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

Markov Traces

A Markov trace on the affine braid group is a trace functional which respects the inclusions 12 where k k+1 b 1 k b 1 k k+1 (5.1) More precisely, a Markov trace on the affine braid group with parameters z,Q1,Q2, is a sequence of functions mtk:k such that

(1) mt1(1)=1,
(2) mtk+1(b)=mtk(b), for bk,
(3) mtk(b1b2)=mtk(b2b1), for b1,b2k,
(4) mtk+1(bTk)=zmtk(b), for bk,
(5) mtk+1(b(Xεk+1)r)=Qrmtk(b), for bk,
where Xεk+1= TkTk-1T2 Xε1T2-1 Tk-1-1 Tk-1= 1 2 k+1

If M is a finite dimensional U=Uh𝔤 module and aEndU(M) the quantum trace of a on M (see [LRa1977, §3] and [CPr1994, Def. 4.2.9]) is the trace of the action of ehρa on M, trq(a)=Tr (ehρa,M), anddimq(M) =trq(idM)= Tr(ehρ,M) (5.2) is the quantum dimension of M. The first step of the standard argument for proving Weyl’s dimension formula [BtD1985, VI Lemma 1.19], shows that the quantum dimension of the finite dimensional irreducible Uh𝔤-module L(μ) is dimq(L(μ)) =Tr(ehρ,L(μ)) = α>0 eh2μ+ρ,α- e-h2μ+ρ,α eh2ρ,α- e-h2ρ,α = α>0 [μ+ρ,α] [ρ,α] , (5.3) where q=eh/2 and [d]=(qd-q-d)/(q-q-1) for a positive integer d.

Let μ,νP+ be dominant integral weights. Let M=L(μ) and V=L(ν) and let Φk be the representation of k defined in Proposition 3.1. Then the functions mtk: k b trq(Φk(b)) dimq(M)dimq(V)k form a Markov trace on the affine braid group with parameters z= qν,ν+2ρ dimq(V) and Qr=λ qr(λ,λ+2ρ-μ,μ+2ρ-ν,ν+2ρ) dimq(L(λ))cμνλ dimq(L(μ)) dimq(L(ν)) , where the positive integers cμνλ and the sum in the expression for Qr are as in the tensor product decomposition L(μ)L(ν)= λL(λ)cμνλ.


The fact that mtk as defined in the statement of the Theorem satisfies (1)–(4) in the definition of a Markov trace follows exactly as in [LRa1977, Theorem 3.10c]. The formula for the parameter z is derived in [LRa1977, (3.9) and Thm. 3.10(2)].

It remains to check (5). The proof is a combination of the argument used in [Ore1999, Theorem 5.3] and the argument in the proof of [LRa1977, Theorem 3.10c]. Let εk:EndU(MVk)EndU(MV(k-1)) be given by εk(z)= (idMV(k-1)ě) (zid) (5.4) where ě: VV* xϕ dimq(V)-1ϕ(ehρx). If V is simple then ě is the unique Uh𝔤-invariant projection onto the invariants in VV*. Pictorially, εk ( z 1 k ) = z = εk(z) 1 k-1 . The argument of [LRa1977, Theorem 3.10b] shows that mtk(b)= mtk-1 (εk-1(b)) ,ifbk. (5.5) Since ε1((Xε1)r) is a Uh𝔤-module homomorphism from M to M and, since M is simple, Schur’s lemma implies that rloops { =ε1((Xε1)r) =ξ·idM,for some ξ. Let Ri=idV(i-1)ŘVMidV(k+1)-i. Then (Xεk+1)r= (RkR1)-1 (Xε1)r (RkR1) and mtk+1(b(Xεk+1)r) = mtk(εk(b(Xεk+1)r)) = mtk ( εk ( b (RkR1)-1 (Xε1)r (RkR1) ) ) = mtk ( b(RkR1)-1 ε1((Xε1)r) (RkR1) ) = mtk ( b(RkR1)-1 ξ·idM RkR1 ) =ξ·mtk(b). This last calculation is more palatable in a pictorial format, mtk+1 ( b (Xεk+1)r 1 k ) = mtk+1 ( b (Xεk+1)r 1 k ) = mtk ( b (Xεk+1)r 1 k ) = ξ·mtk ( b 1 k ) = ξ·mtk ( b 1 k ) . It remains to calculate the constant ξ. By (2.14), (Xε1)r= (Ř02)r= (λqc(λ)Pμνλ)r =λqrc(λ) Pμνλ, where c(λ)=λ,λ+2ρ-μ,μ+2ρ-ν,ν+2ρ and Pμνλ is the projection onto the L(λ)cμνλ component in the decomposition of MV=L(μ)L(ν). Thus ξ = mt0(ξ·idM)= mt1((Xε1)r)= 1dimq(M)dimq(V) trq(λqrc(λ)Pμνλ) = 1dimq(M)dimq(V) trq(λqrc(λ)cμνλidL(λ)) = λqrc(λ) cμνλ dimq(L(λ)) dimq(L(μ)) dimq(L(ν)) .

There is another formula [TWe1217386, Lemma (3.51)], for the constant Q1 in Theorem 5.1, namely, Q1= wW (-1)(w) qμ+ρ,w(ν+ρ) wW (-1)(w) qμ+ρ,wρ , (5.6) where W is the Weyl group of 𝔤.

Let mtk be as in Theorem 5.1 and let 𝒵k=EndU(MVk). Then mtk is the restriction of the linear functional mtk: 𝒵k a trq(a) dimq(M) dimq(Vk) (5.7) to Φk(k). Since MVk is a finite dimensional semisimple module 𝒵k is a finite dimensional semisimple algebra. The weights of the Markov trace mt are the constants tλ/μ defined by mtk=λ tλ/μ χ𝒵kλ/μ, (5.8) where χ𝒵kλ/μ are the irreducible characters of 𝒵k.

Let M=L(μ) and V=L(ν) be finite dimensional irreducible Uh𝔤-modules. The weights of the Markov trace on the affine braid group defined in Theorem 5.1 are tλ/μ= dimq(L(λ)) dimq(L(μ)) dimq(V)k .


Since MVk is finite dimensional and semisimple the algebra 𝒵k=EndU(MVk) is a finite dimensional semisimple algebra. Schur’s lemma can be used to show that, as a (Uh𝔤,𝒵k) bimodule, MVk λL(λ) λ/μ, (5.9) where the λ/μ are the irreducible 𝒵k modules. In the notation of (4.1), λ/μ=Fλ(L(μ)) and χ𝒵kλ/μ is the character of λ/μ. Taking the quantum trace on both sides of (5.9) gives tr1(a)=Tr (e-hρa)= λTr (e-hρ,L(λ)) χ𝒵kλ/μ (a)=λtr (L(λ)) χ𝒵kλ/μ (a). The result follows by dividing both sides by dimq(L(μ))dimq(V)k.

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