## Affine and degenerate affine BMW algebras: The center

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 ${ℋ}_{k}$ 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 ${U}_{q}𝔤\text{.}$ 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 $\left\{1,\dots ,k\right\}\text{.}$ 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 $tu (u∈Sk), κ0,κ1, andy1,…, yk, (2.1)$ with relations $tutv= tuv, yiyj=yjyi, κ0κ1=κ1 κ0,κ0yi= yiκ0,κ1 yi=yiκ1, (2.2) κ0tsi= tsiκ0, κ1ts1κ1 ts1=ts1 κ1ts1κ1 ,andκ1 tsj=tsj κ1, for j≠1, (2.3) tsi(yi+yi+1) =(yi+yi+1) tsi,and yjtsi=tsiyj, for j≠i,i+1, (2.4)$ and $tsitsi+1 γi,i+1 tsi+1tsi= γi+1,i+2, whereγi,i+1= yi+1-tsi yitsi for i=1,…,k-2. (2.5)$

In the degenerate affine braid algebra ${ℬ}_{k}$ let ${c}_{0}={\kappa }_{0}$ and $cj=κ0+2 (y1+…+yj), so thatyj=12 (cj-cj-1) ,for j=1,…,k. (2.6)$ Then ${c}_{0},\dots ,{c}_{k}$ commute with each other, commute with ${\kappa }_{1},$ and the relations (2.4) are equivalent to $tsicj=cj tsi,for j≠i. (2.7)$

The degenerate affine braid algebra ${ℬ}_{k}$ has another presentation by generators $tu, for u∈ Sk,κ0,…, κkand γi,j,for 0≤i,j≤k with i≠j, (2.8)$ and relations $tutv= tuv, twκi tw-1= κw(i), twγi,j tw-1= γw(i),w(j), (2.9) κiκj= κjκi, κiγℓ,m= γℓ,mκi, (2.10) γi,j=γj,i, γp,rγℓ,m =γℓ,mγp,r, andγi,j (γi,r+γj,r) =(γi,r+γj,r) γi,j, (2.11)$ 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 $[κr,γℓ,m]=0, [γi,j,γℓ,m]=0, and [γi,j,γi,m]= [γi,m,γj,m], (2.12)$ 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 $κ0=κ0, κ1=κ1, tw=tw, (2.13) γ0,1=y1- 12κ1,and γj,j+1= yj+1-tsj yjtsj, for j=1,…,k-1, (2.14)$ and $κm=tuκ1 tu-1, γ0,m=tu γ0,1tu-1 andγi,j= tvγ1,2 tv-1, (2.15)$ 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 $κ0=κ0, κ1=κ1, tw=tw,and yj=12κj+ ∑0≤ℓ 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 Using the relations in (2.10) and (2.11), $[yi,yj] = [ 12κi+∑ℓ (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 $ts1γ1,2ts1- γ1,2= (ts1y2ts1-y1) -y2+ts1y1ts1 =ts1(y1+y2) ts1-(y1+y2)=0. (2.17)$ 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 $\left[{\kappa }_{1},{\kappa }_{2}\right]=0\text{.}$ Using this and the relations in (2.2), $[κ1,γ0,2+γ1,2]= [ κ1, (y2-12κ2-γ1,2) +γ1,2 ] =[κ1,12κ2]=0, (2.18)$ and $[γ0,1,γ0,2+γ1,2]= [y1-12κ1,y2-12κ2]= 14[κ1,κ2]=0, (2.19)$ so that $[γ0,1,κ2]= [γ0,1,2y2-2(γ0,2+γ1,2)]= [γ0,1,2y2]= [y1-12κ1,2y2]= -[κ1,y2]=0.$ Conjugating the last relation by ${t}_{{s}_{1}}$ gives $[κ1,γ0,2]=0, and thus [κ1,γ1,2]=0,$ by (2.18). By the third and fourth relations in (2.2), $[κ0,γ0,1]= [κ0,y1-12κ1]=0, and [κ1,γ0,1]= [κ1,y1-12κ1]=0.$ By the relations in (2.3) and (2.2), $[κ0,γ1,2]= [κ0,y2-ts1y1ts1]=0 and[κ1,γ2,3] =[κ1,y3-ts2y2ts2]=0.$ 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) $γ1,2γ3,4= (y2-ts1y1ts1) (y4-ts3y3ts3)= (y4-ts3y3ts3) (y2-ts1y1ts1)= γ3,4γ1,2.$ By the last relation in (2.2) and the last relation in (2.3), $[γ0,1,γ2,3]= [y1-12κ1,y3-ts2y2ts2]=0.$ 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 $\left[{\gamma }_{0,2},{\gamma }_{0,3}+{\gamma }_{2,3}\right]=0,$ and this and the (already established) relations in (2.10) and the first set of relations in (2.11) provide $0 = [y2,y3]= [ 12κ2+γ0,2+ γ1,2,12κ3+ γ0,3+γ1,3+ γ2,3 ] = [ γ0,2+γ1,2, γ0,3+γ1,3+γ2,3 ] = [ γ1,2,γ0,3+ γ1,3+γ2,3 ] =[γ1,2,γ1,3+γ2,3].$ Note also that $[γ1,2,γ1,0+γ2,0] = [γ1,2,γ0,1+γ0,2]= -[γ0,1,γ1,2]+ [γ1,2,γ0,2] = [γ0,1,γ0,2]+ [γ1,2,γ0,2]= ts1[γ0,2+γ1,2,γ0,1] ts1=0,$ 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), $cj=∑i=0j κi+2 ∑0≤ℓ

### 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 $tsiyi= yi+1tsi- (1-ei), for i=1,2,…,k-1, (2.21)$ so that, with ${\gamma }_{i,i+1}$ as in (2.5), $γi,i+1tsi =1-ei. (2.22)$

Fix constants $ε=±1and z0(ℓ)∈C, for ℓ∈ℤ≥0.$ 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 $eitsi= tsiei= εei, eitsi-1 ei=eitsi+1 ei=εei, (2.23) e1y1ℓe1= z0(ℓ)e1, ei (yi+yi+1) =0=(yi+yi+1) ei. (2.24)$ Conjugating (2.21) by ${t}_{{s}_{i}}$ and using the first relation in (2.23) gives $yitsi=tsi yi+1-(1-ei). (2.25)$ Then, by (2.22) and (2.5), $γi,i+1=tsi -εei,and ei+1=tsi tsi+1ei tsi+1tsi. (2.26)$ 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 $eiei+1ei= eitsitsi+1 eitsi+1tsi ei=eitsi+1 eitsi+1ei= εeitsi+1ei =ei,$ so that $eiei±1ei =ei.Note that ei2=z0(0) ei (2.27)$ is a special case of the first identity in (2.24). The relations $ei+1ei= ei+1tsi tsi+1, tiei+1= tsi+1tsi ei+1, (2.28) tsiei+1ei =tsi+1ei, andei+1ei tsi+1=ei+1 tsi (2.29)$ result from $ei+1tsitsi+1 = εei+1tsiei+1 tsitsi+1=ei+1 tsi+1tsiei+1tsi tsi+1=ei+1ei, tsi+1tsiei+1 = εtsi+1tsi ei+1tsiei+1= tsi+1tsi ei+1tsi tsi+1ei+1= eiei+1, tsiei+1ei = εtsiei+1tsi ei=εtsi+1ei tsi+1ei= tsi+1ei,and ei+1eitsi+1 = εei+1tsi+1ei tsi+1=ε ei+1tsiei+1 tsi=ei+1tsi.$

A consequence (see (3.7)) of the defining relations of ${𝒲}_{k}$ is the equation $(z0(-u)-(12+εu)) (z0(u)-(12-εu)) e1=(12-εu) (12+εu)e1,$ where ${z}_{0}\left(u\right)$ is the generating function $z0(u)= ∑ℓ∈ℤ≥0 z0(ℓ)u-ℓ.$ 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 $C= { z∈ℂ[h1,…,hr]Sr | z(h1,…,hr)= z(-h1,h2,…,hr) } ,$ 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 $z0(u) equal, up to a normalization, to ∏i=1r (u+12+hi) (u+12-hi) (u-12-hi) (u-12+hi) .$ 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 $𝒲k=𝒲k-1 ek-1𝒲k-1+ 𝒲k-1tsk-1 𝒲k-1+ ∑ℓ∈ℤ≥0 𝒲k-1ykℓ 𝒲k-1,$ and provides that $ek𝒲kek= 𝒲k-1ek, and 𝒲k ⟶ 𝒲k-1ek b ⟼ ekbek$ is a $\left({𝒲}_{k-1},{𝒲}_{k-1}\right)\text{-bimodule}$ 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 ${ℋ}_{k}$ is the quotient of ${𝒲}_{k}$ by the relations $ei=0,for i =1,…,k-1. (2.30)$ Fix ${b}_{1},\dots ,{b}_{r}\in ℂ\text{.}$ The degenerate cyclotomic BMW algebra ${𝒲}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is the degenerate affine BMW algebra with the additional relation $(y1-b1)⋯ (y1-br)=0. (2.31)$ The degenerate cyclotomic Hecke algebra ${ℋ}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is the degenerate affine Hecke algebra ${ℋ}_{k}$ with the additional relation (2.31).

Since the composite map $C\left[{y}_{1},\dots ,{y}_{k}\right]\to {ℬ}_{k}\to {𝒲}_{k}\to {ℋ}_{k}$ is injective (see [Kle2165457, Theorem 3.2.2]) and the last two maps are surjections, it follows that the polynomial ring $C\left[{y}_{1},\dots ,{y}_{k}\right]$ is a subalgebra of ${ℬ}_{k}$ and ${𝒲}_{k}\text{.}$

A consequence of the relation (2.31) in ${𝒲}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is $(z0(u)+u-12) e1=(u-12(-1)r) ( ∏i=1r u+bi u-bi ) e1. (2.32)$ 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 $TiTj = TjTi, if j≠i±1, (2.33) TiTi+1Ti = Ti+1Ti Ti+1, for i=1,2,…,k-2, (2.34) Xε1T1 Xε1T1 = T1Xε1 T1Xε1, (2.35) Xε1Ti = TiXε1, for i=2,3,…,k-1. (2.36)$ 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 For $i=1,\dots ,k$ define The pictorial computation 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 $q,z∈Cand Z0(ℓ)∈C, for ℓ∈ℤ,$ with $q$ and $z$ invertible. Let ${Y}_{i}=z{X}^{{\epsilon }_{i}}$ so that $Y1=zXε1, Yi=Ti-1 Yi-1Ti-1, andYiYj= YjYi,for 1 ≤i,j≤k. (2.39)$ In the affine braid group $TiYiYi+1= YiYi+1Ti. (2.40)$ Assume $q-{q}^{-1}$ is invertible in $C$ and define ${E}_{i}$ in the group algebra of the affine braid group by $TiYi=Yi+1 Ti-(q-q-1) Yi+1(1-Ei). (2.41)$ 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 $EiTi±1= Ti±1Ei= z∓1Ei, EiTi-1±1 Ei=Ei Ti+1±1Ei =z±1Ei, (2.42) E1Y1ℓE1= Z0(ℓ)E1, EiYiYi+1 =Ei=Yi Yi+1Ei. (2.43)$ 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 $YiTi=Ti Yi+1- (q-q-1) (1-Ei) Yi+1. (2.44)$ Left multiplying (2.41) by ${Y}_{i+1}^{-1}$ and using the second identity in (2.39) shows that (2.41) is equivalent to ${T}_{i}-{T}_{i}^{-1}=\left(q-{q}^{-1}\right)\left(1-{E}_{i}\right),$ so that $Ei=1- Ti-Ti-1 q-q-1 and TiTi+1 Ei Ti+1-1 Ti-1= Ei+1. (2.45)$ 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 $EiEi+1Ei= EiTiTi+1 EiTi+1-1 Ti-1Ei=Ei Ti+1Ei Ti+1-1Ei =zEiTi+1-1 Ei=Ei,$ so that $EiEi±1Ei =Ei ,andEi2= (1+z-z-1q-q-1) Ei (2.46)$ 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), $Z0(0)=1+ z-z-1 q-q-1 and (Ti-z-1) (Ti+q-1) (Ti-q)=0, (2.47)$ since $\left({T}_{i}-{z}^{-1}\right)\left({T}_{i}+{q}^{-1}\right)\left({T}_{i}-q\right){T}_{i}^{-1}=\left({T}_{i}-{z}^{-1}\right)\left({T}_{i}^{2}-\left(q-{q}^{-1}\right){T}_{i}-1\right){T}_{i}^{-1}=\left({T}_{i}-{z}^{-1}\right)\left({T}_{i}-{T}_{i}^{-1}-\left(q-{q}^{-1}\right)\right)=\left({T}_{i}-{z}^{-1}\right)\left(q-{q}^{-1}\right)\left(-{E}_{i}\right)=-\left({z}^{-1}-{z}^{-1}\right)\left(q-{q}^{-1}\right)=0\text{.}$ The relations ${E}_{i+1}Ei= Ei+1Ti Ti+1, EiEi+1= Ti+1-1 Ti-1 Ei+1, (2.48) TiEi+1Ei =Ti+1-1 Ei,and Ei+1Ei Ti+1= Ei+1Ti-1, (2.49)$ follow from the computations $Ei+1TiTi+1= z(Ei+1Ti-1Ei+1) TiTi+1=z (z-1Ei+1Ti+1-1) Ti-1Ei+1Ti Ti+1=Ei+1Ei, Ti+1-1 Ti-1 Ei+1= Ti+1-1 Ti-1 (z-1Ei+1TiEi+1) =Ti+1-1 Ti-1z-1 Ei+1Tiz Ti+1Ei+1= EiEi+1, TiEi+1Ei= TiEi+1 (Ti-1Eiz-1) =z-1 Ti+1-1Ei Ti+1Eiz-1 =Ti+1-1z Eiz-1= Ti+1-1Ei, and Ei+1EiTi+1 =Ei+1Ti+1-1 zEiTi+1=z Ei+1TiEi+1 Ti-1=zz-1 Ei+1Ti-1= Ei+1Ti-1.$

A consequence (see (3.26)) of the defining relations of ${W}_{k}$ is the equation $( Z0-- zq-q-1- u2u2-1 ) ( Z0-+ z-1q-q-1- u2u2-1 ) E1= -(u2-q2) (u2-q-2) (u2-1) (q-q-1)2 E1,$ where ${Z}_{0}^{+}$ and ${Z}_{0}^{-}$ are the generating functions $Z0+= ∑ℓ∈ℤ≥0 Z0(ℓ)u-ℓ andZ0-= ∑ℓ∈ℤ≥0 Z0(ℓ)u-ℓ.$ 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 $C= { z∈ℂ [L1±1,…,Lr±1]Sr | z(L1,L2,…,Lr)= z(L1-1,L2,…,Lr) } ,$ 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 $∏i=1r (u-qLi) (u-q-1Li) · (u-qLi-1) (u-q-1Li-1) and ∏i=1r (u-q-1Li) (u-qLi) · (u-q-1Li-1) (u-qLi-1) ,$ 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 $Wk=Wk-1Ek-1 Wk-1+Wk-1 Tk-1Wk-1+ Wk-1Tk-1-1 Wk-1+∑ℓ∈ℤ Wk-1Ykℓ Wk-1$ (see [GHa0411155, Prop. 3.16] or [Här1834081]), and provides that $EkWkEk= Wk-1Ekand Wk ⟶ Wk-1Ek b ⟼ EkbEk$ is a $\left({W}_{k-1},{W}_{k-1}\right)\text{-bimodule}$ 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 $Ei=0,for i= 1,…,k-1. (2.50)$ Fix ${b}_{1},\dots ,{b}_{r}\in C\text{.}$ The cyclotomic BMW algebra ${W}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is the affine BMW algebra ${W}_{k}$ with the additional relation $(Y1-b1)⋯ (Y1-br)=0. (2.51)$ The cyclotomic Hecke algebra ${H}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is the affine Hecke algebra ${H}_{k}$ with the additional relation (2.51).

Since the composite map $C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]\to {CB}_{k}\to {W}_{k}\to {H}_{k}$ is injective and the last two maps are surjections, it follows that the Laurent polynomial ring $C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$ is a subalgebra of ${CB}_{k}$ and ${W}_{k}\text{.}$

A consequence of the relation (2.51) in ${W}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is $( Z0++ z-1q-q-1- u2u2-1 ) E1= ( zq-q-1+ uzu2-1 ) ( ∏j=1r u-bj-1 u-bj ) E1. (2.52)$ 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.