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: 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 𝒲k and proved a remarkable recursion for them. This recursion was generalized to analogous central elements in the affine BMW algebra Wk 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 𝒲k and Wk (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 ek and Ek, 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 ei 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 𝒲i+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 tsi 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).

Introduce notation zi-1()ei and the generating function zi-1(u)ei by zi-1(u)ei= 0 zi-1()eiu- =ei(0yiu-) ei=ei11-yiu-1ei, (3.5) By [AMR0506467, Lemma 4.15], or the identity (3.9) below, zi-1()𝒲i-1 for 0. 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 zi-1() and zi-1(u) be as defined in (3.5). Then zi-1()Z(𝒲i-1), ( 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 ts1,,tsi-2,e1,ei-2 and y1,,yi-1 of 𝒲i-1 all commute with ei and yi it follows that zi-1()Z(𝒲i-1).

Multiply (3.3) on the right by ei to get (3.7), since (12-u)(12+u)= (12-εu)(12+εu).

Multiplying (3.4) on the left and right by ei+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).

Using the expansion 1u-a= u-11-au-1 =1 a-1u-, and taking the coefficient of u-(+1) on each side of the relations in (3.2) gives tsiyi = yi+1tsi- ( yi+1-1 (1-ei)+ yi+1-2 (1-ei)yi ++(1-ei) yi-1 ) ,and (3.10) tsiyi+1 = yitsi+ yi-1 (1-ei)+ yi-2 (1-ei)yi+1 ++(1-ei) yi+1-1, (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+1, gives ( 2zi-1(2+1) +zi-1(2) - ( zi-1(2) zi-1(0) -zi-1(2-1) zi-1(1) ++ zi-1(0) zi-1(2) ) 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 Ei 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 Yi and using the relation TiYi= Yi+1Ti-1.

Let Q=q-q-1. Then, in the affine BMW algebra Wi+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 Ti-1Ui+1+= Ui+Ti-1+ QUi+(1-Ei) (Ui+1++1) as Ti-1Ui+1+ -Q(Ui+1++1) Ui+=Ui+ Ti-1-QUi+ Ei(Ui+1++1), and multiply on the left by Ti to get Ui+1+-QTi (Ui+1++1) Ui+=TiUi+ Ti-1-QTi Ui+Ei (Ui+1++1). (3.20) Then, since Ti=Ti-1+Q(1-Ei), 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).

Introduce notation Zi-1()ei and generating functions Zi-1+Ei and Zi-1-Ei by Zi-1+Ei= 0 Zi-1()Ei u-=Ei ( 0 Yiu- ) Ei=Ei 11-Yiu-1 Ei, (3.22) Zi-1-Ei= 0 Zi-1(-)Ei u-=Ei ( 0 Yi-u- ) Ei=Ei 11-Yi-1u-1 Ei. (3.23) By [GHa0411155, Lemma 3.15(1)], or the identity (3.28) below, Zi-1()Wi-1 for . 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 Zi-1() and the generating functions Zi-1+ and Zi-1- be as defined in (3.22) and (3.23). Then Zi-1()Z(Wi-1), ( 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 T1,,Ti-2, E2,,Ei-2 and Y1,,Yi-1 of Wi-1 all commute with Ei and Yi, it follows that Zi-1()Z(Wi-1).

Multiply (3.18) on the right by Wi and use Zi-1(0)=1+ (z-z-1)/(q-q-1) to get (3.26).

Multiply (3.19) on the left and right by Ei+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 Zi(0)= Zi-1(0)=1+ (z-z-1)/ (q-q-1). Finally, relation (3.28) follows, by induction, from (3.27).

Taking the coefficient of u-(+1) on each side of (3.14) and (3.15) gives TiYi= Yi+1Ti- (q-q-1) ( Yi+1 (1-Ei)+ Yi+1-1 (1-Ei)Yi ++Yi+1 (1-Ei) Yi-1 ) , (3.29) TiYi+1= YiTi+ (q-q-1) ( Yi-1 (1-Ei) Yi+1+ Yi-2 (1-Ei) Yi+12+ +(1-Ei) Yi+1 ) (3.30) respectively, for 0. Therefore, TiYi-= Yi+1- Ti+(q-q-1) ( Yi+1-(-1) (1-Ei) Yi-1++ (1-Ei) Yi- ) , (3.31) TiYi+1-= Yi- Ti-(q-q-1) ( Yi- (1-Ei)++ Yi-1 (1-Ei) Yi+1-(-1) ) . (3.32)

Combining (3.26) and (3.28) yields a formula for Zk-1- in terms of Z0+ and Y1,Y2,,Yk-1. Using Zi-1(0)=1+z-z-1q-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- in (3.26) to get ( zZi-1(-) -z-1Zi-1() ) Ei =(q-q-1) ( Zi-1(-2)+ Zi-1(-4)++ Zi-1(-(-2)) - ( Zi-1(-1) Zi-1(-1)+ Zi-1(-2) Zi-1(-2)++ Zi-1(1) Zi-1(-(-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.

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