## The affine braid group action

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

$TiTj = TjTi, ifj≠i±1, (1.18) TiTi+1 = Ti+1Ti Ti+1, fori=1,2,…, k-2, (1.19) Xε1T1 Xε1T1 = T1Xε1 T1Xε1, (1.20) Xε1Ti = TiXε1, fori=2,3,…, k-1. (1.21)$

The generators ${T}_{i}$ and ${X}^{{\epsilon }_{1}}$ can be identified with the diagrams

$Ti= i i+1 andXε1= . (1.22)$

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

$Xεi= Ti-1Ti-2 …T2T1Xε1 T1T2…Ti-2 Ti-1= i . (1.23)$

The pictorial computation

$XεjXεi= i i = i i =XεiXεj$

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 $U={U}_{h}𝔤$ be the Drinfel'd-Jimbo quantum group corresponding to $𝔤\text{.}$ The quantum group $U$ is a ribbon Hopf algebra with invertible $ℛ\text{-matrix}$

$ℛ=∑ℛR1⊗R2 inU⊗U, and ribbon elementv=e-hρ u,$

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

$R∨MN: M⊗N ⟶ N⊗M m⊗n ⟼ ∑ℛR2n⊗R1 m M ⊗ N N ⊗ M (1.24)$

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

$M ⊗ N ⊗ P P ⊗ N ⊗ M = M ⊗ N ⊗ P P ⊗ N ⊗ M ( R∨MN⊗idP ) ( idN⊗ R∨MP ) ( R∨NP⊗idM ) = ( idM⊗ R∨NP ) ( R∨MP ⊗idN ) ( idP⊗ R∨MN ) .$

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate $\text{ad}\text{-invariant}$ bilinear form, let $U={U}_{h}𝔤$ be the corresponding Drinfeld-Jimbo quantum group and let $C=Z\left(U\right)$ be the center of ${U}_{h}𝔤\text{.}$ 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

$Φ: CBk ⟶ EndU ( M⊗V⊗k ) Ti ⟼ R∨i, Xε1 ⟼ R∨02, z ⟼ zM, 1≤i≤k-1, (1.25)$

where ${z}_{M}={\text{id}}_{V}^{\otimes k},$

$R∨i=idM⊗ idV⊗(i-1)⊗ R∨VV⊗ idV⊗(k-i-1) and R∨020 ( R∨MV R∨VM ) ⊗idV⊗(k-1),$

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: $R∨i R∨i+1 R∨i = = = R∨i+1 R∨i R∨i+1$ and $R∨02 R∨1 R∨02 R∨1 = = = = = R∨1 R∨02 R∨1 R∨02$ $\square$

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

$CM:M⟶M m⟼vm so that CM⊗N= ( R∨MN R∨NM ) -1 (CM⊗CN) (1.26)$

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

$CM= q - ⟨ λ,λ+2ρ ⟩ idM,whereq= eh/2 (1.27)$

(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 ${U}_{h}𝔤\text{-modules}$ 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

$L(μ)⊗L(ν)= ⨁λ L(λ) ⊕cμνλ by the scalar q ⟨ λ,λ+ 2ρ ⟩ - ⟨ μ,μ+ 2ρ ⟩ - ⟨ ν,ν+ 2ρ ⟩ . (1.28)$

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

$Φ (Xεi) = R∨ M⊗ V⊗(i-1) ,V R∨ V,M⊗ V⊗(i-1) = , (1.29)$

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.