The affine braid group action

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

Last update: 15 October 2012

The affine braid group action

The affine braid group $B_k$ is the group given by generators $T_1,T_2,\dots ,T_{k-1}$ and $X^{{\epsilon }_1},$ with relations

$$\begin{array}{ccccc}{T}_i{T}_j& =& T_j{T}_i,& \text{if}j\ne i\pm 1,& \text{(1.18)}\\ T_i{T}_{i+1}& =& T_{i+1}{T}_i{T}_{i+1},& \text{for}i=1,2,\dots ,k-2,& \text{(1.19)}\\ X^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}{T}_1& =& T_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1},& & \text{(1.20)}\\ X^{{\epsilon }_1}{T}_i& =& T_i{X}^{{\epsilon }_1},& \text{for}i=2,3,\dots ,k-1\text{.}& \text{(1.21)}\end{array}$$

The generators $T_i$ and $X^{{\epsilon }_1}$ can be identified with the diagrams

$$\begin{array}{cc}\begin{array}{ccccc}{T}_i=& i i+1 & \text{and}{X}^{{\epsilon }_1}=& & \text{.}\end{array}& \text{(1.22)}\end{array}$$

For $i=1,\dots ,k$ define

$$\begin{array}{cc}\begin{array}{ccc}{X}^{{\epsilon }_i}=T_{i-1}{T}_{i-2}\dots T_2{T}_1{X}^{{\epsilon }_1}{T}_1{T}_2\dots T_{i-2}{T}_{i-1}=& i & \text{.}\end{array}& \text{(1.23)}\end{array}$$

The pictorial computation

$$\begin{array}{ccccc}{X}^{{\epsilon }_j}{X}^{{\epsilon }_i}=& i i & =& i i & =X^{{\epsilon }_i}{X}^{{\epsilon }_j}\end{array}$$

shows that the $X^{{\epsilon }_i}$ pairwise commute.

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

$$ℛ=\sum _{ℛ}{R}_1\otimes R_2\text{in}U\otimes U,\text{and ribbon element}v=e^{-h\rho }u,$$

where $u=\sum _{ℛ}S\left(R_2\right)R_1$ (see, [LRa1997, Corollary (2.15)]). For $U\text{-modules}$ $M$ and $N,$ the map

$$\begin{array}{cc}\begin{array}{cc}\begin{array}{cccc}{\stackrel{\vee }{R}}_{MN}:& M\otimes N& ⟶& N\otimes M\\ & m\otimes n& ⟼& \sum _{ℛ}{R}_{2}n\otimes R_{1}m\end{array}& M ⊗ N N ⊗ M \end{array}& \text{(1.24)}\end{array}$$

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

$$\begin{array}{ccc} M ⊗ N ⊗ P P ⊗ N ⊗ M & =& M ⊗ N ⊗ P P ⊗ N ⊗ M \\ ({\stackrel{\vee }{R}}_{MN}\otimes {\text{id}}_P)({\text{id}}_N\otimes {\stackrel{\vee }{R}}_{MP})({\stackrel{\vee }{R}}_{NP}\otimes {\text{id}}_M)& =& ({\text{id}}_M\otimes {\stackrel{\vee }{R}}_{NP})({\stackrel{\vee }{R}}_{MP}\otimes {\text{id}}_N)({\text{id}}_P\otimes {\stackrel{\vee }{R}}_{MN})\text{.}\end{array}$$

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate $\text{ad}\text{-invariant}$ bilinear form, let $U=U_h𝔤$ be the corresponding Drinfeld-Jimbo quantum group and let $C=Z\left(U\right)$ be the center of $U_h𝔤\text{.}$ Let $M$ and $V$ be $U\text{-modules}\text{.}$ Then $M\otimes V^{\otimes k}$ is a $C{B}_k\text{-module}$ with action given by

$$\begin{array}{cc}\begin{array}{cc}\begin{array}{cccc}\Phi :& C{B}_k& ⟶& {\text{End}}_U(M\otimes V^{\otimes k})\\ & T_i& ⟼& {\stackrel{\vee }{R}}_i,\\ & X^{{\epsilon }_1}& ⟼& {\stackrel{\vee }{R}}_0^2,\\ & z& ⟼& z_M,\end{array}& 1\le i\le k-1,\end{array}& \text{(1.25)}\end{array}$$

where $z_M={\text{id}}_V^{\otimes k},$

$${\stackrel{\vee }{R}}_i={\text{id}}_M\otimes {\text{id}}_V^{\otimes (i-1)}\otimes {\stackrel{\vee }{R}}_{VV}\otimes {\text{id}}_V^{\otimes (k-i-1)}\text{and}{\stackrel{\vee }{R}}_0^{2}0\left({\stackrel{\vee }{R}}_{MV}{\stackrel{\vee }{R}}_{VM}\right)\otimes {\text{id}}_V^{\otimes (k-1)},$$

with ${\stackrel{\vee }{R}}_{MV}$ as in (1.24). The $C{B}_k$ action commutes with the $U\text{-action}$ on $M\otimes V^{\otimes k}\text{.}$

		Proof.
	The relations (1.18) and (1.21) are consequences of the definition of the action of $T_i$ and $X^{{\epsilon }_1}\text{.}$ The relations (1.19) and (1.20) follow from to following computations: $$\begin{array}{ccccccc}{\stackrel{\vee }{R}}_i{\stackrel{\vee }{R}}_{i+1}{\stackrel{\vee }{R}}_i& =& & =& & =& {\stackrel{\vee }{R}}_{i+1}{\stackrel{\vee }{R}}_i{\stackrel{\vee }{R}}_{i+1}\end{array}$$ and $$\begin{array}{ccccccccccc}{\stackrel{\vee }{R}}_0^2{\stackrel{\vee }{R}}_1{\stackrel{\vee }{R}}_0^2{\stackrel{\vee }{R}}_1& =& & =& & =& & =& & =& {\stackrel{\vee }{R}}_1{\stackrel{\vee }{R}}_0^2{\stackrel{\vee }{R}}_1{\stackrel{\vee }{R}}_0^2\end{array}$$ $\square$

Let $v=e^{-h\rho }u$ be the ribbon element in $U=U_h𝔤\text{.}$ For a $U_h𝔤\text{-module}$ $M$ define

$$\begin{array}{cc}\begin{array}{cccc}{C}_M:& M& ⟶& M\\ & m& ⟼& vm\end{array}\text{so that}{C}_{M\otimes N}={\left({\stackrel{\vee }{R}}_{MN}{\stackrel{\vee }{R}}_{NM}\right)}^{-1}(C_M\otimes C_N)& \text{(1.26)}\end{array}$$

(see [Dri1970, Prop. 3.2]). If $M$ is a $U_h𝔤\text{-module}$ generated by a highest weight vector $v_{\lambda }^{+}$ of weight $\lambda ,$ then

$$\begin{array}{cc}{C}_M=q^{-⟨\lambda ,\lambda +2\rho ⟩}{\text{id}}_M,\text{where}q=e^{h/2}& \text{(1.27)}\end{array}$$

(see [LRa1997, Prop. 2.14] or [Dri1970, Prop. 5.1]). From (1.27) and the relation (1.26) it follows that if $M=L\left(\mu \right)$ and $N=L\left(\nu \right)$ are finite-dimensional irreducible $U_h𝔤\text{-modules}$ of highest weights $\mu$ and $\nu$ respectively, then ${\stackrel{\vee }{R}}_{MN}{\stackrel{\vee }{R}}_{NM}$ acts on the $L\left(\lambda \right)\text{-isotypic}$ component ${L\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}$ of the decomposition

$$\begin{array}{cc}L\left(\mu \right)\otimes L\left(\nu \right)=\underset{\lambda }{⨁}{L\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}\text{by the scalar}{q}^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩}\text{.}& \text{(1.28)}\end{array}$$

By the definition of $X^{{\epsilon }_i}$ in (1.23),

$$\begin{array}{cc}\begin{array}{ccccc}\Phi \left(X^{{\epsilon }_i}\right)& =& {\stackrel{\vee }{R}}_{M\otimes V^{\otimes (i-1)},V}{\stackrel{\vee }{R}}_{V,M\otimes V^{\otimes (i-1)}}& =& ,\end{array}& \text{(1.29)}\end{array}$$

so that, by (1.26), the eigenvalues of $\Phi \left(X^{{\epsilon }_i}\right)$ 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.

page history