Affine and degenerate affine BMW algebras: Actions on tensor space

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

Last update: 13 October 2013

Actions of general type tantalizers

Our goal in Section 2 is to provide a tensor space action of the affine Birman-Murakami-Wenzl (BMW) algebra $W_k$ and its degenerate version ${𝒲}_k$ by way of the group algebra of the affine braid group $C{B}_k$ and the degenerate affine braid algebra ${ℬ}_k,,$ respectively. The definition of the degenerate affine braid algebra ${ℬ}_k$ makes the Schur-Weyl duality framework completely analogous in both the affine and degenerate affine cases.

In this section, we define $C{B}_k$ and ${ℬ}_k$ and show that both act on tensor space of the form $M\otimes V^{\otimes k}\text{.}$ In the degenerate affine case this action commutes with complex reductive Lie algebras $𝔤\text{;}$ in the affine case this action commutes with the Drinfeld-Jimbo quantum group $U_h𝔤\text{.}$ As we will see in Section 2, the affine and degenerate affine BMW algebras arise when $𝔤$ is orthogonal or symplectic and $V$ is the defining representation; similarly, the degenerate affine Hecke algebras arise when $𝔤$ is of type ${𝔤𝔩}_n$ or ${𝔰𝔩}_n$ and $V$ is the defining representation. In the case when $M$ is the trivial representation and is ${𝔰𝔬}_n,$ the elements $y_1,\dots ,y_k$ in ${ℬ}_k$ become the Jucys-Murphy elements for the Brauer algebras used in [Naz1996]; in the case that $𝔤={𝔤𝔩}_n,$ these become the classical Jucys-Murphy elements in the group algebra of the symmetric group.

The action of the degenerate affine braid algebra ${ℬ}_k$ and the action of the affine braid group $B_k$ on $M\otimes V^{\otimes k}$ each provide Schur functors $$\begin{array}{cc}\begin{array}{cccc}{F}_V^{\lambda }:& \left\{U\text{-modules}\right\}& ⟶& \{\begin{array}{cc}\left\{{ℬ}_k\text{-modules}\right\},& \text{if}\hspace{0.17em}U=U𝔤,\\ \left\{B_k\text{-modules}\right\},& \text{if}\hspace{0.17em}U=U_h𝔤\text{.}\end{array}\\ & M& ⟼& {\text{Hom}}_U(M\left(\lambda \right),M\otimes V^{\otimes k})\end{array}& \text{(1.1)}\end{array}$$ where in each case ${\text{Hom}}_U(M\left(\lambda \right),M\otimes V^{\otimes k})$ is the vector space of highest weight vectors of weight $\lambda$ in $M\otimes V^{\otimes k}\text{.}$ These ubiquitous functors transfer representation theoretic information back and forth either between $U𝔤$ and ${ℬ}_k$ or between $U_h𝔤$ and $C{B}_k\text{.}$

The degenerate affine braid algebra action

Let $C$ be a commutative ring and let $S_k$ denote the symmetric group on $\{1,\dots ,k\}\text{.}$ For $i\in \{1,\dots ,k\},$ write $s_i$ for the transposition in $S_k$ that switches $i$ and $i+1\text{.}$ The degenerate affine braid algebra is the algebra ${ℬ}_k$ over $C$ generated by $$\begin{array}{cc}{t}_u(u\in S_k),{\kappa }_0,{\kappa }_1,\text{and}{y}_1,\dots ,y_k,& \text{(1.2)}\end{array}$$ with relations $$\begin{array}{cc}{t}_u{t}_v=t_{uv},y_i{y}_j=y_j{y}_i,{\kappa }_0{\kappa }_1={\kappa }_1{\kappa }_0,{\kappa }_0{y}_i=y_i{\kappa }_0,{\kappa }_1{y}_i=y_i{\kappa }_1,& \text{(1.3)}\\ {\kappa }_0{t}_{s_i}=t_{s_i}{\kappa }_0,{\kappa }_1{t}_{s_1}{\kappa }_1{t}_{s_1}=t_{s_1}{\kappa }_1{t}_{s_1}{\kappa }_1,\text{and}{\kappa }_1{t}_{s_j}=t_{s_j}{k}_1,\hspace{0.17em}\text{for}\hspace{0.17em}j\ne 1,& \text{(1.4)}\\ t_{s_i}(y_i+y_{i+1})(y_i+y_{i+1})t_{s_i},\text{and}{y}_j{t}_{s_i}=t_{s_i}{y}_j\text{for}\hspace{0.17em}j\ne i,i+1,& \text{(1.5)}\end{array}$$ and, for $i=1,\dots ,k-2,$ $$\begin{array}{cc}{t}_{s_i}{t}_{s_{i+1}}{\gamma }_{i,i+1}{t}_{s_{i+1}}{t}_{s_i}={\gamma }_{i+1,i+2},\text{where}{\gamma }_{i,i+1}=y_{i+1}-t_{s_i}{y}_i{t}_{s_i}\text{.}& \text{(1.6)}\end{array}$$ This presentation highlights the “Jucys-Murphy” elements $y_1,\dots ,y_k$ for the degenerate BMW algebra ${𝒲}_k$ as in [Naz1996]. However, the algebra ${ℬ}_k$ also admits the following presentation, which highlights its natural action on tensor space (as we will see in Theorem 1.2).

[DRV1105.4207, Theorem 2.1] The degenerate affine braid algebra ${ℬ}_k$ has another presentation by generators $$\begin{array}{cc}{t}_u(u\in S_k),{\kappa }_0,\dots ,{\kappa }_k\text{and}{\gamma }_{i,j},\text{for}\hspace{0.17em}0\le i,j\le k\hspace{0.17em}\text{with}\hspace{0.17em}i\ne j,& \text{(1.7)}\end{array}$$ and relations $$\begin{array}{cc}{t}_u{t}_v=t_{uv},t_w{\kappa }_i{t}_{w^{-1}}={\kappa }_{w\left(i\right)},t_w{\gamma }_{i,j}{t}_{w^{-1}}={\gamma }_{w\left(i\right),w\left(j\right)},& \text{(1.8)}\\ {\kappa }_i{\kappa }_j={\kappa }_j{\kappa }_i,{\kappa }_i{\gamma }_{\ell ,m}={\gamma }_{\ell ,m}{\kappa }_i,& \text{(1.9)}\\ {\gamma }_{i,j}={\gamma }_{j,i},{\gamma }_{p,r}{\gamma }_{\ell ,m}={\gamma }_{\ell ,m}{\gamma }_{p,r},\text{and}{\gamma }_{i,j}({\gamma }_{i,r}+{\gamma }_{j,r})=({\gamma }_{i,r}+{\gamma }_{j,r}){\gamma }_{i,j},& \text{(1.10)}\end{array}$$ for $p\ne \ell$ and $p\ne m$ and $r\ne \ell$ and $r\ne m$ and $i\ne j,$ $i\ne r$ and $j\ne r\text{.}$

The conversion between the two presentations is given by the formulas ${\kappa }_0={\kappa }_0,$ ${\kappa }_1={\kappa }_1,$ $t_w=t_w,$ $$\begin{array}{cc}{y}_j=\frac{1}{2}{\kappa }_j+\sum _{0\le \ell <j}{\gamma }_{\ell ,j},& \text{(1.11)}\end{array}$$ and the formulas in (1.8). Set $$\begin{array}{cc}{c}_0={\kappa }_0\text{and}{c}_j={\kappa }_0+2(y_1+\dots +y_j),& \text{(1.12)}\end{array}$$ for $j=1,2,\dots ,k\text{.}$ Then $c_0,\dots ,c_k$ pairwise commute, $$\begin{array}{cc}{y}_j=\frac{1}{2}(c_j-c_{j-1})\text{and}{c}_j=\sum _{i=0}^j{\kappa }_i+2\sum _{0\le \ell <m\le j}{\gamma }_{\ell ,m}\text{.}& \text{(1.13)}\end{array}$$

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form i.e., $$⟨,⟩:𝔤\otimes 𝔤\to ℂ,\text{with}⟨[x_1,x_2],x_3⟩+⟨x_2,[x_1,x_3]⟩=0$$ and $⟨x_1,x_2⟩=⟨x_2,x_1⟩,$ for $x_1,x_2,x_3\in 𝔤\text{.}$ Let $B=\{b_1,\dots ,b_n\}$ be a basis for $𝔤$ and let $\{b_1^{*},\dots ,b_n^{*}\}$ be the dual basis with respect to $⟨,⟩\text{.}$ The Casimir is $$\begin{array}{cc}\kappa =b_1{b}_1^{*}+\cdots +b_n{b}_n^{*}=\sum _{b\in B}b{b}^{*}\text{and}\kappa \in Z\left(U𝔤\right),& \text{(1.14)}\end{array}$$ where $Z\left(U𝔤\right)$ is the center of the enveloping algebra $U𝔤$ (see [Bou1990, I §3 Prop. 11]). Since the coproduct on $𝔤$ is defined by $\Delta \left(x\right)=x\otimes 1+1\otimes x$ for $x\in 𝔤,$ $$\begin{array}{cc}\Delta \left(\kappa \right)=\kappa \otimes 1+1\otimes \kappa +2\gamma ,\text{where}\gamma =\sum _{b\in B}b\otimes b^{*}\text{.}& \text{(1.15)}\end{array}$$

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate $\text{ad}\text{-invariant}$ bilinear form $⟨,⟩,$ and let $U𝔤$ be the universal enveloping algebra. Let $C=Z\left(U𝔤\right)$ be the center of $U𝔤,$ $\kappa$ be the Casimir in $C,$ and $\gamma ={\sum }_{b}b\otimes b^{*}$ as in (1.15). Let $M$ and $V$ be $𝔤\text{-modules}$ and let $s_1·(u\otimes v)=v\otimes u,$ for $u,v\in V\text{.}$

The degenerate affine braid algebra ${ℬ}_k$ acts on $M\otimes V^{\otimes k}$ via $\Phi :{ℬ}_k\to \text{End}(M\otimes V^{\otimes k})$ defined by $\Phi \left(t_{s_i}\right)={\text{id}}_M\otimes {\text{id}}_V^{\otimes i-1}\otimes s_1\otimes {\text{id}}_V^{\otimes k-i-1}$ so that $S_k$ by permuting tensor factors of $V,$
$\Phi \left({\kappa }_i\right)$ is $\kappa$ acting in the $i\text{th}$ factor of $V$ in $M\otimes V^{\otimes k}$ and $\Phi \left({\kappa }_0\right)$ is $\kappa$ acting on $M,$
$\Phi \left({\gamma }_{\ell ,m}\right)$ is $\gamma$ acting in the $\ell \text{th}$ and $m\text{th}$ factors of $V$ in $M\otimes V^{\otimes k},$
$\Phi \left({\gamma }_{0,\ell }\right)$ is $\gamma$ acting on $M$ and the $\ell \text{th}$ factor of $V$ in $M\otimes V^{\otimes k},$
$\Phi \left(c_i\right)$ is $\kappa$ acting on $M$ and the first $i$ factors of $V,$
$\Phi \left(z\right)=z\otimes {\text{id}}_V^{\otimes k}$ for $z\in C\text{.}$ This action of ${ℬ}_k$ commutes with the $U𝔤\text{-action}$ on $M\otimes V^{\otimes k}\text{.}$

		Proof.
	Since $\kappa \in Z\left(U𝔤\right)$ the operators $\Phi \left({\kappa }_i\right)$ are in ${\text{End}}_{𝔤}(M\otimes V^{\otimes k})\text{.}$ From the relation $2\gamma =\Delta \left(\kappa \right)-k\otimes 1-1\otimes k$ it also follows that $\Phi \left({\gamma }_{\ell ,m}\right)\in {\text{End}}_{𝔤}(M\otimes V^{\otimes k})\text{.}$ All of the relations in Theorem 1.1 except the last relations in (1.10) follow from consideration of the tensor factors being acted upon. The last relations in (1.10) are established by the computation $${\gamma }_{1,2}({\gamma }_{1,3}+{\gamma }_{2,3})(v\otimes w\otimes z)={\gamma }_{1,2}(\sum _i{b}_{i}v\otimes w\otimes b_i^{*}z+v\otimes b_{i}w\otimes b_i^{*}z)={\gamma }_{1,2}(\sum _i\Delta \left(b_i\right)\otimes b_i^{*})(v\otimes w\otimes z)=(\sum _i\Delta \left(b_i\right)\otimes b_i^{*}){\gamma }_{1,2}(v\otimes w\otimes z)=({\gamma }_{1,3}+{\gamma }_{2,3}){\gamma }_{1,2}(v\otimes w\otimes z),$$ for $v,w,z\in V\text{.}$ Recursively applying the coproduct formula from (1.15) connects the action of $c_i$ with the action of ${\kappa }_i$ and ${\gamma }_{\ell ,m}$ as in the second formula in (1.13). $\square$

For a $U𝔤\text{-module}$ $M$ let $$\begin{array}{cccc}{\kappa }_M:& M& ⟶& M\\ & m& ⟼& \kappa m\end{array}\text{where}\hspace{0.17em}\kappa \hspace{0.17em}\text{is the Casimir}$$ as in (1.14). If $M$ is a $U𝔤\text{-module}$ generated by a highest weight vector $v_{\lambda }^{+}$ of weight $\lambda$ then $$\begin{array}{cc}{\kappa }_M=⟨\lambda ,\lambda +2\rho ⟩{\text{id}}_M,\text{where}\rho =\frac{1}{2}\sum _{\alpha \in R^{+}}\alpha & \text{(1.16)}\end{array}$$ is the half-sum of the positive roots (see [Bou1990, VIII §2 no. 3 Prop. 6 and VIII §6 no. 4 Cor. to Prop. 7]). By equation (1.15), if $M=L\left(\mu \right)$ and $N=L\left(\nu \right)$ are finite-dimensional irreducible $U𝔤\text{-modules}$ of highest weights $\mu$ and $\nu$ respectively, then $\gamma$ acts on the $L\left(\lambda \right)\text{-isotypic}$ component of the decomposition $L\left(\mu \right)\otimes L\left(\nu \right)\cong {⨁}_{\lambda }L{\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}$ by the constant $$\begin{array}{cc}\frac{1}{2}(⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩)\text{.}& \text{(1.17)}\end{array}$$ Pictorially, By (1.17) and (1.12), the eigenvalues of $y_j$ are functions of the eigenvalues of the Casimir.

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 $$\begin{array}{cccccc}{T}_i{T}_j& =& T_j{T}_i,& \text{if}\hspace{0.17em}j\ne i\pm 1,& & \text{(1.18)}\\ T_i{T}_{i+1}{T}_i& =& T_{i+1}{T}_i{T}_{i+1},& \text{for}\hspace{0.17em}i=1,2,\dots ,k-2,& & \text{(1.19)}\\ X^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}{T}_1& =& T_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1},& & & \text{(1.20)}\\ X^{{\epsilon }_1}{T}_i& =& t_i{X}^{{\epsilon }_1},& \text{for}\hspace{0.17em}i=2,3,\dots ,k-1\text{.}& & \text{(1.21)}\end{array}$$ The generators $T_i$ and $X^{{\epsilon }_1}$ can be identified with the diagrams $$\begin{array}{cc}{T}_i=\begin{array}{c} i i+1 \end{array}\text{and}{X}^{{\epsilon }_1}=\begin{array}{c} \end{array}\text{.}& \text{(1.22)}\end{array}$$ For $i=1,\dots ,k$ define $$\begin{array}{cc}{X}^{{\epsilon }_i}=T_{i-1}{T}_{i-2}\cdots T_2{T}_1{X}^{{\epsilon }_1}{T}_1{T}_2\cdots T_{i-2}{T}_{i-1}=\begin{array}{c} i \end{array}\text{.}& \text{(1.23)}\end{array}$$ The pictorial computation $$X^{{\epsilon }_j}{X}^{{\epsilon }_i}=\begin{array}{c} i i \end{array}=\begin{array}{c} i i \end{array}=X^{{\epsilon }_i}{X}^{{\epsilon }_j}$$ shows that the $X^{{\epsilon }_i}$ pairwise commute.

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant 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 $ℛ\text{-matrix}$ $$ℛ=\sum _{ℛ}{R}_1\otimes R_2\text{in}U\otimes U,\text{and ribbon element}\hspace{0.17em}v=e^{-h\rho }u,$$ where $u={\sum }_{ℛ}S\left(R_2\right)R_1$ (see [LRa1977, Corollary (2.15)]). For $U\text{-modules}$ $M$ and $N,$ the map $$\begin{array}{cc}\begin{array}{cccc}{Ř}_{MN}:& M\otimes N& ⟶& N\otimes M\\ & m\otimes n& ⟼& \sum _{ℛ}{R}_{2}n\otimes R_{1}m\end{array}\begin{array}{c} M ⊗ N N ⊗ M \end{array}& \text{(1.24)}\end{array}$$ is a $U\text{-module}$ isomorphism. The quasitriangularity of a ribbon Hopf algebra provides the braid relation (see, for example, [ORa0401317, (2.12)]), $$\begin{array}{ccc}\begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}& =& \begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}\\ ({Ř}_{MN}\otimes {\text{id}}_P)({\text{id}}_N\otimes {Ř}_{MP})({Ř}_{NP}\otimes {\text{id}}_M)& =& ({\text{id}}_M\otimes {Ř}_{NP})({Ř}_{MP}\otimes {\text{id}}_N)({\text{id}}_P\otimes {Ř}_{MN})\text{.}\end{array}$$

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-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 $U\text{-modules.}$ Then $M\otimes V^{\otimes k}$ is a $C{B}_k\text{-module}$ with action given by $$\begin{array}{cc}\begin{array}{cccc}\Phi :& C{B}_k& ⟶& {\text{End}}_U(M\otimes V^{\otimes k})\\ & T_i& ⟼& {Ř}_i,& 1\le i\le k-1,\\ & X^{{\epsilon }_1}& ⟼& {Ř}_0^2,\\ & z& ⟼& z_M,\end{array}& \text{(1.25)}\end{array}$$ where $z_M=z\otimes {\text{id}}_V^{\otimes k},$ $${Ř}_i={\text{id}}_M\otimes {\text{id}}_V^{\otimes (i-1)}\otimes {Ř}_{VV}\otimes {\text{id}}_V^{\otimes (k-i-1)}\text{and}{Ř}_0^2=\left({Ř}_{MV}{Ř}_{VM}\right)\otimes {\text{id}}_V^{\otimes (k-1)},$$ with ${Ř}_{MV}$ 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 the following computations: $${Ř}_i{Ř}_{i+1}{Ř}_i=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}={Ř}_{i+1}{Ř}_i{Ř}_{i+1}$$ and $${Ř}_0^2{Ř}_1{Ř}_0^2{Ř}_1=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}={Ř}_1{Ř}_0^2{Ř}_1{Ř}_0^2\text{.}$$ $\square$

Let $v=e^{-h\rho }u$ be the ribbon element in $U=U_h𝔤\text{.}$ For a $U_h𝔤\text{-module}$ $M$ define $$\begin{array}{cc}\begin{array}{cccc}{C}_M:& M& ⟶& M\\ & m& ⟼& vm\end{array}\text{so that}{C}_{M\otimes N}={\left({Ř}_{MN}{Ř}_{NM}\right)}^{-1}(C_m\otimes C_n)& \text{(1.26)}\end{array}$$ (see [Dri1990, 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 [LRa1977, Prop. 2.14] or [Dri1990, 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 ${Ř}_{MN}{Ř}_{NM}$ acts on the $L\left(\lambda \right)\text{-isotypic}$ component $L{\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}$ of the decomposition $$\begin{array}{cc}L\left(\mu \right)\otimes L\left(\nu \right)=\underset{\lambda }{⨁}L{\left(\lambda \right)}^{c_{\mu \nu }^{\lambda }}\text{by the scalar}{q}^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩\text{.}}& \text{(1.28)}\end{array}$$ By the definition of $X^{{\epsilon }_i}$ in (1.23), $$\begin{array}{cc}\Phi \left(X^{{\epsilon }_i}\right)={Ř}_{M\otimes V^{\otimes (i-1)},V}{Ř}_{V,M\otimes V^{\otimes (i-1)}}=\begin{array}{c} i \end{array},& \text{(1.29)}\end{array}$$ 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 a typed exert of the paper Affine and degenerate affine BMW algebras: Actions on tensor space by Zajj Daugherty, Arun Ram and Rahbar Virk.

page history