Affine Braids, Markov Traces and the Category π’ͺ

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 22 December 2013

Affine Braid Group Representations and the Functors FΞ»

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 β„¬βˆΌk formed by the affine braids with k strands. The affine braid group β„¬βˆΌk can be presented by generators T1,T2,…,Tk-1 and XΞ΅1 Ti= i i+1 andXΞ΅1= (3.1) with relations (3.2a) TiTj=TjTi, if |i,j|>1, (3.2b) TiTi+1Ti=Ti+1TiTi+1, for 1≀i≀k-2, (3.2c) XΞ΅1T1XΞ΅1T1 =T1XΞ΅1T1XΞ΅1, (3.2d) XΞ΅1Ti=Ti XΞ΅1,for 2 ≀i≀k-1. Define XΞ΅i=Ti-1 Ti-2β‹―T2T1 XΞ΅1T1T2β‹― Ti-1,1≀i≀k. (3.3) By drawing pictures of the corresponding affine braids it is easy to check that the XΞ΅i all commute with each other and so X=⟨XΞ΅i | 1≀i≀k⟩ is an abelian subgroup of β„¬βˆΌk. Let Lβ‰…β„€k be the free abelian group generated by Ξ΅1,…,Ξ΅k. Then L={Ξ»1Ξ΅1+β‹―+Ξ»kΞ΅k | λiβˆˆβ„€} andX={Xλ |β€‰Ξ»βˆˆL}, (3.4) where XΞ»=(XΞ΅1)Ξ»1(XΞ΅2)Ξ»2β‹―(XΞ΅k)Ξ»k, for λ∈L.

The β„¬βˆΌk Module MβŠ—VβŠ—k

Let Uh𝔀 be the Drinfeld-Jimbo quantum group associated to a finite dimensional complex semisimple Lie algebra 𝔀. Let M be a Uh𝔀-module in the category π’ͺ and let V be a finite dimensional Uh𝔀 module. Define Ři, 1≀i≀k-1, and Ř02 in EndUh𝔀(MβŠ—VβŠ—k) by Ři=idMβŠ— idVβŠ—(i-1) βŠ—Ε˜VVβŠ— idVβŠ—(k-i-1) andŘ02= (ŘMVŘVM) βŠ—idVβŠ—(k-1).

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

The map defined by Ξ¦: β„¬βˆΌk ⟢ EndUh𝔀(MβŠ—VβŠ—k) Ti ⟼ Ři, 1≀i≀k-1, XΞ΅1 ⟼ Ř02, makes MβŠ—VβŠ—k into a right β„¬βˆΌk module.

Proof.

It is necessary to show that

(a) ŘiŘj=ŘjŘi, if |i-j|>1,
(b) Ř02Ři=ŘiŘ02, i>2,
(c) ŘiŘi+1Ři= Ři+1ŘiŘi+1, 1≀i≀k-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 = (ŘMVŘVMβŠ—id) (idβŠ—Ε˜VV) (ŘMVŘVMβŠ—id) (idβŠ—Ε˜VV) = (ŘMVβŠ—id) (idβŠ—Ε˜MV) (ŘVVβŠ—id) (idβŠ—Ε˜VM) ⏟ (ŘVMβŠ—id) (idβŠ—Ε˜VV) = (idβŠ—Ε˜VV) (ŘMVβŠ—id) (idβŠ—Ε˜MV) ⏟ (ŘVVβŠ—id) (idβŠ—Ε˜VM) (ŘVMβŠ—id) ⏟ = (idβŠ—Ε˜VV) (ŘMV ŘVMβŠ—id) (idβŠ—Ε˜VV) (ŘMV ⏞ ŘVMβŠ—id) = Ř1Ř02 Ř1Ř02, or equivalently, Ř02Ř1Ř02Ř1= = = = =Ř1Ř02Ř1Ř02.

β–‘

A β„¬βˆΌk 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 XΞ΅1,…,XΞ΅k.

If M and V are finite dimensional Uh𝔀 modules then the β„¬βˆΌk module MβŠ—VβŠ—k defined in Proposition 3.1 is calibrated.

Proof.

Let P+ be the set of dominant integral weights. Since M and V are finite dimensional the Uh𝔀-module MβŠ—VβŠ—i is semisimple for every 1≀i≀k and MβŠ—VβŠ—i= ⨁λ∈P+ (MβŠ—VβŠ—i)[Ξ»] β‰…β¨Ξ»βˆˆP+ L(Ξ»)βŠ•mΞ», where mΞ»βˆˆβ„€β‰₯0 and (MβŠ—VβŠ—i)[Ξ»]β‰…L(Ξ»)βŠ•mΞ». Given a basis of MβŠ—VβŠ—(i-1) which respects the decomposition MβŠ—VβŠ—(i-1)=⨁μ(MβŠ—VβŠ—(i-1))[ΞΌ] one can construct a basis of MβŠ—VβŠ—i which respects the decomposition MβŠ—VβŠ—i= (MβŠ—VβŠ—(i-1)) βŠ—V=⨁λ,ΞΌ,Ξ½ ((MβŠ—VβŠ—(i-1))[ΞΌ]βŠ—V[Ξ½])[Ξ»]. Since ((MβŠ—VβŠ—(i-1))[ΞΌ]βŠ—V[Ξ½])[Ξ»]βŠ†(MβŠ—VβŠ—i)[Ξ»] this new basis respects the decomposition MβŠ—VβŠ—i=⨁λ(MβŠ—VβŠ—i)[Ξ»]. This procedure produces, inductively, a basis B of MβŠ—VβŠ—k which respects the decompositions MβŠ—VβŠ—k= (MβŠ—VβŠ—i)βŠ— VβŠ—(k-i)= ⨁λ(MβŠ—VβŠ—i)[Ξ»] βŠ—VβŠ—(k-i), for all 0≀i≀k. The central element e-hρu in Uh𝔀 acts on (MβŠ—VβŠ—i)[Ξ»] by the constant q-⟨λ,Ξ»+2ρ⟩. From (2.10), (2.11) and (2.14) it follows that XΞ΅i acts on MβŠ—VβŠ—k by Ři-1β‹―Ε˜1 Ř02Ř1β‹― Ři-1 = ŘMβŠ—VβŠ—(i-1),V ŘV,MβŠ—VβŠ—(i-1) βŠ—idVβŠ—(k-i) = (CMβŠ—VβŠ—(i-1)βŠ—CV) CMβŠ—VβŠ—i-1 βŠ—idVβŠ—(k-i) = βˆ‘Ξ»,ΞΌ,Ξ½ q⟨λ,Ξ»+2ρ⟩-⟨μ,ΞΌ+2ρ⟩-⟨ν,Ξ½+2ρ⟩ PΞΌΞ½Ξ»βŠ—idVβŠ—(k-i) where PΞΌΞ½Ξ»:MβŠ—idVβŠ—iβ†’MβŠ—idVβŠ—i is the projection onto ((MβŠ—VβŠ—(i-1))[ΞΌ]βŠ—V[Ξ½])[Ξ»]. Thus XΞ΅i acts diagonally on the basis B.

β–‘

Define an anti-involution on β„¬βˆΌk by θ∼(Ti)=Ti andθ∼(XΞ») =XΞ», for 1≀i≀k-1 and λ∈L. A contravariant form on a β„¬βˆΌk module N is a symmetric bilinear form ⟨,⟩:NΓ—Nβ†’β„‚ such that ⟨bn1,n2⟩= ⟨n1,θ∼(b)n2⟩ for n1,n2∈N,  bβˆˆβ„¬βˆΌk. Suppose M is a Uh𝔀-module in the category π’ͺ and V is a finite dimensional Uh𝔀 module. Let ⟨,⟩M and ⟨,⟩V be Uh𝔀-contravariant forms on M and V respectively. By (2.16), ⟨ (v1βŠ—v2) ŘVV,v1β€² βŠ—v2β€² ⟩ = ⟨ v1βŠ—v2, (v1β€²βŠ—v2β€²) ŘVV ⟩ , for v1,v2,v1β€²,v2β€²βˆˆV, and ⟨ (mβŠ—v)ŘMV ŘVM,mβ€²βŠ—vβ€² ⟩ = ⟨ (mβŠ—v)ŘMV, (mβ€²βŠ—vβ€²)ŘMV ⟩ = ⟨ mβŠ—v,(mβ€²βŠ—vβ€²) ŘMVŘVM ⟩ , for m,mβ€²βˆˆM, v,vβ€²βˆˆV. Thus it follows that the form ⟨,⟩ on MβŠ—VβŠ—k given by ⟨ mβŠ—v1βŠ—β‹―βŠ—vk, mβ€²βŠ—v1β€²βŠ—β‹―βŠ— vkβ€² ⟩ = ⟨m,mβ€²βŸ©M ⟨v1,v1β€²βŸ©V ⟨v2,v2β€²βŸ©V β‹― ⟨vk,vkβ€²βŸ©V, (3.5) for m,mβ€²βˆˆM, vi,viβ€²βˆˆV is a β„¬βˆΌk contravariant form on the β„¬βˆΌk module MβŠ—VβŠ—k.

The Functor FΞ»

Fix a finite dimensional Uh𝔀 module V and an integrally dominant weight Ξ» in π”₯*. Let π’ͺ∼k be the category of finite dimensional β„¬βˆΌk modules and define a functor FΞ»: π’ͺ ⟢ π’ͺ∼k M ⟼ HomUh𝔀(M(Ξ»),MβŠ—VβŠ—k) . (3.5) Since an element of HomUh𝔀(M(Ξ»),MβŠ—VβŠ—k) is determined by the image of a generating highest weight vector of M(Ξ»), the space FΞ»(M) can be identified with the vector space of highest weight vectors of weight Ξ» in MβŠ—VβŠ—k. If Ξ» is integrally dominant the highest possible weight of (MβŠ—VβŠ—k)[Ξ»] is Ξ». Thus, viewing FΞ»(M) as the space of highest weight vectors of weight Ξ» in MβŠ—VβŠ—k, HomUh𝔀(M(Ξ»),MβŠ—VβŠ—k)β‰… ((MβŠ—VβŠ—k)[Ξ»])Ξ»β‰… ( MβŠ—VβŠ—k 𝔫-(MβŠ—VβŠ—k) ) Ξ» where we use the notation 𝔫- (MβŠ—VβŠ—k)= βˆ‘iYi (MβŠ—VβŠ—k), (3.7) where the Yi are the Chevalley generators of 𝔫-. In the case of U𝔀-modules, the notation 𝔫-(MβŠ—VβŠ—k) is self explanatoryβ€”the notation in (3.7) is simply a way to define the same object for the quantum group Uh𝔀.

The functor FΞ» is the composition of two functors: the functor Β·βŠ—VβŠ—k and the functor HomU(M(Ξ»),Β·). The first is exact since VβŠ—k is finite dimensional and the second is exact because when Ξ» is integrally dominant M(Ξ») is projective, see [Jan1980, p. 72]. Thus if λ is integrally dominant, the functor  Fλ is exact. (3.8)

Notes and references

This is a typed version of the paper Affine Braids, Markov Traces and the Category π’ͺ 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.

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