Title AFFINE AND DEGENERATE AFFINE BMW ALGEBRAS

28 Pages·2012·0.52 MB·English

AFFINE AND DEGENERATE AFFINE BMW ALGEBRAS : THE
Title
CENTER
Author(s) Daugherty, Zaji; Ram, Arun; Virk, Rahbar
Citation Osaka Journal of Mathematics. 51(1) p.257-p.283
Issue Date 2014-01
oaire:version VoR
URL https://doi.org/10.18910/29192
rights
Note
OOssaakkaa UUnniivveerrssiittyy KKnnoowwlleeddggee AArrcchhiivvee :: OOUUKKAA
https://ir.library.osaka-u.ac.jp/
Osaka University
Daugherty, Z., Ram, A. and Virk, R.
Osaka J. Math.
51 (2014), 257–283
AFFINE AND DEGENERATE AFFINE BMW ALGEBRAS:
THE CENTER
ZAJJ DAUGHERTY, ARUN RAM and RAHBAR VIRK
(Received October 2, 2011, revised August 7, 2012)
Abstract
The degenerate affine and affine BMW algebras arise naturally in the context of
Schur–Weyl duality for orthogonal and symplectic Lie algebras and quantum groups,
respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke
algebras, and their degenerate versions are quotients. In this paper the theory is uni-
fied by treating the orthogonal and symplectic cases simultaneously; we make an ex-
act parallel between the degenerate affine and affine cases via a new algebra which
takes the role of the affine braid group for the degenerate setting. A main result
of this paper is an identification of the centers of the affine and degenerate affine
BMW algebras in terms of rings of symmetric functions which satisfy a “cancel-
lation property” or “wheel condition” (in the degenerate case, a reformulation of a
result of Nazarov). Miraculously, these same rings also arise in Schubert calculus,
as the cohomology and K-theory of isotropic Grassmannians and symplectic loop
Grassmannians. We also establish new intertwiner-like identities which, when pro-
jected to the center, produce the recursions for central elements given previously by
Nazarov for degenerate affine BMW algebras, and by Beliakova–Blanchet for affine
BMW algebras.
1. Introduction
The degenerate affine BMW algebras W and the affine BMW algebras W arise
k k
naturally in the context of Schur–Weyl duality and the application of Schur functors to
modules in category O for orthogonal and symplectic Lie algebras and quantum groups
(using the Schur functors of [41], [1], and [28]). The degenerate algebras W were in-
k
troduced in [27] and the affine versions W appeared in [28], following foundational
k
work of [17]–[19]. The representation theory of W and W contains the represen-
k k
tation theory of any quotient: in particular, the degenerate cyclotomic BMW algebras
W , the cyclotomic BMW algebras W , the degenerate affine Hecke algebras H ,
r,k r,k k
the affine Hecke algebras H , the degenerate cyclotomic Hecke algebras H , and the
k r,k
cyclotomic Hecke algebras H as quotients. In [31,34,35,2,8] and other works, the
r,k
representation theory of the affine BMW algebra is derived by cellular algebra tech-
niques. As indicated in [28], the Schur–Weyl duality also provides a path to the repre-
sentation theory of the affine BMW algebras as an image of the representation theory
of category O for orthogonal and symplectic Lie algebras and their quantum groups
2010 Mathematics Subject Classification. 17B37.
258 Z. DAUGHERTY, A. RAM AND R. VIRK
in the same way that the affine Hecke algebras arise in Schur–Weyl duality with the
enveloping algebra of gl and its Drinfeld–Jimbo quantum group.
n
In the literature, the algebras W and W have often been treated separately. One
k k
of the goals of this paper is to unify the theory. To do this we have begun by adjusting
the definitions of the algebras carefully to make the presentations match, relation by re-
lation. In the same way that the affine BMW algebra is a quotient of the group algebra
of the affine braid group, we have defined a new algebra, the degenerate affine braid
algebra which has the degenerate affine BMW algebra and the degenerate affine Hecke
algebras as quotients. We have done this carefully, to ensure that the Schur–Weyl du-
ality framework is completely analogous for both the degenerate affine and the affine
cases. We have also added a parameter (which takes values 1) so that both the
(cid:15) (cid:6)
orthogonal and symplectic cases can be treated simultaneously. Our new presentations
of the algebras W and W are given in Section 2.
k k
In Section 3 we consider some remarkable recursions for generating central elem-
ents in the algebras W and W . These recursions were given by Nazarov [27] in the
k k
degenerate case, and then extended to the affine BMW algebra by Beliakova–Blanchet
[4]. Another proof in the affine cyclotomic case appears in [35, Lemma 4.21] and,
in the degenerate case, in [2, 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 provides new insight into where these
recursions are coming from. Moreover, the proof is exactly analogous in both the de-
generate and the affine cases, and includes the parameter , so that both the orthogonal
(cid:15)
and symplectic cases are treated simultaneously.
In Section 4 we identify the center of the degenerate and affine BMW algebras. In
the degenerate case this has been done in [27]. Nazarov stated that the center of the
degenerate affine BMW algebra is the subring of the ring of symmetric functions gen-
erated by the odd power sums. We identify the ring in a different way, as the subring
of symmetric functions with the Q-cancellation property, in the language of Pragacz
[29]. This is a fascinating ring. Pragacz identifies it as the cohomology ring of orthog-
onal and symplectic Grassmannians; the same ring appears again as the cohomology
of the loop Grassmannian for the symplectic group in [24,22]; and references for the
relationship of this ring to the projective representation theory of the symmetric group,
the BKP hierarchy of differential equations, representations of Lie superalgebras, and
twisted Gelfand pairs are found in [25, Chapter II §8]. For the affine BMW algebra,
the Q-cancellation property can be generalized well to provide a suitable description of
the center. From our perspective, one would expect that the ring which appears as the
center of the affine BMW algebra should also appear as the K-theory of the orthogonal
and symplectic Grassmannians and as the K-theory of the loop Grassmannian for the
symplectic group, but we are not aware that these identifications have yet been made
in the literature.
AFFINE BMW ALGEBRAS: THE CENTER 259
The recent paper [31] classifies the irreducible representations of W by multiseg-
k
ments, and the recent paper [6] adds to this program of study by setting up commuting
actions between the algebras W and W and the enveloping algebras of orthogonal and
k k
symplectic Lie algebras and their quantum groups, showing how the central elements
whichariseintheNazarovrecursionscoincidewithcentralelementsstudiedinBaumann
[3], and providing an approach to admissibility conditions by providing “universal ad-
missible parameters” in an appropriate ground ring (arising naturally, from Schur–Weyl
duality, as the center of the enveloping algebra, or quantum group). We would also like
to mention the recent paper of A. Sartori [36] which establishes similar results in the
case of the degenerate affine walled Brauer algebra and the recent work of M. Ehrig
and C. Stroppel [7] which studies these algebras in the context of categorification.
2. 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 . We have adjusted the definitions to unify the theory. In
k
particular, in Section 2.2, we define a new algebra, the degenerate affine braid algebra
B , which has the degenerate affine BMW algebras W and the degenerate affine Hecke
k k
algebras H as quotients. The motivation for the definition of B is that the affine
k k
BMW algebras W and the affine Hecke algebras H are quotients of the group algebra
k k
of affine braid group CB .
k
The definition of the degenerate affine braid algebra B also makes the Schur–Weyl
k
duality framework completely analogous in both the affine and degenerate affine cases.
Both B and CB are designed to act on tensor space of the form M V k. In the de-
k k (cid:10)
(cid:10)
generate 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 g. The degenerate affine and affine BMW algebras arise when g is so or sp
q n n
and V is the first fundamental representation and the degenerate affine and affine Hecke
algebras arise when g is gl or sl and V is the first fundamental representation. In the
n n
case when M is the trivial representation and g is so , the “Jucys–Murphy” elements
n
y , , y in B become the “Jucys–Murphy” elements for the Brauer algebras used
1 k k
:::
in [27] and, in the case that g gl , these become the classical Jucys–Murphy elem-
n
D
ents in the group algebra of the symmetric group. The Schur–Weyl duality actions are
explained in [6].
2.1. The affine BMW algebra W . Theaffinebraidgroup B isthegroupgiven
k k
by generators T1,T2, , Tk 1 and X"1, with relations
::: (cid:0)
(2.1) TT T T, if j i 1,
i j j i
D ¤ (cid:6)
(2.2) TT T T TT , for i 1,2, ,k 2,
i i 1 i i 1 i i 1
C D C C D ::: (cid:0)
(2.3) X"1T1X"1T1 T1X"1T1X"1,
D
(2.4) X"1Ti TiX"1, for i 2,3, ,k 1.
D D ::: (cid:0)
260 Z. DAUGHERTY, A. RAM AND R. VIRK
Let C be a commutative ring and let CB be the group algebra of the affine braid
k
group. Fix constants
q,z C and Z(l) C, for l ,
2 0 2 2Z
with q and z invertible. Let Yi zX"i so that
D
(2.5) Y1 zX"1, Yi Ti 1Yi 1Ti 1, and YiYj YjYi, for 1 i, j k.
D D (cid:0) (cid:0) (cid:0) D (cid:20) (cid:20)
In the affine braid group
(2.6) TY Y Y Y T.
i i i 1 i i 1 i
C D C
Assume that q q 1 is invertible in C and define E in the group algebra of the affine
(cid:0) i
(cid:0)
braid group by
(2.7) TY Y T (q q 1)Y (1 E ).
i i i 1 i (cid:0) i 1 i
D C (cid:0) (cid:0) C (cid:0)
The affine BMW algebra W is the quotient of the group algebra CB of the affine
k k
braid group B by the relations
k
(2.8) E T 1 T 1E z 1E , E T 1E E T 1E z 1E ,
i i(cid:6) D i(cid:6) i D (cid:7) i i i(cid:6)(cid:0)1 i D i i(cid:6)C1 i D (cid:6) i
(2.9) E YlE Z(l)E , E Y Y E Y Y E .
1 1 1 D 0 1 i i iC1 D i D i iC1 i
The affine Hecke algebra H is the affine BMW algebra W with the additional relations
k k
(2.10) E 0, for i 1, ,k 1.
i
D D ::: (cid:0)
Fix b , ,b C. The cyclotomic BMW algebra W (b , ,b ) is the affine BMW
1 r r,k 1 r
::: 2 :::
algebra W with the additional relation
k
(2.11) (Y b ) (Y b ) 0.
1 1 1 r
(cid:0) (cid:1)(cid:1)(cid:1) (cid:0) D
The cyclotomic Hecke algebra H (b , ,b ) is the affine Hecke algebra H with the
r,k 1 r k
:::
additional relation (2.11).
Since the composite map C[Y 1, ,Y 1] CB W H is injective and the
1(cid:6) ::: k(cid:6) ! k ! k ! k
last two maps are surjections, it follows that the Laurent polynomial ring C[Y 1, ,
1(cid:6) :::
Y 1] is a subalgebra of CB and W .
k(cid:6) k k
Proposition 2.1. The affine BMW algebra W coincides with the one defined
k
in [28].
Proof. In [28] the affine BMW algebra is defined as the quotient of the group
algebra of the affine braid group by the relations in (2.8) which are [28, (6.3b) and
AFFINE BMW ALGEBRAS: THE CENTER 261
(6.3c)], the first relation in (2.9) which is [28, (6.3d)], the second relation in (2.9) for
i 1 which is [28, (6.3e)], and the second relation in (2.15) below where, in [28], the
D
element E is defined by the first equation in (2.13) below.
i
Working in W , since Y 1(TY )Y Y 1Y Y T Y T, conjugating (2.7) by
Y 1 gives k i(cid:0)C1 i i iC1 D i(cid:0)C1 i iC1 i D i i
i(cid:0)1
C
(2.12) Y T TY (q q 1)(1 E )Y .
i i i i 1 (cid:0) i i 1
D C (cid:0) (cid:0) (cid:0) C
Left multiplying (2.7) by Y 1 and using the second identity in (2.5) shows that (2.7)
i(cid:0)1
is equivalent to T T 1 C(q q 1)(1 E ), so that
i (cid:0) i(cid:0) D (cid:0) (cid:0) (cid:0) i
T T 1
(2.13) Ei D1(cid:0) qi(cid:0)(cid:0)q(cid:0)i(cid:0)1 and TiTiC1EiTi(cid:0)C11Ti(cid:0)1 D EiC1.
Thus the E in W coincides with the E used in [28].
i k i
Multiply the second relation in (2.13) on the left and the right by E , and then use
i
the relations in (2.8) to get
E E E E TT E T 1T 1E E T E T 1E zE T 1E E ,
i iC1 i D i i iC1 i i(cid:0)C1 i(cid:0) i D i iC1 i i(cid:0)C1 i D i i(cid:0)C1 i D i
so that
(cid:18) z z 1(cid:19)
(2.14) E E E E , and E2 1 (cid:0) (cid:0) E
i i(cid:6)1 i D i i D C q(cid:0)q(cid:0)1 i
is obtained by multiplying the first equation in (2.13) by E and using (2.8). As one
i
can construct representations on which E acts non-trivially, the first relation in (2.9)
1
implies
z z 1
(2.15) Z0(0) D1C q(cid:0)q(cid:0)1 and (Ti (cid:0)z(cid:0)1)(Ti Cq(cid:0)1)(Ti (cid:0)q)D0,
(cid:0) (cid:0)
since (T z 1)(T q 1)(T q)T 1 (T z 1)(T2 (q q 1)T 1)T 1 (T
i (cid:0) (cid:0) i C (cid:0) i (cid:0) i(cid:0) D i (cid:0) (cid:0) i (cid:0) (cid:0) (cid:0) i (cid:0) i(cid:0) D i (cid:0)
z 1)(T T 1 (q q 1)) (T z 1)(q q 1)( E ) (z 1 z 1)(q q 1)E 0.
(cid:0) i (cid:0) i(cid:0) (cid:0) (cid:0) (cid:0) D i (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) i D(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) i D
This shows that the relation [28, (6.3a)] follows from the relations in W .
k
To complete the proof let us show that the relations in W follow from [28,
k
(6.3a-e)]. Given the equivalence of the definitions of E as established in (2.13), and
i
the coincidences of the relations [28, (6.3b-e)] with the relations in (2.8) and (2.9) it
only remains to show that the second set of relations in (2.9), for i 1 follow from
>
[28, (6.3a-e)]. But this is established by [28, (6.12)] and the identity
E Y Y (cid:18)1 Ti (cid:0)Ti(cid:0)1(cid:19)Y Y Y Y (cid:18)1 Ti (cid:0)Ti(cid:0)1(cid:19) Y Y E ,
i i iC1 D (cid:0) q(cid:0)q(cid:0)1 i iC1 D i iC1 (cid:0) q(cid:0)q(cid:0)1 D i iC1 i
which follows from (2.6).
262 Z. DAUGHERTY, A. RAM AND R. VIRK
The relations
(2.16) E E E TT , E E T 1T 1E ,
iC1 i D iC1 i iC1 i iC1 D i(cid:0)C1 i(cid:0) iC1
(2.17) T E E T 1E , and E E T E T 1,
i iC1 i D i(cid:0)C1 i iC1 i iC1 D iC1 i(cid:0)
are consequences of (2.8), and the second relation in (2.13).
2.2. The degenerate affine braid algebra B . Let C be a commutative ring,
k
and let S denote the symmetric group on {1, ,k}. For i {1, ,k}, write s for
k i
::: 2 :::
the transposition in S that switches i and i 1. The degenerate affine braid algebra
k
C
is the algebra B over C generated by
k
(2.18) t (u S ), , , and y , , y ,
u k 0 1 1 k
2 (cid:20) (cid:20) :::
with relations
(2.19) t t t , y y y y , , y y , y y ,
u u i j j i 0 1 1 0 0 i i 0 1 i i 1
v D v D (cid:20) (cid:20) D(cid:20) (cid:20) (cid:20) D (cid:20) (cid:20) D (cid:20)
(2.20) t t , t t t t , and t t , for j 1,
(cid:20)0 si D si(cid:20)0 (cid:20)1 s1(cid:20)1 s1 D s1(cid:20)1 s1(cid:20)1 (cid:20)1 sj D sj(cid:20)1 ¤
(2.21) t (y y ) (y y )t , and y t t y , for j i,i 1,
si i C iC1 D i C iC1 si j si D si j ¤ C
(2.22) t y t t y t ,
(cid:20)1 s1 1 s1 D s1 1 s1(cid:20)1
and
t t t t ,
(2.23) si siC1(cid:13)i,iC1 siC1 si D(cid:13)iC1,iC2
where y t yt for i 1, ,k 2.
(cid:13)i,iC1 D iC1(cid:0) si i si D ::: (cid:0)
In the degenerate affine braid algebra B let c and
k 0 0
D(cid:20)
c 2(y y ),
j 0 1 j
D(cid:20) C C(cid:1)(cid:1)(cid:1)C
(2.24) 1
so that y (c c ), for j 1, ,k.
j j j 1
D 2 (cid:0) (cid:0) D :::
Then c , ,c commute with each other, commute with , and the relations (2.21)
0 k 1
::: (cid:20)
are equivalent to
(2.25) t c c t , for j i.
si j D j si ¤
Theorem 2.2. The degenerate affine braid algebra B has another presentation
k
by generators
t , for u S , , , and ,
u k 0 k i,j
(2.26) 2 (cid:20) ::: (cid:20) (cid:13)
for 0 i, j k with i j,
(cid:20) (cid:20) ¤
AFFINE BMW ALGEBRAS: THE CENTER 263
and relations
(2.27) t t t , t t , t t ,
u u i 1 (i) i,j 1 (i), (j)
v D v w(cid:20) w(cid:0) D(cid:20)w w(cid:13) w(cid:0) D(cid:13)w w
(2.28) , ,
i j j i i l,m l,m i
(cid:20) (cid:20) D(cid:20) (cid:20) (cid:20) (cid:13) D(cid:13) (cid:20)
(2.29) , , and ( ) ( ) ,
i,j j,i p,r l,m l,m p,r i,j i,r j,r i,r j,r i,j
(cid:13) D(cid:13) (cid:13) (cid:13) D(cid:13) (cid:13) (cid:13) (cid:13) C(cid:13) D (cid:13) C(cid:13) (cid:13)
for p l and p m and r l and r m and i j, i r and j r.
¤ ¤ ¤ ¤ ¤ ¤ ¤
The commutation relations between the and the can be rewritten in the form
i i,j
(cid:20) (cid:13)
(2.30) [ , ] 0, [ , ] 0, and [ , ] [ , ],
r l,m i,j l,m i,j i,m i,m j,m
(cid:20) (cid:13) D (cid:13) (cid:13) D (cid:13) (cid:13) D (cid:13) (cid:13)
for all r and all i l and i m and j l and j m.
¤ ¤ ¤ ¤
Proof of Theorem 2.2. The generators in (2.26) are written in terms of the gen-
erators in (2.18) by the formulas
(2.31) , , t t ,
0 0 1 1
(cid:20) D(cid:20) (cid:20) D(cid:20) w D w
1
(2.32) y , and y t y t , for j 1, ,k 1,
(cid:13)0,1 D 1(cid:0) 2(cid:20)1 (cid:13)j,jC1 D jC1(cid:0) sj j sj D ::: (cid:0)
and
(2.33) t t , t t and t t ,
m u 1 u 1 0,m u 0,1 u 1 i,j 1,2 1
(cid:20) D (cid:20) (cid:0) (cid:13) D (cid:13) (cid:0) (cid:13) D v(cid:13) v(cid:0)
for u, S such that u(1) m, (1) i and (2) j.
k
v 2 D v D v D
The generators in (2.18) are written in terms of the generators in (2.26) by the
formulas
1
(2.34) , , t t , and y X .
0 0 1 1 j j l,j
(cid:20) D(cid:20) (cid:20) D(cid:20) w D w D 2(cid:20) C (cid:13)
0 l j
(cid:20)<
Let us show that relations in (2.19)–(2.22) follow from the relations in (2.27)–(2.29).
(a) The relation t t t in (2.19) is the first relation in (2.27).
u u
v D v
(b) The relation y y y y in (2.19): Assume that i j. Using the relations in (2.28)
i j j i
D <
and (2.29),
"1 1 # " #
[y , y ] X , X X , X
i j i l,i j m,j l,i m,j
D 2(cid:20) C (cid:13) 2(cid:20) C (cid:13) D (cid:13) (cid:13)
l i m j l i m j
< < < <
" # 2 3
X , X X ,( ) X 0.
D (cid:13)l,i (cid:13)m,j D 4(cid:13)l,i (cid:13)l,j C(cid:13)i,j C (cid:13)m,j5D
l i m j l i m j
< < < m l,<m i
¤ ¤
264 Z. DAUGHERTY, A. RAM AND R. VIRK
(c) The relation in (2.19) is part of the first relation in (2.28), and the
0 1 1 0
(cid:20) (cid:20) D (cid:20) (cid:20)
relations y y and y y in (2.19) follow from the relations
0 i i 0 1 i i 1 i j j i
(cid:20) D (cid:20) (cid:20) D (cid:20) (cid:20) (cid:20) D (cid:20) (cid:20)
and in (2.28).
i l,m l,m i
(cid:20) (cid:13) D(cid:13) (cid:20)
(d) The relations t t and t t for j 1 from (2.20) follow from the
(cid:20)0 si D si(cid:20)0 (cid:20)1 sj D sj(cid:20)1 ¤
relation t t 1 in (2.27), and the relation t t t t from (2.20)
follows frwo(cid:20)mi w(cid:0) D (cid:20)w(i) , which is part of the first r(cid:20)e1las1ti(cid:20)o1ns1inD(2.s21(cid:20)81).s1(cid:20)1
1 2 2 1
(cid:20) (cid:20) D(cid:20) (cid:20)
(e) The relations in (2.21) and (2.23) all follow from the relations t t and
i 1 (i)
w(cid:20) w(cid:0) D(cid:20)w
t t in (2.27).
i,j 1 (i), (j)
w(cid:13) w(cid:0) D(cid:13)w w
(f) By second relation in (2.28) and the (already established) second relation
in (2.20)
(cid:20) (cid:18) 1 (cid:19) 1 (cid:21)
[ ,t y t ] ,t y t t t
(cid:20)1 s1 1 s1 D (cid:20)1 s1 1(cid:0) 2(cid:20)1 s1 C 2 s1(cid:20)1 s1
(cid:20) 1 (cid:21)
, t t 0,
D (cid:20)1 (cid:13)02C 2 s1(cid:20)1 s1 D
which establishes (2.22).
To complete the proof let us show that the relations of (2.27)–(2.29) follow from
the relations in (2.19)–(2.22).
(a) The relation t t t in (2.27) is the first relation in (2.19).
u u
v D v
(b) The relations t t in (2.27) follow from the first and last relations in
i 1 (i)
w(cid:20) w(cid:0) D (cid:20)w
(2.20) (and force the definition of in (2.33)).
m
(cid:20)
(c) Since y (1 2) , the relations t t in (2.28) follow from
0,1 1 1 0,j 1 0, (j)
(cid:13) D (cid:0) = (cid:20) w(cid:13) w(cid:0) D (cid:13) w
the last relation in each of (2.20) and (2.21) (and force the definition of in (2.33)).
0,m
(cid:13)
(d) Since y t y t , the first relation in (2.21) gives that since
(cid:13)1,2 D 2(cid:0) s1 1 s1 (cid:13)2,1 D(cid:13)1,2
t t (t y t y ) y t y t
(2.35) s1(cid:13)1,2 s1 (cid:0)(cid:13)1,2 D s1 2 s1 (cid:0) 1 (cid:0) 2C s1 1 s1
t (y y )t (y y ) 0.
D s1 1C 2 s1 (cid:0) 1C 2 D
The relations t t in (2.27) then follow from (2.35) and the last rela-
1,2 1 (1), (2)
w(cid:13) w(cid:0) D(cid:13)w w
tion in (2.21) (and force the definitions t t in (2.33)).
i,j 1,2 1
(cid:13) D v(cid:13) v(cid:0)
(e) The third relation in (2.19) is and the second relation in (2.20) gives
0 1 1 0
(cid:20) (cid:20) D(cid:20) (cid:20)
. The relations in (2.28) then follow from the second set of
1 2 2 1 i j j i
(cid:20) (cid:20) D (cid:20) (cid:20) (cid:20) (cid:20) D (cid:20) (cid:20)
relations in (2.27).
(f) The second relation in (2.20) gives [ , ] 0. Multiplying (2.22) on the left and
1 2
(cid:20) (cid:20) D
right by t gives [y , ] [y ,t t ] 0. Using these and the relations in (2.19),
s1 1 (cid:20)2 D 1 s1(cid:20)1 s1 D
(cid:20) (cid:18) 1 (cid:19) (cid:21) (cid:20) 1 (cid:21)
(2.36) [ , ] , y , 0,
1 0,2 1,2 1 2 2 1,2 1,2 1 2
(cid:20) (cid:13) C(cid:13) D (cid:20) (cid:0) 2(cid:20) (cid:0)(cid:13) C(cid:13) D(cid:0) (cid:20) 2(cid:20) D
and
(cid:20) 1 1 (cid:21) 1
(2.37) [ , ] y , y [ , ] 0,
0,1 0,2 1,2 1 1 2 2 1 2
(cid:13) (cid:13) C(cid:13) D (cid:0) 2(cid:20) (cid:0) 2(cid:20) D 4 (cid:20) (cid:20) D
AFFINE BMW ALGEBRAS: THE CENTER 265
so that
[ , ] [ ,2y 2( )] [ ,2y ]
0,1 2 0,1 2 0,2 1,2 0,1 2
(cid:13) (cid:20) D (cid:13) (cid:0) (cid:13) C(cid:13) D (cid:13)
(cid:20) 1 (cid:21)
y ,2y [ , y ] 0.
1 1 2 1 2
D (cid:0) 2(cid:20) D(cid:0) (cid:20) D
Conjugating the last relation by t gives
s1
[ , ] 0, and thus [ , ] 0,
1 0,2 1 1,2
(cid:20) (cid:13) D (cid:20) (cid:13) D
by (2.36). By the third and fourth relations in (2.19),
(cid:20) 1 (cid:21) (cid:20) 1 (cid:21)
[ , ] , y 0, and [ , ] , y 0.
0 0,1 0 1 1 1 0,1 1 1 1
(cid:20) (cid:13) D (cid:20) (cid:0) 2(cid:20) D (cid:20) (cid:13) D (cid:20) (cid:0) 2(cid:20) D
By the relations in (2.20) and (2.19),
[ , ] [ , y t y t ] 0 and [ , ] [ , y t y t ] 0.
(cid:20)0 (cid:13)1,2 D (cid:20)0 2(cid:0) s1 1 s1 D (cid:20)1 (cid:13)2,3 D (cid:20)1 3(cid:0) s2 2 s2 D
Putting these together with the (already established) relations in (2.27) provides the sec-
ond set of relations in (2.28).
(g) From the commutativity of the y and the second relation in (2.21)
i
(y t y t )(y t y t ) (y t y t )(y t y t ) .
(cid:13)1,2(cid:13)3,4 D 2(cid:0) s1 1 s1 4(cid:0) s3 3 s3 D 4(cid:0) s3 3 s3 2(cid:0) s1 1 s1 D(cid:13)3,4(cid:13)1,2
By the last relation in (2.19) and the last relation in (2.20),
(cid:20) 1 (cid:21)
[ , ] y , y t y t 0.
(cid:13)0,1 (cid:13)2,3 D 1(cid:0) 2(cid:20)1 3(cid:0) s2 2 s2 D
Together with the (already established) relations in (2.27), we obtain the first set of
relations in (2.29).
(h) Conjugating (2.37) by t t t gives [ , ] 0, and this and the (already
s2 s1 s2 (cid:13)0,2 (cid:13)0,3C(cid:13)2,3 D
established) relations in (2.28) and the first set of relations in (2.29) provide
(cid:20)1 1 (cid:21)
0 [y , y ] ,
2 3 2 0,2 1,2 3 0,3 1,3 2,3
D D 2(cid:20) C(cid:13) C(cid:13) 2(cid:20) C(cid:13) C(cid:13) C(cid:13)
[ , ] [ , ] [ , ].
0,2 1,2 0,3 1,3 2,3 1,2 0,3 1,3 2,3 1,2 1,3 2,3
D (cid:13) C(cid:13) (cid:13) C(cid:13) C(cid:13) D (cid:13) (cid:13) C(cid:13) C(cid:13) D (cid:13) (cid:13) C(cid:13)
Note also that
[ , ] [ , ] [ , ] [ , ]
1,2 1,0 2,0 1,2 0,1 0,2 0,1 1,2 1,2 0,2
(cid:13) (cid:13) C(cid:13) D (cid:13) (cid:13) C(cid:13) D(cid:0) (cid:13) (cid:13) C (cid:13) (cid:13)
[ , ] [ , ] t [ , ]t 0,
D (cid:13)0,1 (cid:13)0,2 C (cid:13)1,2 (cid:13)0,2 D s1 (cid:13)0,2C(cid:13)1,2 (cid:13)0,1 s1 D
by (two applications of) (2.37). The last set of relations in (2.29) now follow from the
last set of relations in (2.27).