a n d 5r,T1(xndx. ) â. ( q r - l _ q - 1 ) x n + 1. When q is not a root of unity the complexes -0,k*(D ,) (r> 0) are exact. Hence, in this case the Hochschild and cyclic ...
413
J. Math. Kyoto U n iv . (JMKYAZ)
35-3 (1995) 413-422
Hochschild and cyclic homology of Q-difference operators By Jorge A. GUCCIONE and Juan J. GUCCIONE
Introduction Let k be a n a rb itra ry field and A =k Ex, x ]. T he ring D o f formal difand the fe re n tia l o p e ra to rs o n A i s th e k -a lg e b ra g e n e ra te d b y x , x , Heisenberg relation ax—xa= 1. T h i s algebra w as studied in [K ], w here the a u th o r b u ilt u p a n explicit 2-cocycle o n th e L ie algebra underlying D, which re s tric ts to th e 2 -c o cy c le defining th e V ira so ro a lg e b r a . T o c a r r y o u t his plan K assel computed the Hochschild homology of D b y u sin g a complex simpler than the canonical o n e . I n this sam e w ork, the q-analogue of D, which is th e algebra D, (q E k\ ICA) of q-difference operators on A , has been studied. By definition D, is the algebra generated by x, x , a and the relation ax—q xa = 1 (w hich is the q-analogue of the Heisenberg r e la t io n ) . One of the main results proved in [K ] is that the Hochschild homology of D , is th e homology of a complex R*,* (A ) sim p le r th a n th e canonical o n e o f H o c h sc h ild . B y using this complex Kassel obtained some partial results about the Hochschild homology HH 1 (DO. In this paper follow ing th e results o f [K ] w e m ake a fu rth e r ste p in the sense that we can compute the Hochschild and cyclic homologies of D ( f o r definitions, basic notions and notations about these theories w e rem it to [L , C hi, 2 a n d 3 ] ) . W hen q = 1 o r qn 1 fo r all n E N th ese hom ologies are known ( s e e [W ], [K ] a n d [G - G ] ) . S o , w e c a n assum e th a t q i s a prim itive m -th root of u n ity w ith m > 1 . T he results that w e obtain resem ble the case when qn* 1 for a ll n E N . In th is sense, the case q = 1 is a special one. T h e p a p e r is d iv id e d in tw o s e c tio n s . I n th e f ir s t one w e com pute the Hochschild homology o f D , b y m e an s o f th e stu d y o f a n a tu ra l filtration of R*,* (De ). T h e graded com plex associated to this filtration gives th e Hoch-
1
I
-
-
1
a
1
g
a
schild homology of the k-algebra generated by x, x , and the relation ax= q x a . T he first tim e that this hom ology (that plays a b a sic role in our work) has been studied w as in [T ], a n d has been fully com puted in [ G - G ] . I n th e second section w e give a n explicit form ula for the m orphism s B*: HE) (De) HH (D ) (s e e [L , 2 .1 .7 .4 ]). T h is fact together HHi (D ) and B*: HHi (D ) -
g
q
2
g
Communicated by Prof. M. Jimbo, November 14,1994
1
Jorge A . Guccione and Juan J. Guccione
414
w ith th e study of the G ysin-C onnes exact sequence allows u s to com pute the cyclic homology of D . g
1. Hochschild homology of D
g
Let k be an a rb itra ry field and q E k\101 . In th is section we compute the Hochschild homology of the k-algebra D g generated by x, 0 an d th e relation ax qxa= 1. W e use the following sta n d a rd n o ta tio n s. Given q E k \ 0} —
w e p u t
(n )g
—
qn —1 for all n E Z (of course, (n) q- 1
n7=11 (n j) q f o r all n E N a n d ( n. ] fined by
n) , (0)
!q
=
1,
(n) !g=
for the G aussian binomial coefficients de-
—
q
if
—
jE N ,n E Z
if j = 0 ,n E Z
(n i +1) —
q
if j EIST, ;T
(i)q From the equality
Z
ax qx5=1 it follows that —
a
n=E
(n (
r+ s= i
s
) ! q q nr- srx n- sar
s
r
In particular, if q * 1 is a root of unity of order m , then axm=xma and amx= In [K ] it w as proved that the Hochschild homology of Da is the homology of the bicomplex
xam .
51,1
Dg
R*,*(D ):
liq L d .i/ k
00
q
01 5
A4
0,1
—
,
1
-4
3
dA/k
r
where 0 0 (x na,) = ( 1 _ q n) x na)+1 ( a j ) xn
n
)
x
n-i a
dx + (n) q X n - l a j dX 50,1(X n a i d i) = [X n a j x ] = (q1 - 1)xn + 1 5 '± (f ) yri n a 1 - 1
01(x naidx ) =
51,1(X n a j dX ) = ( x nal x —
+ l
(q.1 ( r 1 ) x n + 1 + ( i) xn ai
[a,
a , 5 ] , w h e re A = I t is e a s y t o s e e t h a t D g i s t h e O re 's extension A -1 k [x , x ] a n d a: A A i s th e isom orphism that sends x to q x a n d 5 the a-derivative that sends x to 1. So, the complex R*,*(D ) is a particular case of the one obtained in P roposition 3.2 of [G - G ] fo r a n O re 's extension of a s m o o th a lg e b ra . U s in g th is proposition to g e th e r w ith T h e o re m 1 .4 of the —
q
415
Hochschild and cyclic homology sam e paper, w e o b ta in th e quasi isom orphism r*: Tot (R*,* (D ) ) where C*(D q ) is the normalized Hochschild complex, defined by -
g
4
—C * (D ), g
(x n a id x ) = x n a i e x (x n ai) = x n ai o a , To=id gx . Y2 (x n a jd x ) = x n a j ox o a — q i x n a i
a
-
(DO
•••gR*,*(D ) of R*,*(D g ) defined by rflo(pg)= 14) ; A .ai and I I I (D q ) = (ED;Z 4A.0-9S2,1 (i= 0 , 1) . The r- th component U*,* (D q ) = Tri, (D ) ffr* 7 ( D ) of the graded complex (D ) g P*,*
Let us consider the filtration
q
(Dq)
q
-
q
1/k
q
o
associated to this filtration is A
Win, when r=0, and
4 —
1 Ar —
1,1,1
A
Oô
ni .A /k
I 01 56',1
A
when r>0, where
OP(X ) = (1 '
n
—
q n )X n , 0 1 ' (X n d X ) =
( ti n
q -1)
54-,1 (x n dx) = (q n — 1) xn+ 1 a n d 5r,T1(xndx
x
nd x
) —
(q r - l_ q - 1 )x n + 1
W hen q i s n o t a r o o t o f u n ity th e co m p lex e s 0- ,k * (D ,) (r > 0 ) are exact. Hence, in th is case the Hochschild and cyclic homologies of Dg a n d A coincide ( s e e [G-G, Exam ple 3 . 5 ] ) . O n th e o th e r h a n d t h e Hochschild a n d cyclic homologies of D I w ere calculated i n [W ] when k is a characteristic zero field a n d in [K ] in the general c a se (in spite of th a t Kassel w orks w ith the field C o f th e com plex num bers his m ethod w orks fo r a rb itra ry c h a ra c te ris tic ). So, w e assume that q is a prim itive m-th root of unity w ith m > 1 . T o carry out th e computation o f th e Hochschild homology of D g w e w ill n e e d th e following result )s-1 E Tj.,0
(D ) , f 0
m -l
We have that x 1 man- 1
Z.
1 . 1 . L em m a . Let r be a multiple of m and u E
urn-s ar sd x
ar dx E .F■91 (D 0 ) and E'sn=i( 1
-
Iti(pg)
are
cycles of Tot (R*, * (D 0 ) ) .
Proof. It is im m ediate that 0 (x " a r - 1 ) = 0 a n d 50.1(e n i - l ar dx) = O . In o r d e r t o p r o v e t h a t (51,1 ± 01) ( ET-i. (T%rx u n i - s a n - s d x ) = 0 it su ffic e s to observe that 0
(51,1+ 0 1 ) (f m
-s
-S
a r - s d x )= (q '— 1) x
UM- S a r
-
S
which follows by direct computation.
q
-i) x um-s+1 ar-s ( _ x
s
q x UM
-
S -1
s
) q x um-s or
a r - Sd x
416
Jorge A . Guccione and Juan J. Guccione
1 .2 . T heorem . L et m > 1 an d q be a prim itiv e m - t h root of unity. W e have: H1-10 (13,) = IED k .x i e
(130 k . x " a ' „ez,>o
EZ
HT1 (1) ) = 1
k. (x i O x ) E D e l k.Y i.,u,ve
4
k.Y2,u,v
, z
HH2 (1) ) =
k.173,u,v
HH (D ) =0 V
n> 2
4
q
n
where Yi,u,v
= x i' a '
1
1
a
Y 2 ,u ,v = x urn- i a v m x and
,
rn .m_s a vm _ s . x Y3,u,v--- E ( 1 1 q ) (x
a _ q _i x . . _ s a v . „
a ox )
s= 1
It is clear that HH (D ) = 0 for all n > 2 . If r>0 is not a multiple of m, then H ( D q ) ) = 0 for all 1t 0. Hence Hn(ñjc,*(D )) = Hn (A:1(D )) for all n Now let r>0 be a m u ltip le o f m . In this case Proof.
n
q
,
n
q
q
Ho (Ur:lc,*(D ) ) =
k .x '
4
EZ
k .x u 'i d x
H i ( , * ( 1 ) 4 ) ) = 0ED k .x "ED eZ
H2 (j!k*
(Dq)
EZ
.
)= EZ
Moreover, the canonical maps (- ,* (DO )
(P ,* (DO ) -
a n d re: H 2 (P,*
(Dq)
) —> H2 (6-1 4,* (Dq) )
a r e epimorphisms (this follows easily from Lem m a 1 .1 ). Hence, from the
(D ) )
long exact sequence o f homology associated to 0—>
4
(134) —>
->
(D4 ) — ) 0, we obtain
Ho (F- -' ',9 (D ) ) =Ho (T'1174!(DO ) e E D k.xum a r EZ 6
Hi
-
q
-,
( 0 (Da) ) =
(D q )
D
r1EDE k .x um _i a rd x k..zuma —
)e
uez
„Ez rn
H2 (ft,*
(Dq)
)
=
H2 (AcT)It (Dq) ) I E D l
k.E (, 5= 1 -
So,
k.x1 e IEBO k.xurnavni
Ho (R*,* (1) ) ) 4
, z
q
e m -s a r-d x
.
417
Hochschild and cyclic homology — IE D k .x " - l avniclx k .x id x e IED kxumavm-i. to
(R *,* (D. ) ) = 2
iEz
H 2 (R * ,* (D ,))=
/rEZ,v>0
EZ,I,>0
1
E D
k .E ( vez,v>o s=1
xum-savm-sdx
19
Now, the proof can be finished applying r* to this homologies.
2.
Cyclic homology of D g
In th is section we compute the cyclic homology of D , . A s in section 1 we assume th a t q is a prim itive m-th root o f u n ity w ith m > 1 . In th is case we w ill c a r r y o u t th e prom ised com putation giving a n explicit form ula for the morphisms B*: HH 0 (D 4 ) H H 1 (D ,) a n d B * : H H 2 ( D 4) and studying th e Gysin-Connes exact sequence associated to D , . W e first establish som e results that w ill be needed later.
Preliminaries
2 . 1 . N otation. L e t (Ci, d * ) b e a chain c o m p le x . Follow ing [M ] we w ill say that tw o cycles z and o f Cn a r e homologous if they define th e same element in Hn (C * ) . T hat is to say if z z ' is a boundary. —
2 . 2 . Proposition. Let q
be a primitive m - th root of unity with m> 1, u E
Z and v E N . We havh: 1) B (x
2)
The cycles u (x um
a
v m -1
( u - l) m
a y m
- 1
0 e
0
a_
x (u - l)m
a v m -1
0 a
0 x m
) and
0 a ) of C * (D ,) are homolrgous. o am ox_xum-ia(v-um Ox ® u ) and
The cycles v(xuni-la(v-1)m
B (x "
- l
a v m O x ) of ( J ( D 4 ) are homologous.
Proof. 1 ) Let D = k [x , x
-
1
, y ] be the k-algebra generated by x, x - 1 , y and
the relation x y = y x . It is easy to see that th e morphism 02:
D 0/7'
S2i)ik ,
ad,
ad1 02(d 00 d 1 0 4 )— d 0 ax a y
induces a n isomorphism from H H 2 (D ) onto Snik, w h ic h is the inverse of the v m -1 o x canonical isomorphism 6 2 : S21)/k HH2 (D ) . Let T be the cycle u (xu-ly y y x\) B y ) o f U* (D ) . A s 0 2 ( T ) = 0, T is a b o u n d a ry . L e t f: D —> D, be th e morphism of algebras that sends x to xm and y to a . T h e p r o o f c a n b e f in is h e d b y o b s e r v in g t h a t 1 1 2 (f) (T ) = u (x (u- Dmavm- 1 x m a _ x (u_i),n a v.-1 0 a x m ) _ B (x umavm- i 0 a ) 2) is similar.
2 . 3 . Proposition. Let
m > 1 and q be a primitive m - t h root o f unity.
418
Jorge A. Guccione and Juan J. Guccione
The cycles am-i
®
m qE 'st i W ( q x gous.
m -s
a
a — am-i a ® ., x m -i ® . am a - ® and ex 0 — a ' s 0 a e x ) of C *(D ) are homolo_
®
m -s
m
- i
q
To prove this proposition we will need the following lemmas
2 . 4 . L em m a . The cycles
)s_,
E(1— ,
a m _s± , a ,n_s x m _s d x
q
_q
and
)
s_,
x m _s a m _s d x
s=1
of R*,*(D q ) are equal. (2
P ro o f U sing that a x = q x a + 1 w e can see that there exist csEk s m ) such that q
)
s-i m-s+iam-s q x m -sdx = q x m -lan i-ldx + sE=2 c s xm am dx -s
-s
s=1
On the other hand, in the proof of Theorem 1.2 we showed that
) s-i x u m -s a 's d x
q
k. E (
Z2 (14,4: ( D q ) = H2 (R*, * (Dq) ) =
1—
q
w here Z2 (14 , * ( D 4 ) ) is th e submodule of the cycles of degree 2 of R *,* (D q ) C onsequently there exist 2„,, (u E Z, y E N ) , w ith I (u, y): 2 u ,v * C q finite, such that
q x m -lam -ldx -I-E csx m -sam -sdx s=2 /7/
s-1
i
' s=1 ( 1 q — q
21.4 v
a EN ,
u m -s
a v m - s dX
From this fact w e can easily deduce that 2 , q = 0 i f (u, y) ± (1.1), 21,1 = q and cs =q( i I,) fo r all r
2 . 5 . L em m a. Let B q be the algebra generated by x ,
a and the relation ax
— qx a = 1 . The cycles mqE's --1(Tj ) ( x e x o a — q -ix m -s a m -s a m m and am ® X 0 a—am-lea 0 X of C* (134 ) are homologous. n
g
s
m -s
m -s
ae x )
-1
P ro o f It is straightforw ard from th e definition o f b , th e fa ct th at q is a primitive m-th root of unity and the equality
n
Erd-s=j( ir) q( nS
(5 )
q nr—srx n— sar
419
Hochschild and cyclic homology that m -1
.; _ q i i . - i -1 ( m_ ,, a m-1 • 05 oz o x x
b (E
--lo x , a c o x
1-1
i o x o a))
®
equals to a m, 0 a o x m _ a m, a r m® a 1( 1 1(
± qI
)
m 1
(
i
s(s+ 1 -i)
(s) !qq
m — s — i. q s
s=. i=0
( q _i x ._ s _ l a m_s_i
o a
q
o x
_x m - 3 - l a m - s - 1
o x
a )
So, to finish the proof it suffices to see that m -1 (
m- 1 \
(s ) , a s(s+i-)_ m (
/ q( s
m — s— i
i= 0
i q
q
1
—
\S
3
Now the result follows easily from Theorems 1 .2 and 2.7.
B*
Hili (D q )
422
Jo rge A . Guccione and Juan J . Guccione DEPARTAMENTO DE MATEMATICA FACULTAD DE CIENCIAS EX ACTAS Y NATURALES PABELLON 1 UNIVERSIDAD DE BUENOS AIRES -
( 1 4 2 8 ) BUENOS AIRES, ARGENTINA
e mail: vander @ mate. dm . uba. ar -
References [G G ] Jorge A. Guccione and Juan J. Guccione, Hochschild and cyclic homology of O re's extensions and some quantum examples, Preprint. [K] C. K assel, C yclic hom ology o f differential operators, th e V iraso ro a lg e b ra a n a q analogue, Comm. Math. Phys., 146 (1992) , 343-356. [L] J. L. Loday, Cyclic Homology, Springer-Verlag, Berlin GOtingen Heidelberg, 1992. [M] S. Mac Lane, Homology, Springer-Verlag, Berlin GOtingen- Heidelberg, 1963. [T] L. A. Takhtadjian, Noncommutative homology of quantum tori, Punkts. Anal. Pril., 23 (1989), 7576; English translation: Funct. Anal. Appl., 23 (1989) , 147-149. [W ] W. Wodzicki, Cyclic homology of differentiial operators, Duke M ath. J., 54 (1987), 641-647. -
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