Affine Braids, Markov Traces and the Category $𝒪$

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 $F_{\lambda }$

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. $$\begin{array}{c} \\ \\ \\ \text{Figure 1.}\end{array}$$ 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 ${\stackrel{\sim }{ℬ}}_k$ formed by the affine braids with $k$ strands. The affine braid group ${\stackrel{\sim }{ℬ}}_k$ can be presented by generators $T_1,T_2,\dots ,T_{k-1}$ and $X^{{\epsilon }_1}$ $$\begin{array}{cc}{T}_i=\begin{array}{c} i i+1 \end{array}\text{and}{X}^{{\epsilon }_1}=\begin{array}{c} \end{array}& \text{(3.1)}\end{array}$$ with relations $$\begin{array}{cc}\text{(3.2a)}& T_i{T}_j=T_j{T}_i,\text{if}\hspace{0.17em}|i,j|>1,\\ \text{(3.2b)}& T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},\text{for}\hspace{0.17em}1\le i\le k-2,\\ \text{(3.2c)}& X^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}{T}_1=T_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1},\\ \text{(3.2d)}& X^{{\epsilon }_1}{T}_i=T_i{X}^{{\epsilon }_1},\text{for}\hspace{0.17em}2\le i\le k-1\text{.}\end{array}$$ Define $$\begin{array}{cc}{X}^{{\epsilon }_i}=T_{i-1}{T}_{i-2}\cdots T_2{T}_1{X}^{{\epsilon }_1}{T}_1{T}_2\cdots T_{i-1},1\le i\le k\text{.}& \text{(3.3)}\end{array}$$ By drawing pictures of the corresponding affine braids it is easy to check that the $X^{{\epsilon }_i}$ all commute with each other and so $X=⟨X^{{\epsilon }_i}\hspace{0.17em}|\hspace{0.17em}1\le i\le k⟩$ is an abelian subgroup of ${\stackrel{\sim }{ℬ}}_k\text{.}$ Let $L\cong {\mathbb{Z}}^k$ be the free abelian group generated by ${\epsilon }_1,\dots ,{\epsilon }_k\text{.}$ Then $$\begin{array}{cc}L=\{{\lambda }_1{\epsilon }_1+\cdots +{\lambda }_k{\epsilon }_k\hspace{0.17em}|\hspace{0.17em}{\lambda }_i\in \mathbb{Z}\}\text{and}X=\left\{X^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in L\},& \text{(3.4)}\end{array}$$ where $X^{\lambda }={\left(X^{{\epsilon }_1}\right)}^{{\lambda }_1}{\left(X^{{\epsilon }_2}\right)}^{{\lambda }_2}\cdots {\left(X^{{\epsilon }_k}\right)}^{{\lambda }_k},$ for $\lambda \in L\text{.}$

The ${\stackrel{\sim }{ℬ}}_k$ Module $M\otimes V^{\otimes k}$

Let $U_h𝔤$ be the Drinfeld-Jimbo quantum group associated to a finite dimensional complex semisimple Lie algebra $𝔤\text{.}$ Let $M$ be a $U_h𝔤\text{-module}$ in the category $𝒪$ and let $V$ be a finite dimensional $U_h𝔤$ module. Define ${\mathit{Ř}}_i,$ $1\le i\le k-1,$ and ${\mathit{Ř}}_0^2$ in ${\text{End}}_{U_h𝔤}(M\otimes V^{\otimes k})$ by $${\mathit{Ř}}_i={\text{id}}_M\otimes {\text{id}}_V^{\otimes (i-1)}\otimes {\mathit{Ř}}_{VV}\otimes {\text{id}}_V^{\otimes (k-i-1)}\text{and}{\mathit{Ř}}_0^2=\left({\mathit{Ř}}_{MV}{\mathit{Ř}}_{VM}\right)\otimes {\text{id}}_V^{\otimes (k-1)}\text{.}$$

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

The map defined by $$\begin{array}{cccc}\Phi :& {\stackrel{\sim }{ℬ}}_k& ⟶& {\text{End}}_{U_h𝔤}(M\otimes V^{\otimes k})\\ & T_i& ⟼& {\mathit{Ř}}_i,& 1\le i\le k-1,\\ & X^{{\epsilon }_1}& ⟼& {\mathit{Ř}}_0^2,\end{array}$$ makes $M\otimes V^{\otimes k}$ into a right ${\stackrel{\sim }{ℬ}}_k$ module.

		Proof.
	It is necessary to show that (a) ${\mathit{Ř}}_i{\mathit{Ř}}_j={\mathit{Ř}}_j{\mathit{Ř}}_i,$ if $|i-j|>1,$ (b) ${\mathit{Ř}}_0^2{\mathit{Ř}}_i={\mathit{Ř}}_i{\mathit{Ř}}_0^2,$ $i>2,$ (c) ${\mathit{Ř}}_i{\mathit{Ř}}_{i+1}{\mathit{Ř}}_i={\mathit{Ř}}_{i+1}{\mathit{Ř}}_i{\mathit{Ř}}_{i+1},1\le i\le k-2,$ (d) ${\mathit{Ř}}_0^2{\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1={\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1{\mathit{Ř}}_0^2\text{.}$ The relations (a) and (b) follow immediately from the definitions of ${\mathit{Ř}}_i$ and ${\mathit{Ř}}_0^2$ and (c) is a particular case of the braid relation (2.12). The relation (d) is also a consequence of the braid relation: $$\begin{array}{ccc}{\mathit{Ř}}_0^2{\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1& =& ({\mathit{Ř}}_{MV}{\mathit{Ř}}_{VM}\otimes \text{id})(\text{id}\otimes {\mathit{Ř}}_{VV})({\mathit{Ř}}_{MV}{\mathit{Ř}}_{VM}\otimes \text{id})(\text{id}\otimes {\mathit{Ř}}_{VV})\\ & =& ({\mathit{Ř}}_{MV}\otimes \text{id})\underbrace{(\text{id}\otimes {\mathit{Ř}}_{MV})({\mathit{Ř}}_{VV}\otimes \text{id})(\text{id}\otimes {\mathit{Ř}}_{VM})}({\mathit{Ř}}_{VM}\otimes \text{id})(\text{id}\otimes {\mathit{Ř}}_{VV})\\ & =& \underbrace{(\text{id}\otimes {\mathit{Ř}}_{VV})({\mathit{Ř}}_{MV}\otimes \text{id})(\text{id}\otimes {\mathit{Ř}}_{MV})}\underbrace{({\mathit{Ř}}_{VV}\otimes \text{id})(\text{id}\otimes {\mathit{Ř}}_{VM})({\mathit{Ř}}_{VM}\otimes \text{id})}\\ & =& (\text{id}\otimes {\mathit{Ř}}_{VV})({\mathit{Ř}}_{MV}\overbrace{{\mathit{Ř}}_{VM}\otimes \text{id})(\text{id}\otimes {\mathit{Ř}}_{VV})({\mathit{Ř}}_{MV}}{\mathit{Ř}}_{VM}\otimes \text{id})\\ & =& {\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1{\mathit{Ř}}_0^2,\end{array}$$ or equivalently, $${\mathit{Ř}}_0^2{\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}={\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1{\mathit{Ř}}_0^2\text{.}$$ $\square$

A ${\stackrel{\sim }{ℬ}}_k$ 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 $X^{{\epsilon }_1},\dots ,X^{{\epsilon }_k}\text{.}$

If $M$ and $V$ are finite dimensional $U_h𝔤$ modules then the ${\stackrel{\sim }{ℬ}}_k$ module $M\otimes V^{\otimes k}$ 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 $U_h𝔤\text{-module}$ $M\otimes V^{\otimes i}$ is semisimple for every $1\le i\le k$ and $$M\otimes V^{\otimes i}=\underset{\lambda \in P^{+}}{⨁}{(M\otimes V^{\otimes i})}^{\left[\lambda \right]}\cong \underset{\lambda \in P^{+}}{⨁}{L\left(\lambda \right)}^{\oplus m_{\lambda }},$$ where $m_{\lambda }\in {\mathbb{Z}}_{\ge 0}$ and ${(M\otimes V^{\otimes i})}^{\left[\lambda \right]}\cong {L\left(\lambda \right)}^{\oplus m_{\lambda }}\text{.}$ Given a basis of $M\otimes V^{\otimes (i-1)}$ which respects the decomposition $M\otimes V^{\otimes (i-1)}=\underset{\mu }{⨁}{\left(M\otimes V^{\otimes (i-1)}\right)}^{\left[\mu \right]}$ one can construct a basis of $M\otimes V^{\otimes i}$ which respects the decomposition $$M\otimes V^{\otimes i}=(M\otimes V^{\otimes (i-1)})\otimes V=\underset{\lambda ,\mu ,\nu }{⨁}{({(M\otimes V^{\otimes (i-1)})}^{\left[\mu \right]}\otimes V^{\left[\nu \right]})}^{\left[\lambda \right]}\text{.}$$ Since ${({(M\otimes V^{\otimes (i-1)})}^{\left[\mu \right]}\otimes V^{\left[\nu \right]})}^{\left[\lambda \right]}\subseteq {(M\otimes V^{\otimes i})}^{\left[\lambda \right]}$ this new basis respects the decomposition $M\otimes V^{\otimes i}=\underset{\lambda }{⨁}{(M\otimes V^{\otimes i})}^{\left[\lambda \right]}\text{.}$ This procedure produces, inductively, a basis $B$ of $M\otimes V^{\otimes k}$ which respects the decompositions $$M\otimes V^{\otimes k}=(M\otimes V^{\otimes i})\otimes V^{\otimes (k-i)}=\underset{\lambda }{⨁}{(M\otimes V^{\otimes i})}^{\left[\lambda \right]}\otimes V^{\otimes (k-i)},$$ for all $0\le i\le k\text{.}$ The central element $e^{-h\rho }u$ in $U_h𝔤$ acts on ${(M\otimes V^{\otimes i})}^{\left[\lambda \right]}$ by the constant $q^{-⟨\lambda ,\lambda +2\rho ⟩}\text{.}$ From (2.10), (2.11) and (2.14) it follows that $X^{{\epsilon }_i}$ acts on $M\otimes V^{\otimes k}$ by $$\begin{array}{ccc}{\mathit{Ř}}_{i-1}\cdots {\mathit{Ř}}_1{\mathit{Ř}}_0^2{\mathit{Ř}}_1\cdots {\mathit{Ř}}_{i-1}& =& {\mathit{Ř}}_{M\otimes V^{\otimes (i-1)},V}{\mathit{Ř}}_{V,M\otimes V^{\otimes (i-1)}}\otimes {\text{id}}_V^{\otimes (k-i)}\\ & =& (C_{M\otimes V^{\otimes (i-1)}}\otimes C_V)C_{M\otimes V^{\otimes i}}^{-1}\otimes {\text{id}}_V^{\otimes (k-i)}\\ & =& \sum _{\lambda ,\mu ,\nu }{q}^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩}{P}_{\mu \nu }^{\lambda }\otimes {\text{id}}_V^{\otimes (k-i)}\end{array}$$ where $P_{\mu \nu }^{\lambda }:M\otimes {\text{id}}_V^{\otimes i}\to M\otimes {\text{id}}_V^{\otimes i}$ is the projection onto ${({(M\otimes V^{\otimes (i-1)})}^{\left[\mu \right]}\otimes V^{\left[\nu \right]})}^{\left[\lambda \right]}\text{.}$ Thus $X^{{\epsilon }_i}$ acts diagonally on the basis $B\text{.}$ $\square$

Define an anti-involution on ${\stackrel{\sim }{ℬ}}_k$ by $$\stackrel{\sim }{\theta }\left(T_i\right)=T_i\text{and}\stackrel{\sim }{\theta }\left(X^{\lambda }\right)=X^{\lambda },$$ for $1\le i\le k-1$ and $\lambda \in L\text{.}$ A contravariant form on a ${\stackrel{\sim }{ℬ}}_k$ module $N$ is a symmetric bilinear form $⟨,⟩:N\times N\to ℂ$ such that $$⟨b{n}_1,n_2⟩=⟨n_1,\stackrel{\sim }{\theta }\left(b\right)n_2⟩\text{for}\hspace{0.17em}{n}_1,n_2\in N,\hspace{0.17em}b\in {\stackrel{\sim }{ℬ}}_k\text{.}$$ Suppose $M$ is a $U_h𝔤\text{-module}$ in the category $𝒪$ and $V$ is a finite dimensional $U_h𝔤$ module. Let ${⟨,⟩}_M$ and ${⟨,⟩}_V$ be $U_h𝔤\text{-contravariant}$ forms on $M$ and $V$ respectively. By (2.16), $$⟨(v_1\otimes v_2){\mathit{Ř}}_{VV},v_1^{\prime }\otimes v_2^{\prime }⟩=⟨v_1\otimes v_2,(v_1^{\prime }\otimes v_2^{\prime }){\mathit{Ř}}_{VV}⟩,$$ for $v_1,v_2,v_1^{\prime },v_2^{\prime }\in V,$ and $$\begin{array}{ccc}⟨(m\otimes v){\mathit{Ř}}_{MV}{\mathit{Ř}}_{VM},m\prime \otimes v\prime ⟩& =& ⟨(m\otimes v){\mathit{Ř}}_{MV},(m\prime \otimes v\prime ){\mathit{Ř}}_{MV}⟩\\ & =& ⟨m\otimes v,(m\prime \otimes v\prime ){\mathit{Ř}}_{MV}{\mathit{Ř}}_{VM}⟩,\end{array}$$ for $m,m\prime \in M,$ $v,v\prime \in V\text{.}$ Thus it follows that the form $⟨,⟩$ on $M\otimes V^{\otimes k}$ given by $$\begin{array}{cc}⟨m\otimes v_1\otimes \cdots \otimes v_k,m\prime \otimes v_1^{\prime }\otimes \cdots \otimes v_k^{\prime }⟩={⟨m,m\prime ⟩}_M{⟨v_1,v_1^{\prime }⟩}_V{⟨v_2,v_2^{\prime }⟩}_V\cdots {⟨v_k,v_k^{\prime }⟩}_V,& \text{(3.5)}\end{array}$$ for $m,m\prime \in M,$ $v_i,v_i^{\prime }\in V$ is a ${\stackrel{\sim }{ℬ}}_k$ contravariant form on the ${\stackrel{\sim }{ℬ}}_k$ module $M\otimes V^{\otimes k}\text{.}$

The Functor $F_{\lambda }$

Fix a finite dimensional $U_h𝔤$ module $V$ and an integrally dominant weight $\lambda$ in ${𝔥}^{*}\text{.}$ Let ${\stackrel{\sim }{𝒪}}_k$ be the category of finite dimensional ${\stackrel{\sim }{ℬ}}_k$ modules and define a functor $$\begin{array}{cc}\begin{array}{cccc}{F}_{\lambda }:& 𝒪& ⟶& {\stackrel{\sim }{𝒪}}_k\\ & M& ⟼& {\text{Hom}}_{U_h𝔤}(M\left(\lambda \right),M\otimes V^{\otimes k})\end{array}\text{.}& \text{(3.5)}\end{array}$$ Since an element of ${\text{Hom}}_{U_h𝔤}(M\left(\lambda \right),M\otimes V^{\otimes k})$ is determined by the image of a generating highest weight vector of $M\left(\lambda \right),$ the space $F_{\lambda }\left(M\right)$ can be identified with the vector space of highest weight vectors of weight $\lambda$ in $M\otimes V^{\otimes k}\text{.}$ If $\lambda$ is integrally dominant the highest possible weight of ${(M\otimes V^{\otimes k})}^{\left[\lambda \right]}$ is $\lambda \text{.}$ Thus, viewing $F_{\lambda }\left(M\right)$ as the space of highest weight vectors of weight $\lambda$ in $M\otimes V^{\otimes k},$ $${\text{Hom}}_{U_h𝔤}(M\left(\lambda \right),M\otimes V^{\otimes k})\cong {\left({(M\otimes V^{\otimes k})}^{\left[\lambda \right]}\right)}_{\lambda }\cong {\left(\frac{M\otimes V^{\otimes k}}{{𝔫}^{-}(M\otimes V^{\otimes k})}\right)}_{\lambda }$$ where we use the notation $$\begin{array}{cc}{𝔫}^{-}(M\otimes V^{\otimes k})=\sum _i{Y}_i(M\otimes V^{\otimes k}),& \text{(3.7)}\end{array}$$ where the $Y_i$ are the Chevalley generators of ${𝔫}^{-}\text{.}$ In the case of $U𝔤\text{-modules,}$ the notation ${𝔫}^{-}(M\otimes V^{\otimes k})$ is self explanatory—the notation in (3.7) is simply a way to define the same object for the quantum group $U_h𝔤\text{.}$

The functor $F_{\lambda }$ is the composition of two functors: the functor $·\otimes V^{\otimes k}$ and the functor ${\text{Hom}}_U(M\left(\lambda \right),·)\text{.}$ The first is exact since $V^{\otimes k}$ is finite dimensional and the second is exact because when $\lambda$ is integrally dominant $M\left(\lambda \right)$ is projective, see [Jan1980, p. 72]. Thus $$\begin{array}{cc}\text{if}\hspace{0.17em}\lambda \hspace{0.17em}\text{is integrally dominant, the functor}\hspace{0.17em}{F}_{\lambda }\hspace{0.17em}\text{is exact.}& \text{(3.8)}\end{array}$$

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.

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