Báo cáo " Non-commutative chern characters of the $C^*-$algebras of the sphers " ppt

In [4]-[5] we computed the non-commutative Chem characters of compact Lie goup C*—algebras and of compact quantum groups, which are also homomorphisms from quantum KK —groups into entire

Non-commutative chern characters

of the c*-algebras of the sphers

Nguyen Quoc Tho*

Department of Mathematics, Vinh University, 12 Le Duan, Vinh city, Vietnam

Received 8 September 2009

Abstract We propose in this paper the construcion of non-commutative Chern characters

of the C*—algebras of spheres and quantum spheres The final computation gives us clear

relation with the ordinary Z/(2)—graded Chern chracters of torsion or their normalizers

Keyworks: Characters of the C* —algebras

1 Introduction

For compact Lie groups the Chern character ch : K*(G)@Q —> Hj p(G; Q) were constructed

In [4]-[5] we computed the non-commutative Chem characters of compact Lie goup C*—algebras and

of compact quantum groups, which are also homomorphisms from quantum KK —groups into entire current periodic cyclic homology of group C *« —algebras (resp., of C*—algebra quantum groups), chợ : K,(C*(G)) —> HE,(C*(G))), (resp., chow : Ky(C2(G)) —+ HE,(C2(G))) We obtained also the corresponding algebraic vesion cheig : K.(C*(G)) —> HP,(C*(G))), which coincides with the Fedosov-Cuntz- Quillen formula for Chem characters [5] When A = C2(G) we first computed the K —groups of C2(G) and the H F,(C2(G)) Thereafter we computed the Chern character chow : K,(C(G)) — HE,(C2(G))) as an isomorphism modulo torsions

Using the results from [4]-[5], in this paper we compute the non-commutative Chern characters chow : K,(A) — HE,(A), for two cases A = C*(S”)), the C*—algebra of spheres and A = C(S”), the C*—algebras of quantum spheres For compact groups G = O(n + 1), the Chern character ch : K*(S”) ®@ Q — H},,p(S”; Q) of the sphere S” = O(n + 1)/O(n) is an isomorphism (se, [15]) In the paper, we describe two Chern character homomorphisms

chow : K,(C*(S")) —+ HE,(C*(S”)),

and

chow : Ky(CI(S")) — HE,(CE(S”))

* E-mail: thonguyenquoc@gmail.com

249

Finally, we show that there is a commutative diagram

cho Eg K,(C*(8")) HE(C*(S"))

K.(C(Mr,)) - “+ HE,(C(Mr,))

K*(Mu,) — “=> HEBg(Nr,))

(Similarly, for A = C*("), we have an analogous commutative diagram with W x S'! of place of

W x S$”), from which we deduce that cho» is an isomorphism modulo torsions

We now briefly review he structure of the paper In section 1, we compute the Chem chracter of the C*—algebras of spheres The computation of Chern character of C*(S”) is based on two crucial points:

i) Because the sphere S” = O(n + 1)/O(n) is a homogeneous space and C*—algebra of S$” is the transformation group C*—algebra, follwing J.Parker [10], we have:

C*(S1) = C*((O(n)) @ K(L*(8")))

ii) Using the stability property theorem K, and HF, in [5], we reduce it to the computation of C*—algebras of subgroup O(n) in O(n + 1) group

In section 2, we compute the Chem character of C*—algebras of quantum spheres For quantum sphere S”, we define the compact quantum C*—algebras C2(S”), where < is a positive real number Thereafter, we prove that:

®

C‡(s")>€(S')@ @B [ - K(H„„)4i,

cZuCVW

where (H„;) 1s the elementary algebra of compact operators in a separable infinite dimensional Hilbert space H.,¢ and W is the Weyl of a maximal torus T, in SO(n)

Similar to Section 1, we first compute the A,(C2(S”) and HE,(Ci(S”), and we prove that

cho: K,(C2(S”)) —_ HE,(Ci(S”))

is a isomorphism modulo torsion

Notes on Notation: For any compact space X, we write K*(X) for the Z/(2)— graded topo- logical K —theory of X We use Swan’s theorem to identify K*(X) with Z/(2)— graded K*(C(X)) For any involution Banach algebra A, K,(A), HF,(A) and H P,(A) are Z/(2)— graded algebraic or topological K —groups of A, enire cyclic homology, and periodic cyclic homology of A, respectively

If T is a maximal torus of a compact group G’, with the corresponding Weyl group W, write C(T) for the algebra of complex valued functions on T We use the standard notation from the root theory such as P, P* for the positive highest weights, etc We denote by Mp the normalizer of T in G, by

N the set of natural numbers, R the fied of real numbers and C the field of complex numbers, ¢7,(N) the standard ¢? space of square integrable sequences of elements from A, and finally by C*(G) we denote the compact quantum algebras, C*(G) the C*—algebra of G

2 Non-commutative Chern characters of C’*—algebras of spheres

In this section, we compute non-commutative Chern characters of C*—algebras of spheres Let A be an involution Banach algebra We construct the non-commutative Chern characters choc : K,(A) — HE,(A), and show in [4] that for C*—algebra C*(G) of compact Lie groups G, the Chern character chc« is an isomorphism

Proposition 2.1 ({5], Theorem 2.6) Let H be a separable Hilbert space and B an arbitrary Banach space We have

K.(K(H)) = K,(©):

K,(B@K(H)) = K,(B)

HE,(KUT)) = HE,(C);

HH,(B@K(H)) ~ HE,(B),

where K(H) is the elementary algebra of compact operators in a separable infinite dimensional Hilbert space H

Proposition 2.2 ([5], Theorem 3.1) Let A be an involution Banach algebra with unity There is a Chern character homomorphism

chow : K,(A) — HE,(A)

Proposition 2.3 ({5], Theorem 3.2) Let G be an compact group and T a fixed maximal torus of

G with Weyl W := Ny/T Then the Chern character chcx : K,(C*(G)) —> HE,(C*(G)) is an isomorphism modulo torsions i.e

chow : K,(C*(G)) @C ———> HE,(C*(G)),

which can be identified with the classical Chern character

chow : K,(C(Nr)) — HE,(C(Np)), that is also an isomorphic modulo torsion, i.e

ch: K.(Np) @C —— Hhp(Nr)

Now, for S” = O(n + 1)/O(n), where O(n), O(n + 1) are the orthogonal matrix groups We denote by T,, a fixed maximal torus of O(n) and Mp, the normalizer of T,, in O(n) Following Proposition 1.2, there a natural Chern character cho» : K,(C(S”)) —>+ HE,(C(S")) Now, we compute first A,(C(S”)) and then HE,(C(S”)) of C*—algebra of the sphere S”

Proposition 2.4

HE,(C(5")) “ HDn(T,))

Proof We have

HE,(C(S")) = HE,(C(O(n + 1)/O(n)))

HE,(C*(O(n)) ® K(L?(O(n + 1)/O(n))))

(in virtue of, the K(L?(O(n + 1)/O(n)))) is a C * —algebra compact operators in a separable Hilbert space L?(O(n + 1)/O(n)))

~ x(Œ(O(n)))_ (by Proposition 1.1)

x(C(Mz,)) (see [5))

Thus, we have HE,(C*(S”)) = HE*(C(Mr, ))

Apart from that, because C(Mp,,) is then commutative C*—algebra, by a Cuntz- Quillen’s result [1], we have an isomorphism

AE

AE

APAC((Ne,,)) = HprWr,))- Moreover, by a result of Khalkhali [8],[9], we have

HP,(C(Nn,)) = HE(C((Ne,,))

We have, hence

HE(C(S")) = HE”(C(Mr,)) HPA(C(NH,))

> Hồn(Ấr,)* HDg(Mm,)- (by [DÌ):

Remark 1 Because H}},,(Nr,,) is the de Rham cohomology of T,,, invariant under the action of the Weyl group W, following Watanabe [15], we have a canonical isomorphism H}},.(T,,) & H*(SOn)) =

A (#3, 27, -.,@2:43), Where v2343 = o*(p;) € H?"'3(S0(n)) and o* : H*(BSO(n),R) — H*(SO(n), R) for a commutative ring R with a unit 1 € R, and p; = o;(t?, t3, ., 1?) ¢ H *(BT,,Z) the Pontryagin classes

Thus, we have

HE,(C*(S”)) =A (x3, UT vey 2/13)

Proposition 2.5

K(C(S")) = K"(H, ))

Proof We have

K(C(S")) =

T 3 so Sed i oO 5 5 © Ww HW "h } 8 =

Thus, Kk,.(C(S”)) = K,(N¢,,)

Remark 2 Following Lemma 4.2 from [5], we have

K,(ẤMr,) = K*(SO(n+ 1))/Tor

= A (BAL), os BAn—3: Ent), where 3 : R(SO(n)) —> K~!(SO(n)) be the homomorphism of Abelian groups assigning to each rep- resentation p : SO(n) —+ U(n +1) the homotopic class 3(p) = [inp] € [SO(n), U] = K-'(SO(n)), where i, : U(n + 1) — U is the canonical one, U(n + 1) and U by the n — th and infinite unitary groups respectively and <,+; € K~'(SO(n+1)) We have, finally

K*(C*(S")) = A (BOI), es BOAn—3s Ent):

Moreover, the Chern character of SU(n + 1) was computed in [14], for all n > 1 Let us recall the result Define a function

o¢:NxNx N—Z, given by

k

Theorem 2.6 Let T,, be a fixed maximal torus of O(n) and T the fixed maximal torus of SO(n), with Weyl groups W := N/T, the Chern character of C*(S”)

cho» + K,(C*(S")) — HE,(C*(S”))

is an isomorphism, given by

n chơœ(6(Ag)) = 3 ((D” °2/¡— 1)19(2n + 1, 202i (k=1, ., n-1);

chev (Enst) — 3D” '2/0i- me 360m 11,208

Proof By Proposition 1.5, we have

K,(O"(S")) = Ky (CWNa,)) = AN, and

HE,(C*(8")) * HE.(C(ẤMr,)) * HPp(ẤMr,) (by Proposition 1.4)

Now, consider the commutative diagram

cho Eg K,(C*(8")) HE(C*(S"))

K,(C(Mr,)) 22+ HBAC(Nr,))

K*Nr,) — + HồN(Ấn,))

Moreover, by the results of Watanabe [15], the Chern character ch : K*(Mr„) @ C — H}pWr,,))

is an isomorphism

Thus, chc« : K,(C*(S”")) —> HE,(C*(S”)) is an isomorphic (Proposition 1.4 and 1.5), given

by

chor(BOk)) = S2((-1) 12/24 — DN GAn+1,k, iar (k=l, « , n-1);

i=1

cho(6sa) = S3((-V* 12/248 = 1)) (55) Yon +1, 28) avai,

where

K*(C*(S")) AN (BAL), +s BAn—3: Ent1)

HE,(C*(S”)) ~ A (33, #7, ., 2+3)

3 Non-commutative Chern characters of C*—algebras of quantum spheres

In this section, we at first recall definition and main properties of compact quantum spheres and their representations More precisely, for S”, we define C*(.S”), the C*—algebras of compact quantum spheres as the C*—completion of the *—algebra F.(S”) with respect to the C*—norm, where F,(S”)

is the quantized Hopf subalgebra of the Hopf algebra, dual to the quantized universal enveloping algebra U(G), generated by matrix elements of the U(G) modules of type 1(see [3]) We prove that

Œ}(S") > oe ŒB N

cZuCVW

where (H„;) 1s the elementary algebra of compact operators in a separable infinite dimensional Hilbert space H,,; and W is the Weyl group of S” with respect to a maximal torus T

After that,we first compute the K—groups K,(C2(S”)) and the HE,(Ci(S”)), respectively Thereafter we define the Chern character of C*—algebras quantum spheres, as a homomorphism from K,(C2(S”)) to HE,(C*(S”)), and we prove that cho» : K,(C*(S")) —> HE,(C%(S”)) is an isomorphism modulo torsion

Let G be a complex algebraic group with Lie algebra G =LieG and « is real number, « £ —1 Definition 3.1 ((3], Definition 13.1) The quantized function algebra F,(G) is the subalgebra

of the Hopf algebra dual to U(G), generated by the matrix elements of the finite-dimensional U-(G)—modules of type 1

For compact quantum groups the unitary representations of F.(G) are parameterized by pairs (w, ), where ¢ is an element of a fixed maximal torus of the compact real form of G and w is a element

of the Weyl group W of T in G

Let \ € Pt,V.(A) be the irreducible U.(G)—module of type 1 with the highest weight À Then V.(A) admits a positive definite hermitian form (.,.) such that wv, v2) = (v1,2*v2) for all v1, v2 € V.(A), @ € U(G) Let {vi} be an orthogonal basis for weight space V.(A),, 4 € PT Then Ute} is an orthogonal basis for V.() Let CA „„(&) — (à?, 05) be the associated matrix elements

of V.(A) Then the matrix elements C}, „„(where A runs throngh P*, while (4,7) and (v,s) runs independently through the index set of a basis of V.(A) form a basis of Z-(Œ)(see [3])

Now very irreducible *—representation of F.(.S.L2(C) is equivalent to a representation belonging

to one of the following two families, each of which is parameterized by S' = {¢ € C\|t| = 1} i) the family of one-dimensional representations ‘7;

ii) the family 7; of representations in €?(N)(see [3])

Moreover, there exists a surjective homomorphism F;(G) —> F-(SL2(C)) induced by the natural inclusion SL2C < G and by composing the representation _, of F.(SL2C) with this homomorphic, we obtain a representation of F.(G) in €?(N) denoted by 7,,, where s; appears in the reduced decomposition w = $;,, Si, ., $;,- More precisely, 7s, : F.(G) —> L(€?(N)) is of class CCR(see [11]),i.c, its image is dense in the ideal of compact operators £(€7(N))

Then representation 7; is one-dimensional and is of the form

TC sr) = Š„sổ„„exp(2rv—1()),

1 = exp(2xv—1(œ) €TT, for œ cLieT(ee [3])

Proposition 3.1 ({3], 13.1.7) Every irreducible unitary representation of f.(ŒG) on a separable Hilbert space is the completion of a unitarizable highest weight representation Moreover, two such representation are equivalent if and only if they have the same highest weight

Proposition 3.2 ((3].13.1.9) Let w = 8;,, S¿;, , S¿„ be a reduced decomposition of an elementu of the Weyl group W of G Then

i) The Hilbert space tensor product pot = Ts, @ Tsiy @ vv Q Ts, @ T, is an irreducible

*—representation of F:(G) which is associated to the Schubert cell S.,;

ii) Up to equivalence, the representation p,,,4 does not depend on the choice of the reduced decomposition of w;

iti) Every irreducible *—representation of F(G) is equivalent to some px

The sphere S”, can be realized as the orbit under the action of the compact group SU(n +1)

of the highest weight vector vo in its natural (n + 1)—dimensional representation V_ of SU(n +1) TIfbrs, O<47,8 <n, are the matrix entries of V_, the algebra of functions on the orbit is generated

by the entries in the “first column” t,o and their complex conjugates In fact,

F(S”) ;— Cltoo, ¬" too, wey Eno] / ~,

”

where” ~” is the following equivalence relation

n tsa ~~ t60 = » = 1

s=0

Proposition 3.3 ([3], 13.2.6) The *—structure on Hopƒ algebra 7„(SL¿()), is given by

a

tr, = (-e)" *qdet(T;s),

where T 5 is the matrix obtained by removing the r*” row and the s‘ column from T

Definition 3.2 ((3],13.2.7) The *—subalgebra of F:(SIn4i1(C)) generated by he elements tz and t*,9, for s = 0, ,, is called the quantized algebra of functions on the sphere S”, and is denoted

by F-(S”) It is a quantum SLyn41(C)—space

We set z, = ts0 from now on Using Proposition 2.4, it is easy to see that the following relations hold in F,(S”):

#„.Z4 —=£ lzzz„ÌlŸr < s

a zeae lzkz, ifrAs

Zp Sk — zy + (E72 +1) a, 25.2% = 0

?t seo Zs:24 — 0 *

(CP)

Hence,F,(S”) has (CP) as its defining relations The construction of irreducible *—representation of F(S”), is given by

Theorem 3.4 ((3],13.2.9) Every irreducible *—representation of F-(S”) is equivalent exactly to one

of the following:

i) the one-dimensional representation poz +t € S', given by po¿(zä) = £} and po¿(zŸ) = 0 ifr >0,

ii) the representation poi, L<r<n, te S', on the Hilbert space tensor product ¢?(N)®", give by

Ør(23)(€k, 6 @ Ck) =

al ter kets)(] — hot +) 26, @ 1 @ ey, @ Chg, $1@ ky yy @- OER, ifs <r

The representation poz is equivalent to the restriction of the representation T, of Fz(SLn41(C))

(cf:2.3); and or r > 0, Prt is equivalent to the restriction of Ts, @ .® Ts, ® 2

From Theorem 2.6, we have

1 ker Pw,t = {0}, (œ,)€W x7

i.e the represenfatlon Œ®„„cựy Ƒ Pw tat is faithful and

dim put = 4

, Oifwfe

We recall now the definition of compact quantum of spheres C*— algebra

Definition 3.3 The C*—algebraic compact quantum sphere C2(S") is he C*—completion of the

*_algebra F(S”) with respect to the C*—norm

fll = sup loll, f € F0S")

where p runs through the*— representations of F:(S”) (cf, Theorem 2.6) and the norm on the right- hand side is the operator

It suffcies to show that || f|| is finite for all f €¢ F.(S”), for it is clear that ||.|| is a C*—norm,

ic || f.f*|| = ||f\]? We now prove that following result about he structure of compact quantum C*—algebra of sphere S”

Theorem 3.5 With notation as above, we have

® C*(s")>€(s')@ @B | K(H,, «dt,

exwew 7 5"

where C(S") is the algebra of complex valued continuous functions on S' and K(H) ideal of compact operators in a separable Hilbert space H

Proof Let w = 8;,.S8j, 8;, be a reduced decomposition of the element w € W into a product of reflections Then by Proposition 2.6, for r > 0, the representation p,, 4 1s equivalent to the restriction of Ts;, @ ++ OTs, ®T,, where 7, is the composition of the homomorphism of F,(G) onto F.(.SL2(C)) and the representation _, of F:(SL2(C)) in the Hilbert space ¢?(N)®"; and the family of one- dimensional representations 7;, given by

Ti(a) = t, Ti(b) = Tile) = 0, Tid) = £ `,

where t € S' and a,b, c,d are give by: AlgebraF.(SL2(C)) is generated by the matrix elements of

type (: ) Hence, by construction the representatlon /ø„„¿ — 1a, @ @ 7g, @ 7¿ Thus, we have

Ts, 1 C2(S") ——> OF(SL2(C)) “+> £02(R)#®").

Now, 7s, is CCR (see, [11]) and so, we have 75,(C2(S”) = K(H.,,z) Moreover T:(C2($”)) = C

Hence,

PutlCZ(S")) — (ts, @ + OTs, @7/)(G2(5”)

= Ts; (CZ(S")) 6 6 Ws, (C2(S")) @ Ti(Cz(S"))

~ K(As,,) @ ® K(As,,) ac

= K(H.„)

where H,,; = Hs, ® ®@ Hs, ® C

Thus, Pw,t(CE(S”)) = K( Hat):

Hence,

® f, mcciorn= ® fw

Now, recall a result of S Sakai from [11]: Let A be a commutative C*—algebra and B be a C* —algebra Then, Co(Q, B) = A @ B, where 2 is the spectrum space of A

Applying this result, for B = K(H.,,4) ~ K and A = C(W x S') be a commutative C*—algebra

Thus, we have

C*(S") & e@ [xu

©

Now, we first compute the K,(C(S”)) and the HE,(C2(S”)) of C*—algebra of quantum sphere S” Proposition 3.6

HE,(O3(S")) © Hip(W x $")

Proof We have

HE,(C*#(S")) = e ® [ox H.,4)dt)

= HE,(C(S') 6 HE, c@ [Kí H.,4)dt))

II HE,(C(W x S')@K_ (by Proposition 1.1)

~ HE, (C(W x S'))

Since C(W x S') is a commutative *—algebra, by Proposition 1.5 §1, we have

HE,(C%(S”)) © HE,(C(W x S')) © Hpp(W x S"))

Proposition 3.7

K,(Œ(S")) > K*(W x 8).

Proof We have

K.(0X(S")) = K,(C()øœ ŒĐ | KH) Ab)

~ K,(C(W x S')@K_ (by Proposition 1.1)

~ K,(C(W x $'))

In result of Proposition 1.5, §1, we have

K,(C(W x 8")) © K,(W x S!)

Theorem 3.8 With notation above, the Chern character of Cx —algebra of quantum sphere C'*,(S”)

chow : K,(C *2 (S") — HE,(C *, (S”))

is an isomorphism

Proof By Proposition 2.9 and 2.10, we have

HE,(C3(S")) © HE(C(W x $")) © Hpr(W x S')),

K,(C2(S")) = K,(C(W x S')) = K*(W x S"))

Now, consider the commutative diagram

ches

K,(C(W x S91)) -22, HE„(CQWV x 59)

Moreover, follwing Watanabe [I5], the ch : K*(W x 61) &Œ —¬ HƑp(W x §}) 1s an isomorphism Thus, chơ+ : K„( *; (S”?) —¬ HE,(C x;¿ (5”)) 1s an Isomorphism

Acknowledgment The author would like to thank Professor Do Ngoc Diep for his guidance and encouragement during this paper

References

[1] J Cuntz, Entice cyclic cohomology of Banach algebra and character of @—summable Fredhom modules, K-Theory., 1 (1998) 519

[2] J Cuntz, D Quillen, The X complex of the unuversal extensions, Preprint Math Inst Uni Heidelbeg, (1993) [3] V Chari, A Pressley, A guide to quantum groups, Cambridge Uni Press, (1995)

[4| D.N Diep, A.O Kuku, N.Q Tho, Non-commutative Chern character of compact Lie group C*—algebras, K- Theory, 17) (1999) 195

[5] D.N Diep, A.O Kuku, N.Q Tho, Non-commutative Chern character of compact quantum group, A- Theory, 17(2) (1999), 178

[6] D.N Diep, N.V Thu, Homotopy ivariance of entire curnt cyclic homology, Vietnam J of Math, 25(3) (1997) 211.