Identities in affine and degenerate affine BMW algebras

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

Last updates: 13 April 2011

The degenerate affine case

Let u be a variable and define zi-1() 𝒲i-1 by the generating function

zi-1(u) ei = 0 zi-1() u- ei = ei ( 0 yi u- ) ei = ei uu-yi ei . (zid)
Let
ui+ = 1 u-yi and ui- = 1 u+yi , and note that (upm)
ui+ ui+1+ = 1 2u-(yi+ yi+1) ui+ + ui+1+ , (uup)
ei ui+1+ = ei ui- , ui+1+ = ui- ei, ei ui± ei = zi-1 ± u u ei, (eup)
where, for i=1, the last identity is a restatement of the first identity in (dbw2). By (dbw3) and the definition of ei in (edb),
(u-yi+1) tsi = tsi (u-yi) -(1-ei) and (u-yi) tsi = tsi (u-yi+1) +(1-ei) ,
which give
tsi ui+ = ui+1+ tsi + ui+1+ ei ui+ - ui+1+ ui+ ,and tsi ui+1+ = ui+ tsi - ui+ ei ui+1+ + ui+1+ ui+ , (tuc)
respectively.

In the degenerate affine BMW algebra 𝒲i+1,

( ei 11- yi+1 - tsi - 1 2u-(yi+ yi+1) ) ( ei 11- yi + tsi - 1 2u-(yi+ yi+1) ) = - (2u- (yi +yi+1) +1) (2u- (yi +yi+1) -1) (2u- (yi +yi+1) )2 , (itw1)
( ui+1+ + tsi - ei 1 2u-(yi+ yi+1) ) - ui+ ( ui+1+ + tsi - ei 1 2u-(yi+ yi+1) ) ui+ ) = ( tsi ui+ tsi + tsi - ei 1 2u-(yi+ yi+1) ) - ui+1+ ( ei ui+ ei +ϵ ei - ei 1 2u-(yi+ yi+1) ) ui+1+ . (itw2)

Proof.

Putting (uup) into the first identity in (tuc) says that if

A = tsi + 1 2u-(yi+ yi+1) and B = ei ui+ + tsi + 1 2u-(yi+ yi+1)
then
A ui+ = ui+1+ B, and Aei = ei A
follows from (dbw1) and (dbw2). So
( ei ui+1+ - tsi - 1 2u-(yi+ yi+1) ) ( ei ui+ + tsi - 1 2u-(yi+ yi+1) ) = ei ui+1+ B -AB = ei A ui+ -AB = A ei ui+ -AB = A( ei ui+ -B) = - ( tsi + 1 2u-(yi+ yi+1) ) ( tsi - 1 2u-(yi+ yi+1) )
and mutiplying out the right hand side gives (itw1).

Multiplying the second relation in (tuc) by tsi

ui+1+ - tsi ui+1+ ui+ = tsi ui+ tsi - tsi ui+ ei ui-
and again using the relations in (tuc) then
ui+1+ - ui+ ( tsi - ei ui+1+ + ui+1+ ) ui+ = tsi ui+ tsi - ui+1+ ( tsi + ei ui+ - ui+ ) ei ui+1+ .
Using (uup) and
adding    tsi - ei ( 1 2u-(yi+ yi+1) ) - 1 2u-(yi+ yi+1) ui+ ei ui+1+     to each side
gives
( ui+1+ + tsi - ei 1 2u-(yi+ yi+1) ) - ui+ ( ui+1+ + tsi - ei 1 2u-(yi+ yi+1) ) ui+ = tsi ui+ tsi + tsi - ei 1 2u-(yi+ yi+1) - ui+1+ ( ei ui+ + tsi - 1 2u-(yi+ yi+1) ) ei ui+1+ = ( tsi ui+ tsi + tsi - ei 1 2u-(yi+ yi+1) ) - ui+1+ ( ei ui+ ei +ϵ ei - ei 1 2u-(yi+ yi+1) ) ui+1+ ,
completing the proof of (itw2).

The identities (4.2), (4.3), and (4.4) of the following theorem are [Naz, Lemma 2.5], [Naz, Prop. 4.2] and [Naz, Lemma 3.8], respectively.

As in (zid), define zi-1() 𝒲i-1 by the generating function

zi-1(u) ei = 0 zi-1() u- ei = ei ( 0 yi u- ) ei = ei uu-yi ei . (zid)
Then
( zi-1 (-u) - ( 12 +ϵu ) ) zi-1 (u) - 12 -ϵu ei = 12 - ϵu 12 +ϵu ei, (zmz)
zi (u) + ϵ u - 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 (zrc)
zk-1 (u) + ϵ u - 12 ei+1 = z0 (u) + ϵ u - 12 i=1 k-1 u+yi-1 u+yi+1 u-yi 2 u+yi 2 u-yi+1 u-yi-1 ei+1. (zgn)

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 (itw1) on the right by ei to get (zmz), since ( 12 - u ) ( 12 +u ) = ( 12 - ϵu ) ( 12 +ϵu ) .

Multiplying (itw2) on the right by ei+1 and using the relations in (dbw5), (dbw6), and (dbw7),

ei+1 tsi ui+ tsi ei+1 = ei+1 tsi tsi+1 ui+ tsi+1 tsi ei+1 = ei+1 ei ui+ ei ei+1 ,  and
ei+1 ui+1+ ei ui+1+ ei+1 = ei+1 ui- ei ui- ei+1 = ui- ei+1 ei ei+1 ui- = ( ui- ) 2 ei+1 ,
gives ( zi (u) u + ϵ - 12u ) 1 - ui+ 2 ei+1 = ( zi-1 (u) u + ϵ - 12u ) 1 - ui- 2 ei+1 . So (zrc) follows from 1 - ui- 2 1 - ui+ 2 = 1 - 1 u+yi 2 1 - 1 u-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 (zgn) follows, by induction, from (zrc).

Remark. Taking the coefficient of u-s on each side of (4.2) gives a trivial identity for even s, but for odd s=2l+1, gives

2 zi(2l+1) + zi(2l) - zi(2l) zi(0) - zi(2l-1) zi(1) + + zi(0) zi(2l) ei = 0 4.10
which is the admissibltiy relation in [AMR, Remark 2.11] (see also [Naz, (4.6)]).

The affine case

Let u be a variable and let
Ui+ = Yi u-Yi and Ui- = Yi-1 u-Yi-1 , (3.21)
and note that
Ui+ Ui+1+ = 1 u2- Yi Yi+1 ( Ui+ + Ui+1+ +1 ) . (3.22)
The second relation in (BW2) and the definition of Zi-1± give
Ei Ui+1+ = Ei Ui- , Ui+1+ Ei = Ui- Ei, Ei Ui± Ei = ( Zi-1± - Zi-1(0) ) Ei, (3.23)
and the relations
Ti Ui+ = Ui+1+ Ti-1 - ( q-q-1 ) Ui+1+ ( 1 - Ei ) Ui+ = Ui+1+ ( Ti-1 - ( q-q-1 ) ( 1 - Ei ) Ui+ ) , and (3.24)
Ti-1 Ui+1+ = Ui+ Ti - ( q-q-1 ) Ui+ Ei Ui- + ( q-q-1 ) Ui+1+ Ui+ = Ui+ ( Ti + ( q-q-1 ) ( 1- Ei ) Ui+1+ ) (3.25)
are obtained by multiplying (3.15) (resp. (3.16)) on the right (resp. left) by Yi and using the relation Ti Yi = Yi+1 Ti-1 .

Let Q=q-q-1. In the affine BMW algebra Wi+1,

( Ei Yi+1 u- Yi+1 - Ti Q - Yi Yi+1 u2- Yi Yi+1 ) ( Ei Yi u-Yi + Ti-1 Q - Yi Yi+1 u2- Yi Yi+1 ) = - ( u2- q2 Yi Yi+1 ) ( u2- q-2 Yi Yi+1 ) Q2 (u2- Yi Yi+1 )2 , (Itw1)
and
( Ui+1+ + Ti Q - Ei Yi Yi+1 u2- Yi Yi+1 ) - Q2 ( Ui+ +1) ( Ui+1+ + Ti Q - Ei Yi Yi+1 u2- Yi Yi+1 ) Ui+ = ( Ti Ui+ Ti-1 + Ti Q - Ei Yi Yi+1 u2- Yi Yi+1 ) - Q2 Ui+1+ ( Ei Ui+ Ei +z Ei Q - Ei Yi Yi+1 u2- Yi Yi+1 ) ( Ui+1+ +1). (Itw2)

Proof. Putting (UUp) into (TUc) says that if

A = Ti Q + Yi Yi+1 u2- Yi Yi+1 and B = Ei Ui+ + Ti-1 Q - Yi Yi+1 u2- Yi Yi+1
then
A Ui+ = Ui+1+ B - Yi Yi+1 u2- Yi Yi+1 . Next, AEi = Ei A
follows from (BW1) and (BW2). So
( Ei Yi+1 u- Yi+1 - Ti Q - Yi Yi+1 u2- Yi Yi+1 ) ( Ei Yi u-Yi + Ti-1 Q - Yi Yi+1 u2- Yi Yi+1 ) = Ei ( Ui+1+ B) -AB = Ei ( AUi+ + Yi Yi+1 u2- Yi Yi+1 ) -AB = A( Ei Ui+ -B) + Ei Yi Yi+1 u2- Yi Yi+1 =- ( Ti Q + Yi Yi+1 u2- Yi Yi+1 ) ( Ti-1 Q - Yi Yi+1 u2- Yi Yi+1 ) + Ei Yi Yi+1 u2- Yi Yi+1
and, by (BW4), multiplying out the right hand side gives (Itw1).

Rewrite Ti-1 Ui+1+ = Ui+ Ti-1 + Q Ui+ (1-Ei) ( Ui+1+ +1) as

Ti-1 Ui+1+ - Q ( Ui+1+ +1) Ui+ = Ui+ Ti-1 - Q Ui+ Ei ( Ui+1+ +1) ,
and multiply on the left by Ti to get
Ui+1+ - Q Ti ( Ui+1+ +1) Ui+ = Ti Ui+ Ti-1 - Q Ti Ui+ Ei ( Ui+1+ +1) .
Then, since Ti = Ti-1 +Q(1-Ei) , equations (3.25) and (3.24) imply
Ti ( Ui+1+ +1) = Q ( Ui+1+ +1) ( Ti Q + (1-Ei) Ui+1+ ) and Ti Ui+ = Q Ui+1+ ( Ti-1 Q - (1-Ei) Ui+ ) .
and so (3.28) is
Ui+1+ - Q2 ( Ui+ +1) ( Ti Q + (1-Ei) Ui+1+ ) Ui+ = Ti Ui+ Ti-1 - Q2 Ui+1+ ( Ti-1 Q - (1-Ei) Ui+ ) Ei ( Ui+1+ +1) . (3.29)
Using (3.22) and
adding    Ti Q - Ei Yi Yi+1 u2- Yi Yi+1 -Q2 Yi Yi+1 u2- Yi Yi+1 ( Ui+ +1) Ei ( Ui+1+ +1)    to each side
of (3.29) gives Ui+1+ + Ti Q - Ei Yi Yi+1 u2- Yi Yi+1 -Q2 ( Ui+ +1) ( Ui+1+ + Ti Q - Ei Yi Yi+1 u2- Yi Yi+1 ) Ui+ = Ti Ui+ Ti-1 + Ti Q - Ei Yi Yi+1 u2- Yi Yi+1 -Q2 Ui+1+ ( Ei Ui+ + Ti-1 Q - Yi Yi+1 u2- Yi Yi+1 ) Ei ( Ui+1+ +1) = Ti Ui+ Ti-1 + Ti Q - Ei Yi Yi+1 u2- Yi Yi+1 -Q2 Ui+1+ ( Ei Ui+ Ei + z Ei Q - Ei Yi Yi+1 u2- Yi Yi+1 ) ( Ui+1+ +1) , completing the proof of (3.27).

The identities (4.13) and (4.14) of the following theorem are found in [GH1, Lemma 2.8(4)] and [BB, Lemma 7.4], respectively.

Define central elements Zi-1() Z (Wi-1) by the generating functions Zi-1+ and Zi-1- given by

Zi-1+ Ei = 0 Zi-1 () u- Ei = Ei ( 0 Yi u- ) Ei = Ei 1 1- Yiu-1 Ei, (Zpd)
Zi-1- Ei = 0 Zi-1(-) u- Ei = Ei ( 0 Yi- u- ) Ei = Ei 1 1- Yi-1 u-1 Ei. (Zmd)
Then
( Zi-1- - z q-q-1 - u2 u2-1 ) ( Zi-1+ + z-1 q-q-1 - u2 u2-1 ) Ei = - (u2-q2) (u2- q-2 ) (u2-1) 2 ( q-q-1) 2 Ei, 4.13
Zi+1+ - Zi+1(0) + z q-q-1 - 1 u2-1 Ei+1 = Zi+ - Zi(1) + z q-q-1 - 1 u2-1 u-Yi 2 u-q-2Yi-1 u-q2Yi-1 u-Yi-1 2 u-q2Yi u-q-2Yi Ei+1, 4.14
Zk+ - Zk(0) + z q-q-1 - 1 u2 -1 Ei+1 = Z1+ - Z1(0) + z q-q-1 - 1 u2-1 i=1 k-1 u-Yi 2 u - q-2 Yi-1 u - q2 Yi-1 u - Yi-1 2 u - q2 Yi u - q-2 Yi Ei+1 4.15

Proof.

Since the generators T1,, Ti-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 (3.26) on the right by Ei and use Zi-1(0) = 1+ (z-z-1) / ( q-q-1) to get (3.32).

Multiply (3.27) on the left and right by Ei+1 and use the relations in (2.42), (2.43), (2.46), and Ei+1 Ti Ui+ Ti-1 Ei+1 = Ei+1 Ti Ti+1 Ui+ Ti+1-1 Ti-1 Ei+1 = Ei+1 Ei Ui+ Ei Ei+1 , to obtain ( Zi+ - Zi(0) + z q-q-1 - 1 u2-1 ) ( 1- (q-q-1 ) 2 Ui+ ( Ui++1 ) ) Ei+1 = ( Zi-1+ - Zi-1(0) + z q-q-1 - 1 u2-1 ) ( 1- (q-q-1 ) 2 Ui- ( Ui-+1 ) ) Ei+1. Then (3.33) follows from 1- (q-q-1) 2 Ui- ( Ui- +1 ) 1- (q-q-1) 2 Ui+ ( Ui+ +1) = 1- ( q-q-1 ) 2 Yi-1 u- Yi-1 Yi-1 u- Yi-1 +1 1- q-q-1 2 Yi u-Yi Yi u- Yi +1 = u- Yi-1 2 - (q-q-1) 2 Yi-1 u 1 u- Yi-1 2 u- Yi 2 - (q-q-1) 2 Yi u 1 u- Yi 2 = u- q-2 Yi-1 u- q2 Yi-1 u- Yi 2 u- q-2 Yi u- q2 Yi u- Yi-1 2 . and Zi(0) = Zi-1(0) = 1+ (z-z-1) / ( q-q-1) . Finally, relation (3.34) follows, by induction, from (3.33).

Remark. Combining (4.13) and (4.15) yields a formula for Zk-1- in terms of Z0+ and Y1, Y2, , Yk-1 . Using Zi-1(0) = 1+ z-z-1 q-q-1 , rewrite (3.32) as

( z Zi-1- - z-1 Zi-1+ - ( z - z-1 ) Zi-1(0) ) Ei = ( q-q-1 ) ( 1 u2-1 ( Zi-1+ + Zi-1- - Zi-1(0) ) - ( Zi-1- - Zi-1(0) ) ( Zi-1+ - Zi-1(0) ) ) Ei, (3.35)
and take the coefficient of u- in (3.32) to get
( z Zi-1(-) - z-1 Zi-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, (3.36)
from [GH1, Lemma 2.8(4)].

Notes and References

This section is based on forthcoming joint work with Z. Daugherty and R. Virk [DRV]. The remarkable recursions for generating central elements which appear in Theorems ??? and ??? were given by Nazarov [Naz] in the degenerate case, and then extended to the affine BMW algebra by Beliakova-Blanchet [BB]. Another proof in the affine cyclotomic case appears in [RX2, Lemma 4.21] and, in the degenerate case, in [AMR, Lemma 4.15]. In all of these proofs, the recursion is obtained by a rather mysterious and tedious computation. We show that there is an "intertwiner like identity in the full algebra which, when "projected to the center" produces the Nazarov recursions. Our approach dramatically simplifies the proof and gives some insight into where these recursions are coming from. Moreover, the proof is exactly analogous in both the degenerate and the affine cases, and includes the parameter ϵ, so that both the orthogonal and symplectic cases are treated simultaneously.

References

[AMR] S. Ariki, A. Mathas, and H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006), 47-134. MR2235339 (2007d:20005)

[BB] A. Beliakova and C. Blanchet, Skein construction of idempotents in Birman-Murakami-Wenzl algebras, Math. Ann. 321 (2001), 347-373. MR1866492 (2002h:57018)

[Bou] N. Bourbaki, Groupes et Alg├Ębres de Lie, Masson, Paris, 1990.

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

[GH1] F. Goodman and H. Hauschild Mosley, Cyclotomic Birman-Wenzl-Murakami algebras. I. Freeness and realization as tangle algebras, J. Knot Theory Ramifications 18 (2009), 1089-1127. MR2554337 (2010j:57014)

[Naz] M. Nazarov, Young's orthogonal form for Brauer's centralizer algebra, J. Algebra 182 (1996), no. 3, 664-693. MR1398116 (97m:20057)

[OR] R. Orellana and A. Ram, Affine braids, Markov traces and the category 𝒪, Algebraic groups and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007. MR2348913 (2008m:17034)

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