Affine and degenerate affine BMW algebras: The center

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

Last update: 18 October 2013

Affine and degenerate affine BMW algebras

In this section, we define the affine Birman-Murakami-Wenzl (BMW) algebra $W_k$ and its degenerate version ${𝒲}_k\text{.}$ We have adjusted the definitions to unify the theory. In particular, in section 2.1, we define a new algebra, the degenerate affine braid algebra ${ℬ}_k,$ which has the degenerate affine BMW algebras ${𝒲}_k$ and the degenerate affine Hecke algebras ${\mathscr{H}}_k$ as quotients. The motivation for the definition of ${ℬ}_k$ is that the affine BMW algebras $W_k$ and the affine Hecke algebras $H_k$ are quotients of the group algebra of affine braid group ${CB}_k\text{.}$

The definition of the degenerate affine braid algebra ${ℬ}_k$ also makes the Schur-Weyl duality framework completely analogous in both the affine and degenerate affine cases. Both ${ℬ}_k$ and ${CB}_k$ are designed to act on tensor space of the form $M\otimes V^{\otimes k}\text{.}$ In the degenerate affine case this is an action commuting with a complex semisimple Lie algebra $𝔤,$ and in the affine case this is an action commuting with the Drinfeld-Jimbo quantum group $U_q𝔤\text{.}$ The degenerate affine and affine BMW algebras arise when $𝔤$ is ${𝔰𝔬}_n$ or ${𝔰𝔭}_n$ and $V$ is the first fundamental representation and the degenerate affine and affine Hecke algebras arise when $𝔤$ is ${𝔤𝔩}_n$ or $f_n$ and $V$ is the first fundamental representation. In the case when $M$ is the trivial representation and $𝔤$ is ${𝔰𝔬}_n,$ the “Jucys-Murphy” elements $y_1,\dots ,y_k$ in ${ℬ}_k$ become the “Jucys-Murphy” elements for the Brauer algebras used in [Naz1996] and, in the case that $𝔤={𝔤𝔩}_n,$ these become the classical Jucys-Murphy elements in the group algebra of the symmetric group. The Schur-Weyl duality actions are explained in [DRV1205.1852v1] and [RamNotes].

The degenerate affine braid algebra ${ℬ}_k$

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 $$\begin{array}{cc}{t}_u(u\in S_k),{\kappa }_0,{\kappa }_1,\text{and}{y}_1,\dots ,y_k,& \text{(2.1)}\end{array}$$ with relations $$\begin{array}{cc}{t}_u{t}_v=t_{uv},y_i{y}_j=y_j{y}_i,{\kappa }_0{\kappa }_1={\kappa }_1{\kappa }_0,{\kappa }_0{y}_i=y_i{\kappa }_0,{\kappa }_1{y}_i=y_i{\kappa }_1,& \text{(2.2)}\\ {\kappa }_0{t}_{s_i}=t_{s_i}{\kappa }_0,{\kappa }_1{t}_{s_1}{\kappa }_1{t}_{s_1}=t_{s_1}{\kappa }_1{t}_{s_1}{\kappa }_1,\text{and}{\kappa }_1{t}_{s_j}=t_{s_j}{\kappa }_1,\hspace{0.17em}\text{for}\hspace{0.17em}j\ne 1,& \text{(2.3)}\\ t_{s_i}(y_i+y_{i+1})=(y_i+y_{i+1})t_{s_i},\text{and}{y}_j{t}_{s_i}=t_{s_i}{y}_j,\text{for}\hspace{0.17em}j\ne i,i+1,& \text{(2.4)}\end{array}$$ and $$\begin{array}{cc}{t}_{s_i}{t}_{s_{i+1}}{\gamma }_{i,i+1}{t}_{s_{i+1}}{t}_{s_i}={\gamma }_{i+1,i+2},\text{where}{\gamma }_{i,i+1}=y_{i+1}-t_{s_i}{y}_i{t}_{s_i}\hspace{0.17em}\text{for}\hspace{0.17em}i=1,\dots ,k-2\text{.}& \text{(2.5)}\end{array}$$

In the degenerate affine braid algebra ${ℬ}_k$ let $c_0={\kappa }_0$ and $$\begin{array}{cc}{c}_j={\kappa }_0+2(y_1+\dots +y_j),\text{so that}{y}_j=\frac{1}{2}(c_j-c_{j-1}),\text{for}\hspace{0.17em}j=1,\dots ,k\text{.}& \text{(2.6)}\end{array}$$ Then $c_0,\dots ,c_k$ commute with each other, commute with ${\kappa }_1,$ and the relations (2.4) are equivalent to $$\begin{array}{cc}{t}_{s_i}{c}_j=c_j{t}_{s_i},\text{for}\hspace{0.17em}j\ne i\text{.}& \text{(2.7)}\end{array}$$

The degenerate affine braid algebra ${ℬ}_k$ has another presentation by generators $$\begin{array}{cc}{t}_u,\hspace{0.17em}\text{for}\hspace{0.17em}u\in S_k,{\kappa }_0,\dots ,{\kappa }_k\text{and}{\gamma }_{i,j},\text{for}\hspace{0.17em}0\le i,j\le k\hspace{0.17em}\text{with}\hspace{0.17em}i\ne j,& \text{(2.8)}\end{array}$$ and relations $$\begin{array}{cc}{t}_u{t}_v=t_{uv},t_w{\kappa }_i{t}_{w^{-1}}={\kappa }_{w\left(i\right)},t_w{\gamma }_{i,j}{t}_{w^{-1}}={\gamma }_{w\left(i\right),w\left(j\right)},& \text{(2.9)}\\ {\kappa }_i{\kappa }_j={\kappa }_j{\kappa }_i,{\kappa }_i{\gamma }_{\ell ,m}={\gamma }_{\ell ,m}{\kappa }_i,& \text{(2.10)}\\ {\gamma }_{i,j}={\gamma }_{j,i},{\gamma }_{p,r}{\gamma }_{\ell ,m}={\gamma }_{\ell ,m}{\gamma }_{p,r},\text{and}{\gamma }_{i,j}({\gamma }_{i,r}+{\gamma }_{j,r})=({\gamma }_{i,r}+{\gamma }_{j,r}){\gamma }_{i,j},& \text{(2.11)}\end{array}$$ 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 commutation relations between the ${\kappa }_i$ and the ${\gamma }_{i,j}$ can be rewritten in the form $$\begin{array}{cc}[{\kappa }_r,{\gamma }_{\ell ,m}]=0,[{\gamma }_{i,j},{\gamma }_{\ell ,m}]=0,\text{and}[{\gamma }_{i,j},{\gamma }_{i,m}]=[{\gamma }_{i,m},{\gamma }_{j,m}],& \text{(2.12)}\end{array}$$ for all $r$ and all $i\ne \ell$ and $i\ne m$ and $j\ne \ell$ and $j\ne m\text{.}$

		Proof.
	The generators in (2.8) are written in terms of the generators in (2.1) by the formulas $$\begin{array}{cc}{\kappa }_0={\kappa }_0,{\kappa }_1={\kappa }_1,t_w=t_w,& \text{(2.13)}\\ {\gamma }_{0,1}=y_1-\frac{1}{2}{\kappa }_1,\text{and}{\gamma }_{j,j+1}=y_{j+1}-t_{s_j}{y}_j{t}_{s_j},\text{for}\hspace{0.17em}j=1,\dots ,k-1,& \text{(2.14)}\end{array}$$ and $$\begin{array}{cc}{\kappa }_m=t_u{\kappa }_1{t}_{u^{-1}},{\gamma }_{0,m}=t_u{\gamma }_{0,1}{t}_{u^{-1}}\text{and}{\gamma }_{i,j}=t_v{\gamma }_{1,2}{t}_{v^{-1}},& \text{(2.15)}\end{array}$$ for $u,v\in S_k$ such that $u\left(1\right)=m,$ $v\left(1\right)=i$ and $v\left(2\right)=j\text{.}$ The generators in (2.1) are written in terms of the generators in (2.8) by the formulas $$\begin{array}{cc}{\kappa }_0={\kappa }_0,{\kappa }_1={\kappa }_1,t_w=t_w,\text{and}{y}_j=\frac{1}{2}{\kappa }_j+\sum _{0\le \ell <j}{\gamma }_{\ell ,j}\text{.}& \text{(2.16)}\end{array}$$ Let us show that relations in (2.2-5) follow from the relations in (2.9-2.11). (a) The relation $t_u{t}_v=t_{uv}$ in (2.2) is the first relation in (2.9). (b) The relation $y_i{y}_j=y_j{y}_i$ in (2.2): Assume that $i<j\text{.}$ Using the relations in (2.10) and (2.11), $$\begin{array}{ccc}[y_i,y_j]& =& [\frac{1}{2}{\kappa }_i+\sum _{\ell <i}{\gamma }_{\ell ,i},\frac{1}{2}{\kappa }_j+\sum _{m<j}{\gamma }_{m,j}]=[\sum _{\ell <i}{\gamma }_{\ell ,i},\sum _{m<j}{\gamma }_{m,j}]\\ & =& \sum _{\ell <i}[{\gamma }_{\ell ,i},\sum _{m<j}{\gamma }_{m,j}]=\sum _{\ell <i}[{\gamma }_{\ell ,i},({\gamma }_{\ell ,j}+{\gamma }_{i,j})+\sum _{\underset{m\ne \ell ,m\ne i}{m<j}}{\gamma }_{m,j}]=0\text{.}\end{array}$$ (c) The relation ${\kappa }_0{\kappa }_1={\kappa }_1{\kappa }_0$ in (2.2) is part of the first relation in (2.10), and the relations ${\kappa }_0{y}_i=y_i{\kappa }_0$ and ${\kappa }_1{y}_i=y_i{\kappa }_1$ in (2.2) follow from the relations ${\kappa }_i{\kappa }_j={\kappa }_j{\kappa }_i$ and ${\kappa }_i{\gamma }_{\ell ,m}={\gamma }_{\ell ,m}{\kappa }_i$ in (2.10). (d) The relations ${\kappa }_0{t}_{s_i}=t_{s_i}{\kappa }_0$ and ${\kappa }_1{t}_{s_j}=t_{s_j}{\kappa }_1$ for $j\ne 1$ from (2.3) follow from the relation $t_w{\kappa }_i{t}_w^{-1}={\kappa }_{w\left(i\right)}$ in (2.9), and the relation ${\kappa }_1{t}_{s_1}{\kappa }_1{t}_{s_1}=t_{s_1}{\kappa }_1{t}_{s_1}{\kappa }_2$ from (2.3) follows from ${\kappa }_1{\kappa }_2={\kappa }_2{\kappa }_1,$ which is part of the first relation in (2.10). (e) The relations in (2.4) and (2.5) all follow from the relations $t_w{\kappa }_i{t}_{w^{-1}}={\kappa }_{w\left(i\right)}$ and $t_w{\gamma }_{i,j}{t}_{w^{-1}}={\gamma }_{w\left(i\right),w\left(j\right)}$ in (2.9). To complete the proof let us show that the relations of (2.9-11) follow from the relations in (2.2-5). (a) The relation $t_u{t}_v=t_{uv}$ in (2.9) is the first relation in (2.2). (b) The relations $t_w{\kappa }_i{t}_{w^{-1}}={\kappa }_{w\left(i\right)}$ in (2.9) follow from the first and last relations in (2.3) (and force the definition of ${\kappa }_m$ in (2.15)). (c) Since ${\gamma }_{0,1}=y_1-\frac{1}{2}{\kappa }_1,$ the relations $t_w{\gamma }_{0,j}{t}_{w^{-1}}={\gamma }_{0,w\left(j\right)}$ in (2.10) follow from the last relation in each of (2.3) and (2.4) (and force the definition of ${\gamma }_{0,m}$ in (2.15)). (d) Since ${\gamma }_{1,2}=y_2-t_{s_1}{y}_1{t}_{s_1},$ the first relation in (2.4) gives $$\begin{array}{cc}{t}_{s_1}{\gamma }_{1,2}{t}_{s_1}-{\gamma }_{1,2}=(t_{s_1}{y}_2{t}_{s_1}-y_1)-y_2+t_{s_1}{y}_1{t}_{s_1}=t_{s_1}(y_1+y_2)t_{s_1}-(y_1+y_2)=0\text{.}& \text{(2.17)}\end{array}$$ The relations $t_w{\gamma }_{1,2}{t}_{w^{-1}}={\gamma }_{w\left(1\right),w\left(2\right)}$ in (2.9) then follow from (2.17) and the last relation in (2.4) (and force the definitions ${\gamma }_{i,j}=t_v{\gamma }_{1,2}{t}_{v^{-1}}$ in (2.15)). (e) The third relation in (2.2) is ${\kappa }_0{\kappa }_1={\kappa }_1{\kappa }_0$ and the second relation in (2.3) gives ${\kappa }_1{\kappa }_2={\kappa }_2{\kappa }_1\text{.}$ The relations ${\kappa }_i{\kappa }_j={\kappa }_j{\kappa }_i$ in (2.10) then follow from the second set of relations in (2.9). (f) The second relation in (2.3) gives $[{\kappa }_1,{\kappa }_2]=0\text{.}$ Using this and the relations in (2.2), $$\begin{array}{cc}[{\kappa }_1,{\gamma }_{0,2}+{\gamma }_{1,2}]=[{\kappa }_1,(y_2-\frac{1}{2}{\kappa }_2-{\gamma }_{1,2})+{\gamma }_{1,2}]=[{\kappa }_1,\frac{1}{2}{\kappa }_2]=0,& \text{(2.18)}\end{array}$$ and $$\begin{array}{cc}[{\gamma }_{0,1},{\gamma }_{0,2}+{\gamma }_{1,2}]=[y_1-\frac{1}{2}{\kappa }_1,y_2-\frac{1}{2}{\kappa }_2]=\frac{1}{4}[{\kappa }_1,{\kappa }_2]=0,& \text{(2.19)}\end{array}$$ so that $$[{\gamma }_{0,1},{\kappa }_2]=[{\gamma }_{0,1},2y_2-2({\gamma }_{0,2}+{\gamma }_{1,2})]=[{\gamma }_{0,1},2y_2]=[y_1-\frac{1}{2}{\kappa }_1,2y_2]=-[{\kappa }_1,y_2]=0\text{.}$$ Conjugating the last relation by $t_{s_1}$ gives $$[{\kappa }_1,{\gamma }_{0,2}]=0,\text{and thus}[{\kappa }_1,{\gamma }_{1,2}]=0,$$ by (2.18). By the third and fourth relations in (2.2), $$[{\kappa }_0,{\gamma }_{0,1}]=[{\kappa }_0,y_1-\frac{1}{2}{\kappa }_1]=0,\text{and}[{\kappa }_1,{\gamma }_{0,1}]=[{\kappa }_1,y_1-\frac{1}{2}{\kappa }_1]=0\text{.}$$ By the relations in (2.3) and (2.2), $$[{\kappa }_0,{\gamma }_{1,2}]=[{\kappa }_0,y_2-t_{s_1}{y}_1{t}_{s_1}]=0\text{and}[{\kappa }_1,{\gamma }_{2,3}]=[{\kappa }_1,y_3-t_{s_2}{y}_2{t}_{s_2}]=0\text{.}$$ Putting these together with the (already established) relations in (2.9) provides the second set of relations in (2.10). (g) From the commutativity of the $y_i$ and the second relation in (2.4) $${\gamma }_{1,2}{\gamma }_{3,4}=(y_2-t_{s_1}{y}_1{t}_{s_1})(y_4-t_{s_3}{y}_3{t}_{s_3})=(y_4-t_{s_3}{y}_3{t}_{s_3})(y_2-t_{s_1}{y}_1{t}_{s_1})={\gamma }_{3,4}{\gamma }_{1,2}\text{.}$$ By the last relation in (2.2) and the last relation in (2.3), $$[{\gamma }_{0,1},{\gamma }_{2,3}]=[y_1-\frac{1}{2}{\kappa }_1,y_3-t_{s_2}{y}_2{t}_{s_2}]=0\text{.}$$ Together with the (already established) relations in (2.9), we obtain the first set of relations in (2.11). (h) Conjugating (2.19) by $t_{s_2}{t}_{s_1}{t}_{s_2}$ gives $[{\gamma }_{0,2},{\gamma }_{0,3}+{\gamma }_{2,3}]=0,$ and this and the (already established) relations in (2.10) and the first set of relations in (2.11) provide $$\begin{array}{ccc}0& =& [y_2,y_3]=[\frac{1}{2}{\kappa }_2+{\gamma }_{0,2}+{\gamma }_{1,2},\frac{1}{2}{\kappa }_3+{\gamma }_{0,3}+{\gamma }_{1,3}+{\gamma }_{2,3}]\\ & =& [{\gamma }_{0,2}+{\gamma }_{1,2},{\gamma }_{0,3}+{\gamma }_{1,3}+{\gamma }_{2,3}]=[{\gamma }_{1,2},{\gamma }_{0,3}+{\gamma }_{1,3}+{\gamma }_{2,3}]=[{\gamma }_{1,2},{\gamma }_{1,3}+{\gamma }_{2,3}]\text{.}\end{array}$$ Note also that $$\begin{array}{ccc}[{\gamma }_{1,2},{\gamma }_{1,0}+{\gamma }_{2,0}]& =& [{\gamma }_{1,2},{\gamma }_{0,1}+{\gamma }_{0,2}]=-[{\gamma }_{0,1},{\gamma }_{1,2}]+[{\gamma }_{1,2},{\gamma }_{0,2}]\\ & =& [{\gamma }_{0,1},{\gamma }_{0,2}]+[{\gamma }_{1,2},{\gamma }_{0,2}]=t_{s_1}[{\gamma }_{0,2}+{\gamma }_{1,2},{\gamma }_{0,1}]t_{s_1}=0,\end{array}$$ by (two applications of) (2.19). The last set of relations in (2.11) now follow from the last set of relations in (2.9). $\square$

By the first formula in (2.6) and the last formula in (2.16), $$\begin{array}{cc}{c}_j=\sum _{i=0}^j{\kappa }_i+2\sum _{0\le \ell <m\le j}{\gamma }_{\ell ,m}\text{.}& \text{(2.20)}\end{array}$$

The degenerate affine BMW algebra ${𝒲}_k$

Let $C$ be a commutative ring and let ${ℬ}_k$ be the degenerate affine braid algebra over $C$ as defined in Section 2.1. Define $e_i$ in the degenerate affine braid algebra by $$\begin{array}{cc}{t}_{s_i}{y}_i=y_{i+1}{t}_{s_i}-(1-e_i),\text{for}\hspace{0.17em}i=1,2,\dots ,k-1,& \text{(2.21)}\end{array}$$ so that, with ${\gamma }_{i,i+1}$ as in (2.5), $$\begin{array}{cc}{\gamma }_{i,i+1}{t}_{s_i}=1-e_i\text{.}& \text{(2.22)}\end{array}$$

Fix constants $$\epsilon =\pm 1\text{and}{z}_0^{\left(\ell \right)}\in C,\text{for}\hspace{0.17em}\ell \in {\mathbb{Z}}_{\ge 0}\text{.}$$ The degenerate affine Birman-Wenzl-Murakami (BMW) algebra ${𝒲}_k$ (with parameters $\epsilon$ and $z_0^{\left(\ell \right)}\text{)}$ is the quotient of the degenerate affine braid algebra ${ℬ}_k$ by the relations $$\begin{array}{cc}{e}_i{t}_{s_i}=t_{s_i}{e}_i=\epsilon e_i,e_i{t}_{s_{i-1}}{e}_i=e_i{t}_{s_{i+1}}{e}_i=\epsilon e_i,& \text{(2.23)}\\ e_1{y}_1^{\ell }{e}_1=z_0^{\left(\ell \right)}{e}_1,e_i(y_i+y_{i+1})=0=(y_i+y_{i+1})e_i\text{.}& \text{(2.24)}\end{array}$$ Conjugating (2.21) by $t_{s_i}$ and using the first relation in (2.23) gives $$\begin{array}{cc}{y}_i{t}_{s_i}=t_{s_i}{y}_{i+1}-(1-e_i)\text{.}& \text{(2.25)}\end{array}$$ Then, by (2.22) and (2.5), $$\begin{array}{cc}{\gamma }_{i,i+1}=t_{s_i}-\epsilon e_i,\text{and}{e}_{i+1}=t_{s_i}{t}_{s_{i+1}}{e}_i{t}_{s_{i+1}}{t}_{s_i}\text{.}& \text{(2.26)}\end{array}$$ Multiply the second relation in (2.26) on the left and the right by $e_i,$ and then use the relations in (2.23) to get $$e_i{e}_{i+1}{e}_i=e_i{t}_{s_i}{t}_{s_{i+1}}{e}_i{t}_{s_{i+1}}{t}_{s_i}{e}_i=e_i{t}_{s_{i+1}}{e}_i{t}_{s_{i+1}}{e}_i=\epsilon e_i{t}_{s_{i+1}}{e}_i=e_i,$$ so that $$\begin{array}{cc}{e}_i{e}_{i\pm 1}{e}_i=e_i\text{.}\text{Note that}{e}_i^2=z_0^{\left(0\right)}{e}_i& \text{(2.27)}\end{array}$$ is a special case of the first identity in (2.24). The relations $$\begin{array}{cc}{e}_{i+1}{e}_i=e_{i+1}{t}_{s_i}{t}_{s_{i+1}},t_i{e}_{i+1}=t_{s_{i+1}}{t}_{s_i}{e}_{i+1},& \text{(2.28)}\\ t_{s_i}{e}_{i+1}{e}_i=t_{s_{i+1}}{e}_i,\text{and}{e}_{i+1}{e}_i{t}_{s_{i+1}}=e_{i+1}{t}_{s_i}& \text{(2.29)}\end{array}$$ result from $$\begin{array}{ccc}{e}_{i+1}{t}_{s_i}{t}_{s_{i+1}}& =& \epsilon e_{i+1}{t}_{s_i}{e}_{i+1}{t}_{s_i}{t}_{s_{i+1}}=e_{i+1}{t}_{s_{i+1}}{t}_{s_i}{e}_{i+1}{t}_{s_i}{t}_{s_{i+1}}=e_{i+1}{e}_i,\\ t_{s_{i+1}}{t}_{s_i}{e}_{i+1}& =& \epsilon t_{s_{i+1}}{t}_{s_i}{e}_{i+1}{t}_{s_i}{e}_{i+1}=t_{s_{i+1}}{t}_{s_i}{e}_{i+1}{t}_{s_i}{t}_{s_{i+1}}{e}_{i+1}=e_i{e}_{i+1},\\ t_{s_i}{e}_{i+1}{e}_i& =& \epsilon t_{s_i}{e}_{i+1}{t}_{s_i}{e}_i=\epsilon t_{s_{i+1}}{e}_i{t}_{s_{i+1}}{e}_i=t_{s_{i+1}}{e}_i,\text{and}\\ e_{i+1}{e}_i{t}_{s_{i+1}}& =& \epsilon e_{i+1}{t}_{s_{i+1}}{e}_i{t}_{s_{i+1}}=\epsilon e_{i+1}{t}_{s_i}{e}_{i+1}{t}_{s_i}=e_{i+1}{t}_{s_i}\text{.}\end{array}$$

A consequence (see (3.7)) of the defining relations of ${𝒲}_k$ is the equation $$(z_0(-u)-(\frac{1}{2}+\epsilon u))(z_0\left(u\right)-(\frac{1}{2}-\epsilon u))e_1=(\frac{1}{2}-\epsilon u)(\frac{1}{2}+\epsilon u)e_1,$$ where $z_0\left(u\right)$ is the generating function $$z_0\left(u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{z}_0^{\left(\ell \right)}{u}^{-\ell }\text{.}$$ This means that, unless the parameters $z_0^{\left(\ell \right)}$ are chosen carefully, it is likely that $e_1=0$ in ${𝒲}_k\text{.}$

From the point of view of the Schur-Weyl duality for the degenerate affine BMW algebra (see [ASu9710037] and [RamNotes]) the natural choice of base ring is the center of the enveloping algebra of the orthogonal or symplectic Lie algebra which, by the Harish-Chandra isomorphism, is isomorphic to the subring of symmetric functions given by $$C=\{z\in ℂ{[h_1,\dots ,h_r]}^{S_r}\hspace{0.17em}|\hspace{0.17em}z(h_1,\dots ,h_r)=z(-h_1,h_2,\dots ,h_r)\},$$ where the symmetric group $S_r$ acts by permuting the variables $h_1,\dots ,h_r\text{.}$ Here the constants $z_0^{\left(\ell \right)}\in C$ are given, explicitly, by setting the generating function $$z_0\left(u\right)\hspace{0.17em}\text{equal, up to a normalization, to}\prod _{i=1}^r\frac{(u+\frac{1}{2}+h_i)(u+\frac{1}{2}-h_i)}{(u-\frac{1}{2}-h_i)(u-\frac{1}{2}+h_i)}\text{.}$$ This choice of $C$ and the $z_0^{\left(\ell \right)}$ are the universal admissible parameters for ${𝒲}_k\text{.}$ This point of view will be explained in [DRV1205.1852v1].

Careful manipulation of the defining relations of ${𝒲}_k$ provides an inductive presentation of ${𝒲}_k$ as $${𝒲}_k={𝒲}_{k-1}{e}_{k-1}{𝒲}_{k-1}+{𝒲}_{k-1}{t}_{s_{k-1}}{𝒲}_{k-1}+\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{𝒲}_{k-1}{y}_k^{\ell }{𝒲}_{k-1},$$ and provides that $$e_k{𝒲}_k{e}_k={𝒲}_{k-1}{e}_k,\text{and}\begin{array}{ccc}{𝒲}_k& ⟶& {𝒲}_{k-1}{e}_k\\ b& ⟼& e_{k}b{e}_k\end{array}$$ is a $({𝒲}_{k-1},{𝒲}_{k-1})\text{-bimodule}$ homomorphism. These structural facts are important to the understanding of ${𝒲}_k$ by “Jones basic constructions”. Under the conditions of Theorem 4.1(a) it is true, but not immediate from the defining relations, that the natural homomorphism ${𝒲}_{k-1}\to {𝒲}_k$ is injective so that ${𝒲}_{k-1}$ is a subalgebra of ${𝒲}_k\text{.}$ These useful structural results for the algebras ${𝒲}_k$ are justified in [AMR0506467].

Quotients of ${𝒲}_k$

The degenerate affine Hecke algebra ${\mathscr{H}}_k$ is the quotient of ${𝒲}_k$ by the relations $$\begin{array}{cc}{e}_i=0,\text{for}\hspace{0.17em}i=1,\dots ,k-1\text{.}& \text{(2.30)}\end{array}$$ Fix $b_1,\dots ,b_r\in ℂ\text{.}$ The degenerate cyclotomic BMW algebra ${𝒲}_{r,k}(b_1,\dots ,b_r)$ is the degenerate affine BMW algebra with the additional relation $$\begin{array}{cc}(y_1-b_1)\cdots (y_1-b_r)=0\text{.}& \text{(2.31)}\end{array}$$ The degenerate cyclotomic Hecke algebra ${\mathscr{H}}_{r,k}(b_1,\dots ,b_r)$ is the degenerate affine Hecke algebra ${\mathscr{H}}_k$ with the additional relation (2.31).

Since the composite map $C[y_1,\dots ,y_k]\to {ℬ}_k\to {𝒲}_k\to {\mathscr{H}}_k$ is injective (see [Kle2165457, Theorem 3.2.2]) and the last two maps are surjections, it follows that the polynomial ring $C[y_1,\dots ,y_k]$ is a subalgebra of ${ℬ}_k$ and ${𝒲}_k\text{.}$

A consequence of the relation (2.31) in ${𝒲}_{r,k}(b_1,\dots ,b_r)$ is $$\begin{array}{cc}(z_0\left(u\right)+u-\frac{1}{2})e_1=(u-\frac{1}{2}{(-1)}^r)\left(\prod _{i=1}^r\frac{u+b_i}{u-b_i}\right)e_1\text{.}& \text{(2.32)}\end{array}$$ This equation makes the data of the values $b_i$ almost equivalent to the data of the $z_0^{\left(\ell \right)}\text{.}$

The affine braid group $B_k$

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 $$\begin{array}{cccccc}{T}_i{T}_j& =& T_j{T}_i,& \text{if}\hspace{0.17em}j\ne i\pm 1,& & \text{(2.33)}\\ T_i{T}_{i+1}{T}_i& =& T_{i+1}{T}_i{T}_{i+1},& \text{for}\hspace{0.17em}i=1,2,\dots ,k-2,& & \text{(2.34)}\\ X^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}{T}_1& =& T_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1},& & & \text{(2.35)}\\ X^{{\epsilon }_1}{T}_i& =& T_i{X}^{{\epsilon }_1},& \text{for}\hspace{0.17em}i=2,3,\dots ,k-1\text{.}& & \text{(2.36)}\end{array}$$ The affine braid group is isomorphic to the group of braids in the thickened annulus, where the generators $T_i$ and $X^{{\epsilon }_1}$ are identified with the diagrams $$\begin{array}{cc}{T}_i=\begin{array}{c} i i+1 \end{array}\text{and}{X}^{{\epsilon }_1}=\begin{array}{c} \end{array}\text{.}& \text{(2.37)}\end{array}$$ For $i=1,\dots ,k$ define $$\begin{array}{cc}{X}^{{\epsilon }_i}=T_{i-1}{T}_{i-2}\cdots T_2{T}_1{X}^{{\epsilon }_1}{T}_1{T}_2\cdots T_{i-2}{T}_{i-1}=\begin{array}{c} i \end{array}\text{.}& \text{(2.38)}\end{array}$$ 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 $X^{{\epsilon }_i}$ all commute with each other.

The affine BMW algebra $W_k$

Let $C$ be a commutative ring and let $C{B}_k$ be the group algebra of the affine braid group. Fix constants $$q,z\in C\text{and}{Z}_0^{\left(\ell \right)}\in C,\text{for}\hspace{0.17em}\ell \in \mathbb{Z},$$ with $q$ and $z$ invertible. Let $Y_i=z{X}^{{\epsilon }_i}$ so that $$\begin{array}{cc}{Y}_1=z{X}^{{\epsilon }_1},Y_i=T_{i-1}{Y}_{i-1}{T}_{i-1},\text{and}{Y}_i{Y}_j=Y_j{Y}_i,\text{for}\hspace{0.17em}1\le i,j\le k\text{.}& \text{(2.39)}\end{array}$$ In the affine braid group $$\begin{array}{cc}{T}_i{Y}_i{Y}_{i+1}=Y_i{Y}_{i+1}{T}_i\text{.}& \text{(2.40)}\end{array}$$ Assume $q-q^{-1}$ is invertible in $C$ and define $E_i$ in the group algebra of the affine braid group by $$\begin{array}{cc}{T}_i{Y}_i=Y_{i+1}{T}_i-(q-q^{-1})Y_{i+1}(1-E_i)\text{.}& \text{(2.41)}\end{array}$$ The affine BMW algebra $W_k$ is the quotient of the group algebra ${CB}_k$ of the affine braid group $B_k$ by the relations $$\begin{array}{cc}{E}_i{T}_i^{\pm 1}=T_i^{\pm 1}{E}_i=z^{\mp 1}{E}_i,E_i{T}_{i-1}^{\pm 1}{E}_i=E_i{T}_{i+1}^{\pm 1}{E}_i=z^{\pm 1}{E}_i,& \text{(2.42)}\\ E_1{Y}_1^{\ell }{E}_1=Z_0^{\left(\ell \right)}{E}_1,E_i{Y}_i{Y}_{i+1}=E_i=Y_i{Y}_{i+1}{E}_i\text{.}& \text{(2.43)}\end{array}$$ Since $Y_{i+1}^{-1}\left(T_i{Y}_i\right)Y_{i+1}=Y_{i+1}^{-1}{Y}_i{Y}_{i+1}{T}_i=Y_i{T}_i,$ conjugating (2.41) by $Y_{i+1}^{-1}$ gives $$\begin{array}{cc}{Y}_i{T}_i=T_i{Y}_{i+1}-(q-q^{-1})(1-E_i)Y_{i+1}\text{.}& \text{(2.44)}\end{array}$$ Left multiplying (2.41) by $Y_{i+1}^{-1}$ and using the second identity in (2.39) shows that (2.41) is equivalent to $T_i-T_i^{-1}=(q-q^{-1})(1-E_i),$ so that $$\begin{array}{cc}{E}_i=1-\frac{T_i-T_i^{-1}}{q-q^{-1}}\text{and}{T}_i{T}_{i+1}{E}_i{T}_{i+1}^{-1}{T}_i^{-1}=E_{i+1}\text{.}& \text{(2.45)}\end{array}$$ Multiply the second relation in (2.45) on the left and the right by $E_i,$ and then use the relations in (2.42) to get $$E_i{E}_{i+1}{E}_i=E_i{T}_i{T}_{i+1}{E}_i{T}_{i+1}^{-1}{T}_i^{-1}{E}_i=E_i{T}_{i+1}{E}_i{T}_{i+1}^{-1}{E}_i=z{E}_i{T}_{i+1}^{-1}{E}_i=E_i,$$ so that $$\begin{array}{cc}{E}_i{E}_{i\pm 1}{E}_i=E_i,\text{and}{E}_i^2=(1+\frac{z-z^{-1}}{q-q^{-1}})E_i& \text{(2.46)}\end{array}$$ is obtained by multiplying the first equation in (2.45) by $E_i$ and using (2.42). Thus from the first relation in (2.43), $$\begin{array}{cc}{Z}_0^{\left(0\right)}=1+\frac{z-z^{-1}}{q-q^{-1}}\text{and}(T_i-z^{-1})(T_i+q^{-1})(T_i-q)=0,& \text{(2.47)}\end{array}$$ since $(T_i-z^{-1})(T_i+q^{-1})(T_i-q)T_i^{-1}=(T_i-z^{-1})(T_i^2-(q-q^{-1})T_i-1)T_i^{-1}=(T_i-z^{-1})(T_i-T_i^{-1}-(q-q^{-1}))=(T_i-z^{-1})(q-q^{-1})(-E_i)=-(z^{-1}-z^{-1})(q-q^{-1})=0\text{.}$ The relations $$\begin{array}{cc}$ E_{i+1}$E_i=E_{i+1}{T}_i{T}_{i+1},E_i{E}_{i+1}=T_{i+1}^{-1}{T}_i^{-1}{E}_{i+1},& \text{(2.48)}\\ T_i{E}_{i+1}{E}_i=T_{i+1}^{-1}{E}_i,\text{and}{E}_{i+1}{E}_i{T}_{i+1}=E_{i+1}{T}_i^{-1},& \text{(2.49)}\end{array}$$ follow from the computations $$\begin{array}{c}{E}_{i+1}{T}_i{T}_{i+1}=z\left(E_{i+1}{T}_i^{-1}{E}_{i+1}\right)T_i{T}_{i+1}=z\left(z^{-1}{E}_{i+1}{T}_{i+1}^{-1}\right)T_i^{-1}{E}_{i+1}{T}_i{T}_{i+1}=E_{i+1}{E}_i,\\ T_{i+1}^{-1}{T}_i^{-1}{E}_{i+1}=T_{i+1}^{-1}{T}_i^{-1}\left(z^{-1}{E}_{i+1}{T}_i{E}_{i+1}\right)=T_{i+1}^{-1}{T}_i^{-1}{z}^{-1}{E}_{i+1}{T}_{i}z{T}_{i+1}{E}_{i+1}=E_i{E}_{i+1},\\ T_i{E}_{i+1}{E}_i=T_i{E}_{i+1}\left(T_i^{-1}{E}_i{z}^{-1}\right)=z^{-1}{T}_{i+1}^{-1}{E}_i{T}_{i+1}{E}_i{z}^{-1}=T_{i+1}^{-1}z{E}_i{z}^{-1}=T_{i+1}^{-1}{E}_i,\text{and}\\ E_{i+1}{E}_i{T}_{i+1}=E_{i+1}{T}_{i+1}^{-1}z{E}_i{T}_{i+1}=z{E}_{i+1}{T}_i{E}_{i+1}{T}_i^{-1}=z{z}^{-1}{E}_{i+1}{T}_i^{-1}=E_{i+1}{T}_i^{-1}\text{.}\end{array}$$

A consequence (see (3.26)) of the defining relations of $W_k$ is the equation $$(Z_0^{-}-\frac{z}{q-q^{-1}}-\frac{u^2}{u^2-1})(Z_0^{-}+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1})E_1=\frac{-(u^2-q^2)(u^2-q^{-2})}{(u^2-1){(q-q^{-1})}^2}{E}_1,$$ where $Z_0^{+}$ and $Z_0^{-}$ are the generating functions $$Z_0^{+}=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Z}_0^{\left(\ell \right)}{u}^{-\ell }\text{and}{Z}_0^{-}=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Z}_0^{\left(\ell \right)}{u}^{-\ell }\text{.}$$ This means that, unless the parameters $Z_0^{\left(\ell \right)}$ are chosen carefully, it is likely that $E_1=0$ in $W_k\text{.}$

From the point of view of Schur-Weyl duality for the affine BMW algebra (see [ORa0401317] and [RamNotes]) the natural choice of base ring is the center of the quantum group corresponding to the orthogonal or symplectic Lie algebra which, by the (quantum version) of the Harish-Chandra isomorphism, is isomorphic to the subring of symmetric Laurent polynomials given by $$C=\{z\in ℂ{[L_1^{\pm 1},\dots ,L_r^{\pm 1}]}^{S_r}\hspace{0.17em}|\hspace{0.17em}z(L_1,L_2,\dots ,L_r)=z(L_1^{-1},L_2,\dots ,L_r)\},$$ where the symmetric group $S_r$ acts by permuting the variables $L_1,\dots ,L_r\text{.}$ Here the constants $Z_0^{\left(\ell \right)}\in C$ are given, explicitly, by setting the generating functions $Z_0^{+}$ and $Z_0^{-}$ equal, up to a normalization, to $$\prod _{i=1}^r\frac{(u-q{L}_i)}{(u-q^{-1}{L}_i)}·\frac{(u-q{L}_i^{-1})}{(u-q^{-1}{L}_i^{-1})}\text{and}\prod _{i=1}^r\frac{(u-q^{-1}{L}_i)}{(u-q{L}_i)}·\frac{(u-q^{-1}{L}_i^{-1})}{(u-q{L}_i^{-1})},$$ respectively. This choice of $C$ and the $Z_0^{\left(\ell \right)}$ are the universal admissible parameters for $W_k\text{.}$ This point of view will be explained in [DRV1205.1852v1].

Careful manipulation of the defining relations of $W_k$ provides an inductive presentation of $w_k$ as $$W_k=W_{k-1}{E}_{k-1}{W}_{k-1}+W_{k-1}{T}_{k-1}{W}_{k-1}+W_{k-1}{T}_{k-1}^{-1}{W}_{k-1}+\sum _{\ell \in \mathbb{Z}}{W}_{k-1}{Y}_k^{\ell }{W}_{k-1}$$ (see [GHa0411155, Prop. 3.16] or [Här1834081]), and provides that $$E_k{W}_k{E}_k=W_{k-1}{E}_k\text{and}\begin{array}{ccc}{W}_k& ⟶& W_{k-1}{E}_k\\ b& ⟼& E_{k}b{E}_k\end{array}$$ is a $(W_{k-1},W_{k-1})\text{-bimodule}$ homomorphism (see [GHa0411155, Prop. 3.17]). These structural facts are important to the understanding of $W_k$ by “Jones basic constructions”. Under the conditions of Theorem 4.4(a) it is true, but not immediate from the defining relations, that the natural homomorphism $W_{k-1}\to W_k$ is injective so that $W_{k-1}$ is a subalgebra of $W_k$ (see [GHa0411155, Cor. 6.15]).

Quotients of $W_k$

The affine Hecke algebra $H_k$ is the affine BMW algebra $W_k$ with the additional relations $$\begin{array}{cc}{E}_i=0,\text{for}\hspace{0.17em}i=1,\dots ,k-1\text{.}& \text{(2.50)}\end{array}$$ Fix $b_1,\dots ,b_r\in C\text{.}$ The cyclotomic BMW algebra $W_{r,k}(b_1,\dots ,b_r)$ is the affine BMW algebra $W_k$ with the additional relation $$\begin{array}{cc}(Y_1-b_1)\cdots (Y_1-b_r)=0\text{.}& \text{(2.51)}\end{array}$$ The cyclotomic Hecke algebra $H_{r,k}(b_1,\dots ,b_r)$ is the affine Hecke algebra $H_k$ with the additional relation (2.51).

Since the composite map $C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]\to {CB}_k\to W_k\to H_k$ is injective and the last two maps are surjections, it follows that the Laurent polynomial ring $C[Y_1^{\pm 1},\dots ,Y_k^{\pm 1}]$ is a subalgebra of ${CB}_k$ and $W_k\text{.}$

A consequence of the relation (2.51) in $W_{r,k}(b_1,\dots ,b_r)$ is $$\begin{array}{cc}(Z_0^{+}+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1})E_1=(\frac{z}{q-q^{-1}}+\frac{uz}{u^2-1})\left(\prod _{j=1}^r\frac{u-b_j^{-1}}{u-b_j}\right)E_1\text{.}& \text{(2.52)}\end{array}$$ This equation makes the data of the values $b_i$ almost equivalent to the data of the $Z_0^{\left(\ell \right)}\text{.}$

Notes and references

This is a typed exert of the paper Affine and degenerate affine BMW algebras: The center by Zajj Daugherty, Arun Ram and Rahbar Virk.

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