) as the space of highest weight vectors of weight λ in MVk, HomUh𝔤(M(λ),MVk) ((MVk)[λ])λ ( MVk 𝔫-(MVk) ) λ where we use the notation 𝔫- (MVk)= iYi (MVk), (3.7) where the Yi are the Chevalley generators of 𝔫-. In the case of U𝔤-modules, the notation 𝔫-(MVk) is self explanatory—the notation in (3.7) is simply a way to define the same object for the quantum group Uh𝔤.

The functor Fλ is the composition of two functors: the functor ·Vk and the functor HomU(M(λ),·). The first is exact since Vk is finite dimensional and the second is exact because when λ is integrally dominant M(λ) is projective, see [Jan1980, p. 72]. Thus ifλis integrally dominant, the functor Fλis exact. (3.8)

Notes and references

This is a typed version of the paper Affine Braids, Markov Traces and the Category 𝒪 by Rosa Orellana and Arun Ram*.

*Research supported in part by National Science Foundation grant DMS-9971099, the National Security Agency and EPSRC grant GR K99015.

This paper is a slightly revised version of a preprint of 2001. We thank F. Goodman, A. Henderson and an anonymous referee for their very helpful comments on the original preprint.

page history