## Affine and degenerate affine BMW algebras: The center

Last update: 21 October 2013

## Identities in affine and degenerate affine BMW algebras

In [Naz1996], Nazarov defined some naturally occurring central elements in the degenerate affine BMW algebra ${𝒲}_{k}$ and proved a remarkable recursion for them. This recursion was generalized to analogous central elements in the affine BMW algebra ${W}_{k}$ by Beliakova-Blanchet [BBl1866492]. In both cases, the recursion was accomplished with an involved computation. In this section, we provide a new proof of the Nazarov and Beliakov-Blanchet recursions by lifting them out of the center, to intertwiner-like identities in ${𝒲}_{k}$ and ${W}_{k}$ (Propositions 3.1 and 3.5). These intertwiner-like identities for the degenerate affine and affine BMW algebras are reminiscent of the intertwiner identities for the degenerate affine and affine Hecke algebras found, for example, in [KRa2002, Prop. 2.5(c)] and [Ram2003, Prop. 2.14(c)], respectively. The central element recursions of [Naz1996] and [BBl1866492] are then obtained by multiplying the intertwiner-like identities by the projectors ${e}_{k}$ and ${E}_{k},$ respectively. We have carefully arranged the proofs so that the degenerate affine and the affine cases are exactly in parallel.

### The degenerate affine case

Let $u$ be a variable, $ui+= 1u-yi, and note that ui+ui+1+= 12u-(yi+yi+1) (ui+ui+1+). (3.1)$ By (2.25) and the definition of ${e}_{i}$ in (2.21), $(u-yi+1)tsi =tsi(u-yi)- (1-ei)and (u-yi)tsi= tsi(u-yi+1) +(1-ei),$ which give $tsiui+=ui+1+ tsi+ui+1+ eiui+-ui+1+ ui+,andtsi ui+1+=ui+tsi -ui+eiui+1++ ui+1+ui+, (3.2)$ respectively.

In the degenerate affine BMW algebra ${𝒲}_{i+1},$ $( ei11-yi+1 -tsi- 12u-(yi+yi+1) ) ( ei11-yi+ tsi- 12u-(yi+yi+1) ) = -(2u-(yi+yi+1)+1) (2u-(yi+yi+1)-1) (2u-(yi+yi+1))2 , (3.3)$ and $( ui+1++tsi-ei 12u-(yi+yi+1) ) -ui+ ( ui+1++tsi-ei 12u-(yi+yi+1) ) ui+ = ( tsiui+tsi+ tsi-ei 12u-(yi+yi+1) ) -ui+1+ ( eiui+ei+εei -ei 12u-(yi+yi+1) ) ui+1+. (3.4)$

 Proof. Putting (3.1) into the first identity in (3.2) says that if $A=tsi+ 12u-(yi+yi+1) andB=eiui++tsi- 12u-(yi+yi+1)$ then $Aui+=ui+1+B, andAei=eiA$ follows from (2.23) and (2.24). So $( eiui+1+-tsi- 12u-(yi+yi+1) ) ( eiui++tsi- 12u-(yi+yi+1) ) =eiui+1+B-AB= eiAui+-AB=Aei ui+-AB=A(eiui+-B) = - ( tsi+ 12u-(yi+yi+1) ) ( tsi- 12u-(yi+yi+1) ) ,$ and multiplying out the right hand side gives (3.3). Multiplying the second relation in (3.2) by ${t}_{{s}_{i}}$ gives $ui+1+-tsi ui+1+ui+= tsiui+tsi -tsiui+ei ui-$ and again using the relations in (3.2) gives $ui+1+-ui+ (tsi-eiui+1++ui+1+) ui+=tsiui+tsi -ui+1+ (tsi+eiui+-ui+) eiui+1+.$ Using (3.1) and $addingtsi-ei (12u-(yi+yi+1))- 12u-(yi+yi+1) ui+eiui+1+ to each side$ gives $( ui+1++tsi-ei 12u-(yi+yi+1) ) +ui+ ( ui+1++tsi-ei 12u-(yi+yi+1) ) ui+ = tsiui+tsi+tsi-ei 12u-(yi+yi+1) -ui+1+ ( eiui++tsi- 12u-(yi+yi+1) ) eiui+1+ = ( tsiui+tsi+tsi-ei 12u-(yi+yi+1) ) -ui+1+ ( eiui+ei+εei-ei 12u-(yi+yi+1) ) ui+1+,$ completing the proof of (3.4). $\square$

Introduce notation ${z}_{i-1}^{\left(\ell \right)}{e}_{i}$ and the generating function ${z}_{i-1}\left(u\right){e}_{i}$ by $zi-1(u)ei= ∑ℓ∈ℤ≥0 zi-1(ℓ)eiu-ℓ =ei(∑ℓ∈ℤ≥0yiℓu-ℓ) ei=ei11-yiu-1ei, (3.5)$ By [AMR0506467, Lemma 4.15], or the identity (3.9) below, ${z}_{i-1}^{\left(\ell \right)}\in {𝒲}_{i-1}$ for $\ell \in {ℤ}_{\ge 0}\text{.}$ If $ui-=1u+yi thenei ui+1+=ei ui-, ui+1+ei= ui-ei, eiui±ei= zi-1(±u)u ei, (3.6)$ where, for $i=1,$ the last identity is a restatement of the first identity in (2.24). The identities (3.7), (3.8), and (3.9) of the following theorem are [Naz1996, Lemma 2.5], [Naz1996, Prop. 4.2] and [Naz1996, Lemma 3.8], respectively.

Let ${z}_{i-1}^{\left(\ell \right)}$ and ${z}_{i-1}\left(u\right)$ be as defined in (3.5). Then ${z}_{i-1}^{\left(\ell \right)}\in Z\left({𝒲}_{i-1}\right),$ $( zi-1(-u)- (12+εu) ) ( zi-1(u)- (12-εu) ) ei=(12-εu) (12+εu)ei, (3.7) (zi(u)+εu-12) ei+1= (zi-1(u)+εu-12) ( ((u+yi)2-1) (u-yi)2 ((u-yi)2-1) (u+yi)2 ) ei+1,and (3.8) (zk-1(u)+εu-12) ei+1= (z0(u)+εu-12) ∏i=1k-1 (u+yi-1) (u+yi+1) (u-yi)2 (u+yi)2 (u-yi+1) (u-yi-1) ei+1. (3.9)$

 Proof. Since the generators ${t}_{{s}_{1}},\dots ,{t}_{{s}_{i-2}},{e}_{1}\dots ,{e}_{i-2}$ and ${y}_{1},\dots ,{y}_{i-1}$ of ${𝒲}_{i-1}$ all commute with ${e}_{i}$ and ${y}_{i}$ it follows that ${z}_{i-1}^{\left(\ell \right)}\in Z\left({𝒲}_{i-1}\right)\text{.}$ Multiply (3.3) on the right by ${e}_{i}$ to get (3.7), since $\left(\frac{1}{2}-u\right)\left(\frac{1}{2}+u\right)=\left(\frac{1}{2}-\epsilon u\right)\left(\frac{1}{2}+\epsilon u\right)\text{.}$ Multiplying (3.4) on the left and right by ${e}_{i+1}$ and using the relations in (2.27), (2.28) and (2.29), $ei+1tsiui+tsi ei+1=ei+1tsi tsi+1ui+tsi+1 tsiei+1=ei+1 eiui+eiei+1, and ei+1ui+1+ei ui+1+ei+1= ei+1ui-eiui- ei+1=ui-ei+1 eiei+1ui-= (ui-)2ei+1,$ gives $( zi(u)u+ ε-12u ) (1-(ui+)2) ei+1= ( zi-1(u)u +ε-12u ) (1-(ui-)2) ei+1.$ So (3.8) follows from $1-(ui-)2 1-(ui+)2 = 1-(1u+yi)2 1-(1u-yi)2 = (u2+2yiu+yi2-1) (u-yi)2 (u2-2yiu+yi2-1) (u+yi)2 = (u+yi-1) (u+yi+1) (u-yi)2 (u-yi-1) (u-yi+1) (u+yi)2 .$ Finally, relation (3.9) follows, by induction, from (3.8). $\square$

Using the expansion $1u-a= u-11-au-1 =∑ℓ∈ℤ≥1 aℓ-1u-ℓ,$ and taking the coefficient of ${u}^{-\left(\ell +1\right)}$ on each side of the relations in (3.2) gives $tsiyiℓ = yi+1ℓtsi- ( yi+1ℓ-1 (1-ei)+ yi+1ℓ-2 (1-ei)yi +⋯+(1-ei) yiℓ-1 ) ,and (3.10) tsiyi+1ℓ = yiℓtsi+ yiℓ-1 (1-ei)+ yiℓ-2 (1-ei)yi+1 +⋯+(1-ei) yi+1ℓ-1, (3.11)$ respectively.

Taking the coefficient of ${u}^{-s}$ on each side of (3.7) gives a trivial identity for even $s$ but, for odd $s=2\ell +1,$ gives $( 2zi-1(2ℓ+1) +zi-1(2ℓ) - ( zi-1(2ℓ) zi-1(0) -zi-1(2ℓ-1) zi-1(1) +⋯+ zi-1(0) zi-1(2ℓ) ) ei=0 ) (3.12)$ which is the admissibility relation inAMR0506467AMR, Remark 2.11] (see also [Naz1996, (4.6)].)

### The affine case

Let $u$ be a variable, $Ui+= Yiu-Yi, and note that Ui+Ui+1+= YiYi+1 u2-YiYi+1 (Ui++Ui+1++1). (3.13)$ By the definition of ${E}_{i}$ in (2.41), $(u-Yi+1)Ti =Ti(u-Yi)- (q-q-1)Yi+1 (1-Ei),$ and, by (2.44), $(u-Yi)Ti= Ti(u-Yi+1) +(q-q-1) (1-Ei)Yi+1,$ so that $Ti1u-Yi = 1u-Yi+1 Ti-(q-q-1) Yi+1u-Yi+1 (1-Ei) 1u-Yi,and (3.14) Ti1u-Yi+1 = 1u-YiTi+ (q-q-1) 1u-Yi (1-Ei) Yi+1u-Yi+1. (3.15)$ The relations $TiUi+ = Ui+1+ Ti-1- (q-q-1) Ui+1+ (1-Ei) Ui+ (3.16) = Ui+1+ ( Ti-1- (q-q-1) (1-Ei) Ui+ ) ,and$ $Ti-1 Ui+1+ = Ui+Ti- (q-q-1) Ui+Ei Ui+1++ (q-q-1) Ui+1+ Ui+ (3.17) = Ui+ ( Ti+(q-q-1) (1-Ei)Ui+1+ )$ are obtained by multiplying (3.14) and (3.15) on the right (resp. left) by ${Y}_{i}$ and using the relation ${T}_{i}{Y}_{i}={Y}_{i+1}{T}_{i}^{-1}\text{.}$

Let $Q=q-{q}^{-1}\text{.}$ Then, in the affine BMW algebra ${W}_{i+1},$ $( EiYi+1u-Yi+1 -TiQ- YiYi+1u2-YiYi+1 ) ( EiYiu-Yi -Ti-1Q- YiYi+1u2-YiYi+1 ) = -(u2-q2YiYi+1) (u2-q-2YiYi+1) Q2 (u2-YiYi+1)2 ,and (3.18) ( Ui+1++ TiQ-Ei YiYi+1 u2-YiYi+1 ) -Q2(Ui++1) ( Ui+1++ TiQ-Ei YiYi+1 u2-YiYi+1 ) Ui+ = ( TiUi+ Ti-1+ TiQ-Ei YiYi+1 u2-YiYi+1 ) -Q2Ui+1+ ( EiUi+Ei+ zEiQ-Ei YiYi+1 u2-YiYi+1 ) (Ui+1++1). (3.19)$

 Proof. Putting (3.13) into (3.16) says that if $A=TiQ+ YiYi+1 u2-YiYi+1 andB=Ei Ui++ Ti-1Q- YiYi+1 u2-YiYi+1$ then $AUi+= Ui+1+B- YiYi+1 u2-YiYi+1 .Next,AEi =EiA$ follows from (2.42) and (2.43). So $( EiYi+1u-Yi+1 -TiQ- YiYi+1 u2-YiYi+1 ) ( EiYiu-Yi +Ti-1Q- YiYi+1 u2-YiYi+1 ) = Ei(Ui+1+B) -AB=Ei ( AUi++ YiYi+1 u2-YiYi+1 ) -AB=A (EiUi+-B) +Ei YiYi+1 u2-YiYi+1 = - ( TiQ+ YiYi+1 u2-YiYi+1 ) ( Ti-1Q- YiYi+1 u2-YiYi+1 ) +Ei YiYi+1 u2-YiYi+1 ,$ and, by (2.45), multiplying out the right hand side gives (3.18). Rewrite ${T}_{i}^{-1}{U}_{i+1}^{+}={U}_{i}^{+}{T}_{i}^{-1}+Q{U}_{i}^{+}\left(1-{E}_{i}\right)\left({U}_{i+1}^{+}+1\right)$ as $Ti-1Ui+1+ -Q(Ui+1++1) Ui+=Ui+ Ti-1-QUi+ Ei(Ui+1++1),$ and multiply on the left by ${T}_{i}$ to get $Ui+1+-QTi (Ui+1++1) Ui+=TiUi+ Ti-1-QTi Ui+Ei (Ui+1++1). (3.20)$ Then, since ${T}_{i}={T}_{i}^{-1}+Q\left(1-{E}_{i}\right),$ equations (3.17) and (3.16) imply $Ti(Ui+1++1)= Q(Ui++1) ( TiQ+(1-Ei) Ui+1+ ) andTiUi+= QUi+1+ ( Ti-1Q- (1-Ei) Ui+ ) ,$ and so (3.20) is $Ui+1+-Q2 (ui++1) ( TiQ+ (1-Ei) Ui+1+ ) Ui+ =Ti Ui+Ti-1 -Q2Ui+1+ ( Ti-1Q- (1-Ei)Ui+ ) Ei(Ui+1++1). (3.21)$ Using (3.13) and $addingTiQ-Ei YiYi+1 u2-YiYi+1 -Q2 YiYi+1 u2-YiYi+1 (Ui++1)Ei (Ui+1++1) to each side$ of (3.21) gives $Ui+1++ TiQ-Ei YiYi+1 u2-YiYi+1 -Q2(Ui++1) ( Ui+1++TiQ -Ei YiYi+1 u2-YiYi+1 ) Ui+ =TiUi+ Ti-1+TiQ -Ei YiYi+1 u2-YiYi+1 -Q2Ui+1+ ( EiUi++ Ti-1Q- YiYi+1 u2-YiYi+1 ) Ei(Ui+1++1) =TiUi+ Ti-1+TiQ -Ei YiYi+1 u2-YiYi+1 -Q2Ui+1+ ( EiUi+Ei+ zEiQ-Ei YiYi+1 u2-YiYi+1 ) (Ui+1++1),$ completing the proof of (3.19). $\square$

Introduce notation ${Z}_{i-1}^{\left(\ell \right)}{e}_{i}$ and generating functions ${Z}_{i-1}^{+}{E}_{i}$ and ${Z}_{i-1}^{-}{E}_{i}$ by $Zi-1+Ei= ∑ℓ∈ℤ≥0 Zi-1(ℓ)Ei u-ℓ=Ei ( ∑ℓ∈ℤ≥0 Yiℓu-ℓ ) Ei=Ei 11-Yiu-1 Ei, (3.22) Zi-1-Ei= ∑ℓ∈ℤ≥0 Zi-1(-ℓ)Ei u-ℓ=Ei ( ∑ℓ∈ℤ≥0 Yi-ℓu-ℓ ) Ei=Ei 11-Yi-1u-1 Ei. (3.23)$ By [GHa0411155, Lemma 3.15(1)], or the identity (3.28) below, ${Z}_{i-1}^{\left(\ell \right)}\in {W}_{i-1}$ for $\ell \in ℤ\text{.}$ If $Ui-= Yi-1 u-Yi-1 then Zi-1(0) =1+ z-z-1 q-q-1 , (3.24)$ by the second relation in (2.46), and $EiUi+1+= EiUi-, Ui+1+Ei =Ui-Ei, EiUi±Ei =(Zi-1±-Zi-1(0)) Ei, (3.25)$ where, for $i=1,$ the last identity is a restatement of the first identity in (2.43). In the following theorem, the identity (3.26) is equivalent to [GHa0411155, Lemma 2.8, parts (2)and (3)] or [GMo0612064, Lemma 2.6(4)] (see Remark 3.8) and the identity (3.27) is found in [BBl1866492, Lemma 7.4].

Let ${Z}_{i-1}^{\left(\ell \right)}$ and the generating functions ${Z}_{i-1}^{+}$ and ${Z}_{i-1}^{-}$ be as defined in (3.22) and (3.23). Then ${Z}_{i-1}^{\left(\ell \right)}\in Z\left({W}_{i-1}\right),$ $( Zi-1-- zq-q-1- u2u2-1 ) ( Zi-1++ z-1q-q-1- u2u2-1 ) Ei= -(u2-q2) (u2-q-2) (u2-1)2 (q-q-1)2 Ei, (3.26) ( Zi++ z-1q-q-1 -u2u2-1 ) Ei+1 = ( Zi-1++ z-1q-q-1- u2u2-1 ) (u-Yi)2 (u-q-2Yi-1) (u-q2Yi-1) (u-Yi-1)2 (u-q2Yi) (u-q-2Yi) Ei+1,and (3.27) ( Zk-1++ z-1q-q-1- u2u2-1 ) Ei+1 = ( Z0++ z-1q-q-1- u2u2-1 ) ( ∏i=1k-1 (u-Yi)2 (u-q-2Yi-1) (u-q2Yi-1) (u-Yi-1)2 (u-q2Yi) (u-q-2Yi) ) Ei+1. (3.28)$

 Proof. Since the generators ${T}_{1},\dots ,{T}_{i-2},$ ${E}_{2},\dots ,{E}_{i-2}$ and ${Y}_{1},\dots ,{Y}_{i-1}$ of ${W}_{i-1}$ all commute with ${E}_{i}$ and ${Y}_{i},$ it follows that ${Z}_{i-1}^{\left(\ell \right)}\in Z\left({W}_{i-1}\right)\text{.}$ Multiply (3.18) on the right by ${W}_{i}$ and use ${Z}_{i-1}^{\left(0\right)}=1+\left(z-{z}^{-1}\right)/\left(q-{q}^{-1}\right)$ to get (3.26). Multiply (3.19) on the left and right by ${E}_{i+1}$ and use the relations in (2.42), (2.43), (2.46), and $Ei+1Ti Ui+Ti-1 Ei+1= Ei+1Ti Ti+1 Ui+ Ti+1-1 Ti-1 Ei+1= Ei+1Ei Ui+Ei Ei+1,$ to obtain $( Zi-1+- Zi(0)+ zq-q-1 -1u2-1 ) ( 1-(q-q-1)2 Ui+(Ui++1) ) Ei+1 = ( Zi-1+- Zi-1(0)+ zq-q-1- 1u2-1 ) ( 1-(q-q-1)2 Ui-(Ui-+1) ) Ei+1.$ Then (3.27) follows from $1-(q-q-1)2 Ui-(Ui-+1) 1-(q-q-1)2 Ui+(Ui++1) = 1-(q-q-1)2 Yi-1u-Yi-1 (Yi-1u-Yi-1+1) 1-(q-q-1)2 Yiu-Yi (Yiu-Yi+1) = ( (u-Yi-1)2 -(q-q-1)2 Yi-1u ) 1(u-Yi-1)2 ( (u-Yi)2 -(q-q-1)2 Yiu ) 1(u-Yi)2 = (u-q-2Yi-1) (u-q2Yi-1) (u-Yi)2 (u-q-2Yi) (u-q2Yi) (u-Yi-1)2$ and ${Z}_{i}^{\left(0\right)}={Z}_{i-1}^{\left(0\right)}=1+\left(z-{z}^{-1}\right)/\left(q-{q}^{-1}\right)\text{.}$ Finally, relation (3.28) follows, by induction, from (3.27). $\square$

Taking the coefficient of ${u}^{-\left(\ell +1\right)}$ on each side of (3.14) and (3.15) gives $TiYiℓ= Yi+1ℓTi- (q-q-1) ( Yi+1ℓ (1-Ei)+ Yi+1ℓ-1 (1-Ei)Yi +⋯+Yi+1 (1-Ei) Yiℓ-1 ) , (3.29) TiYi+1ℓ= YiℓTi+ (q-q-1) ( Yiℓ-1 (1-Ei) Yi+1+ Yiℓ-2 (1-Ei) Yi+12+ ⋯+(1-Ei) Yi+1ℓ ) (3.30)$ respectively, for $\ell \in {ℤ}_{\ge 0}\text{.}$ Therefore, $TiYi-ℓ= Yi+1-ℓ Ti+(q-q-1) ( Yi+1-(ℓ-1) (1-Ei) Yi-1+⋯+ (1-Ei) Yi-ℓ ) , (3.31) TiYi+1-ℓ= Yi-ℓ Ti-(q-q-1) ( Yi-ℓ (1-Ei)+⋯+ Yi-1 (1-Ei) Yi+1-(ℓ-1) ) . (3.32)$

Combining (3.26) and (3.28) yields a formula for ${Z}_{k-1}^{-}$ in terms of ${Z}_{0}^{+}$ and ${Y}_{1},{Y}_{2},\dots ,{Y}_{k-1}\text{.}$ Using ${Z}_{i-1}^{\left(0\right)}=1+\frac{z-{z}^{-1}}{q-{q}^{-1}},$ rewrite (3.26) as $( zZi-1--z-1 Zi-1+- (z-z-1) Zi-1(0) ) Ei =(q-q-1) ( 1u2-1 ( Zi-1++ Zi-1-- Zi-1(0) ) -(Zi-1--Zi-1(0)) (Zi-1+-Zi-1(0)) ) Ei, (3.33)$ and take the coefficient of ${u}^{-\ell }$ in (3.26) to get $( zZi-1(-ℓ) -z-1Zi-1(ℓ) ) Ei =(q-q-1) ( Zi-1(ℓ-2)+ Zi-1(ℓ-4)+⋯+ Zi-1(-(ℓ-2)) - ( Zi-1(ℓ-1) Zi-1(-1)+ Zi-1(ℓ-2) Zi-1(-2)+⋯+ Zi-1(1) Zi-1(-(ℓ-1)) ) ) Ei, (3.34)$ from [GMo0612064, Lemma 2.6(4)].

## 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.