Noncommutative geometry and quantum group symmetries Francesco - - PowerPoint PPT Presentation

Noncommutative geometry and quantum group symmetries

Francesco D’Andrea

International School for Advanced Studies Via Beirut 2-4, I-34014, Trieste, Italy

26th October 2007

Workshop on Noncommutative Manifolds II, 22-26 October 2007, Trieste

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 1 / 23

Abstract

Quantum Groups are deformations of Poisson–Lie groups in the framework of Hopf

• algebras. Quantum groups being geometric objects described by noncommutative

algebras ⇒ it is natural to study them from the point of view of Connes NCG. A comodule algebra for a quantum group (which contains the trivial corep. with

• mult. 1) is interpreted as ‘algebra of coordinates’ on a ‘quantum homogeneous space’.

Are quantum groups, resp. q-spaces, ‘noncommutative spin manifolds’? (Do they satisfy the conditions for a real spectral triple? In the original form or with some modifications?) To answer this, a number of examples were studied. I’ll use one of them — the quantum orthogonal 4-spheres S4

q — to illustrate the situation.

FD, L. D ˛ abrowski and G. Landi, The Isospectral Dirac Operator on the 4D Quantum Orthogonal Sphere,

• Commun. Math. Phys., in press [arXiv:math.QA/0611100] .

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 2 / 23

Outline

1

Preliminary notions of NCG Introduction Spectral triples with symmetries Equivariant projective modules

2

Spectral triples for q-spaces The quantum 4-sphere and its symmetries The isospectral Dirac operator for S4

q 3

The remaining conditions for an NC-manifold Regularity condition Reality and the first order condition The orientation condition

4

Additional examples

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 3 / 23

What is NCG?

Riemann:

“. . . it seems that the empirical notions on which the metrical determinations of space are founded [. . . ] cease to be valid for the infinitely small.”

Archetype of a space described by a noncommutative algebra: phase-space of quantum mechanics, [x, p] = i . What about the geometry of such spaces? From the beginning of Connes’ book:

“The correspondence between geometric spaces and commutative algebras is a familiar and basic idea

• f algebraic geometry. The purpose of this book is to extend the correspondence to the noncommutative

case in the framework of real analysis.”

Aim of NCG: to translate (differential) geometric properties into algebraic ones, that can be studied with algebraic tools and generalized to noncomm. algebras.

Geometry “is dual to” Algebra Compact Haus. top. spaces X Unital comm. C∗-algebras C(X)

( Gel’fand, 1939 )

Vector bundles E over X Finite projective C(X)-modules

( Serre-Swan, 1962 )

. . . . . . Compact spin manifolds

• Comm. real spectral triples

✒ Connes, 1996 Rennie-Varilly, 2006 ✓ Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 4 / 23

Spectral Triples

The datum (A, H, D) is called spectral triple iff: A ⊂ B(H) is a ∗-algebra with 1, H a (separable) Hilbert space; D is a selfadjoint operator on (a dense subspace of) H, (D + i)−1 ∈ K(H) and [D, a] ∈ B(H) ∀ a ∈ A; ⇒ D is p+-summable iff ∃ p ∈ R+ s.t. (D2 + 1)−1/2 ∈ L(p,∞)(H); ⇒ the triple is even if ∃ γ = γ∗, γ2 = 1, such that γD + Dγ = 0 and aγ = γa ∀ a ∈ A. Examples: The prototype: (C∞(M), L2(M, S), D / ). M = compact spinc manifold, S = spinor bundle, D / = Dirac operator. Baby example: (C∞(S1), L2(S1), −i∂θ). A simple NC-example: (A, ℓ2(N), N) with |n can. ortho. basis of ℓ2(N), S |n := |n + 1 the unilateral shift, A the algebra of polynomials in {S, S∗} and N |n := n |n the ‘number’ operator.

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 5 / 23

Spectral triples with symmetries

PROTOTYPE: G = Lie group, K ⊂ G closed subgroup, (under certain conditions) M :≃ G/K has a homogeneous spin structure, i.e.: (i) g := Lie(G) is represented on H∞; even case: [ U(g), γ] = 0 ; (ii) D / commutes with g; then with the whole U(g); (iii) [h, a]ξ = (h ⊲ a)ξ ∀ h ∈ g, a ∈ C∞(M), ξ ∈ H∞; (iii′) h(1)aS(h(1))ξ = (h ⊲ a)ξ ∀ h ∈ U(g), a ∈ C∞(M), ξ ∈ H∞. (∆h = h(1) ⊗ h(2) is the coproduct, h ⊲ a action of U(g) on C∞(M)) FORMALIZATION: Given a Hopf ∗-algebra U and an U-module ∗-algebra A, an equivariant spectral triple over A is a spectral triple (A, H, D) such that there is a ∗-rep. A ⋊ U extending the ∗-rep. of A (and commuting with γ), defined on a dense subspace of H containing the smooth domain of D; in such a subspace, D and U commute. ⇒ the 1st step is to find ∗-representations of A ⋊ U.

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 6 / 23

Where do equivariant reps come from?

On AN := A ⊗ CN there is a natural rep. (a · v)i := avi , (h · v)i := ❳N

j=1(h(1) ⊲ vj)σij(h(2)) ,

with a ∈ A, v = (v1, . . . , vN), h ∈ U and σ : U → MatN(C) any ∗-rep. of U. If ϕ : A → C is an invariant faithful state ⇒ previous rep. of A ⋊ U is a ∗-rep. w.r.t. the inner product v, w := ❳N

i=1 ϕ(v∗ i wi) ,

v, w ∈ AN . Other ∗-reps. comes from equivariant idempotents. Suppose there exists κ ∈ Aut(A) such that ϕ(ab) = ϕ

• b κ(a)

✁ , a, b ∈ A . Example: A = C.M.Q.G. and ϕ = Haar state ⇒ κ = σmod is called modular automorphism. Let e = (eij) ∈ MatN(A) and define π : AN → AN by π(v)j := ❳N

i=1 vieij ,

v ∈ AN , j = 1, . . . , N . Then, π is an orthogonal projection (w.r.t. , ) iff e2 = e = κ(e∗) ; π is a (left) A ⋊ U-module map if h ⊲ e = σ(h(1))t e σ(S−1(h(2)))t , ∀ h ∈ U .

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 7 / 23

Outline

1

Preliminary notions of NCG Introduction Spectral triples with symmetries Equivariant projective modules

2

Spectral triples for q-spaces The quantum 4-sphere and its symmetries The isospectral Dirac operator for S4

q 3

The remaining conditions for an NC-manifold Regularity condition Reality and the first order condition The orientation condition

4

Additional examples

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 8 / 23

The quantum U(so(5)) algebra

Let 0 < q < 1 . The Hopf ∗-algebra U := Uq(so(5)) is generated, as a ∗-algebra, by Ki = K∗

i , K−1 i

, Ei, Fi := E∗

i (i = 1, 2), with relations

[Ei, Fj] = δij K2

j − K−2 j

qj − q−j , KiEiK−1

i

= qiEi , KiEjK−1

i

= q−1Ej if i = j , [. . .] For each l ∈ 1

2N , we have a (finite-dim.) vector space Vl with orthonormal basis

|l, m1, m2; j (labels satisfying suitable constraints) and an irreducible ∗-representation σl : U → End(Vl) given by [Chakrabarti, 1994] σl(K1) |l, m1, m2; j = qm1 |l, m1, m2; j , σl(K2) |l, m1, m2; j = qm2−m1 |l, m1, m2; j , σl(E1) |l, m1, m2; j = ♣ [ j − m1][ j + m1 + 1] |l, m1 + 1, m2; j , σl(E2) |l, m1, m2; j = ♣ [ j − m1 + 1][ j − m1 + 2] al(j, m2) |l, m1 − 1, m2 + 1; j + 1 + ♣ [ j + m1][ j − m1 + 1] bl(j, m2) |l, m1 − 1, m2 + 1; j + ♣ [ j + m1][ j + m1 − 1] cl(j, m2) |l, m1 − 1, m2 + 1; j − 1 . Here [x ] := (qx − q−x)/(q − q−1) is the q-analogue of x.

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 9 / 23

The quantum 4-sphere

The (left) U-module ∗-algebra A := A(S4

q) has 5 generators {x0 = x∗ 0 , x1, x∗ 1 , x2, x∗ 2 }

and can be defined as follows. Let γi be the matrices γ0 =

✵ ❇ ❇ ❇ ❇ ❅ 1 . . . . −q2 . . . . −q2 . . . . q4 ✶ ❈ ❈ ❈ ❈ ❆ ,

γ1 =

✵ ❇ ❇ ❇ ❅ . . −q . . . . q3 . . . . . . . . ✶ ❈ ❈ ❈ ❆ ,

γ−1 =

✵ ❇ ❇ ❇ ❅ . . . . . . . . −q−1 . . . . q . . ✶ ❈ ❈ ❈ ❆ ,

γ2 =

✵ ❇ ❇ ❇ ❇ ❅ . q3 . . . . . . . . . q3 . . . . ✶ ❈ ❈ ❈ ❈ ❆ ,

γ−2 =

✵ ❇ ❇ ❇ ❇ ❅ . . . . q−3 . . . . . . . . . q−3 . ✶ ❈ ❈ ❈ ❈ ❆ ,

and call e = 1

2(1 + γ0x0 + γ1x1 + γ2x2 + γ−1x∗ 1 + γ−2x∗ 2 ) .

Then, with η := diag(q4, q−2, q2, q−4) and σ 1

2 the spin 1/2 irrep of U in matrix form,

Relations defining A

• e2 = e

∗-structure of A

• e∗ = η−1e η

Action of U on A

• h ⊲ e := σ 1

2 (h(1))t e σ 1 2 (S−1(h(2)))t Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 10 / 23

The left regular and spinorial representations

Notice that σmod(e∗) = e , where σmod(a) := K8

1K6 2 ⊲ a is the modular automorphism.

⇒ A4e and A4(1 − e) carry ∗-representations π± of A ⋊ U . With π0 : A ⋊ U → End(A4) the natural rep., we have: π0|U ≃ ▼

l∈N σl ,

π±|U ≃ ▼

l∈N+ 1

2

σl . By means of the representation theory of U, the representations of A are computed in the ‘harmonic’ basis |l, m1, m2; j , and can be completed to bounded reps on Hilbert spaces denoted H0, resp. H±. π0 being the GNS rep associated to the Haar state ϕ, this can be computed via the formula ϕ(a) = 0|π0(a)|0 , for all a ∈ A . In a basis of monomials, we get ϕ(xn0

0 xn1 1 (x∗ 1 )n2xn3 2 ) = δn0,2j δn1,k δn2,k δn3,0 q(2j−1)(j+k)−3j[3] [2j − 1]!! [2k]!!

[2(j + k) + 3]!! , where [0]!! = [−1]!! = 1 and [n]!! = [n] · [n − 2]!! for all n ≥ 1 .

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 11 / 23

The Dirac operator

If H := H+ ⊕ H−, γ := 1 ⊕ −1 is the natural grading, and D |l, m1, m2; j± := (l + 3

2) |l, m1, m2; j∓ ,

then,

Proposition

(A, H, D, γ) is a 4-dimensional equivariant even spectral triple. The Fredholm module (A, H, F) , with F := D|D|−1, is 1-summ. and the Chern number chF

0 ([p]) of [p] ∈ K0(A) ( p an N × N idempotent) is

chF

0 ([p]) := 1 2TrH⊗CN(γF[F, p]) .

Is this map K0(A) → Z trivial? For p = e the above-mentioned projection, chF

0 ([e]) = 1

⇒ [D] = 0 in the K-homology of A .

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 12 / 23

Outline

1

Preliminary notions of NCG Introduction Spectral triples with symmetries Equivariant projective modules

2

Spectral triples for q-spaces The quantum 4-sphere and its symmetries The isospectral Dirac operator for S4

q 3

The remaining conditions for an NC-manifold Regularity condition Reality and the first order condition The orientation condition

4

Additional examples

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 13 / 23

Regular spectral triples

(A, H, D) is called regular if A ∪ [D, A] ⊂ ❭

j∈N dom δ j ,

with δ( . ) := [|D|, . ] unbounded on B(H). Let Ψ0 be the algebra generated by ❙

j∈N δj(A ∪ [D, A]) . Assume (A, H, D) regular,

finite-dimensional, and D invertible. To each T ∈ Ψ0 we associate: ζT(z) := TrH(T|D|−z) , holomorphic for z ∈ C with Re z sufficiently large.

Definition

If (i) for all T ∈ Ψ0, the function ζT(z) has a meromorphic extension to C whose only singularities are poles, (ii) the union of such singularities is a countable set Σ ⊂ C, then we call Σ the dimension spectrum of the spectral triple.

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 14 / 23

Smoothing operators and local computations

Let OP0 := ❚

j∈N dom δ j . The class

OP−∞ := ♥ T ∈ OP0 : |D|nT ∈ OP0 ∀ n ∈ N ♦ is a two-sided ∗-ideal in the ∗-algebra OP0. Since ζ-functions associated to T ∈ OP−∞ are holomorphic on all C, OP−∞ do not contribute to Σ and one has to look at the image of Ψ0 in OP0/OP−∞ only. This motivates the following: A (multi)linear map ϕ : (Ψ0)N → C is called local if insensitive to smoothing pertur- bations of the arguments, “ϕ|OP−∞ = 0”. More generally, a two-sided ideal Jr in OP0 is given by Jr := ✟ T ∈ OP0 ☞ ☞ T|D|−p ∈ L1(H) ∀ p > r ✠ , and if T ∈ Jr, then ζT(z) has no singularities for Re z > r .

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 15 / 23

Analytic properties of S4

q

We define an auxiliary Hilbert space ˆ H containing H, and operators α ||l, m1, m2; j ± = ♣ 1 − q2(j+m1+1) ||l + 1

2, m1 + 1 2, m2; j + 1 2

± , β ||l, m1, m2; j ± = q j+m1 ||l + 1

2, m1 − 1 2, m2; j + 1 2

± , A ||l, m1, m2; j ± = ql−j+m2−ǫ ||l, m1, m2; j ± , B ||l, m1, m2; j ± = ♣ 1 − q2(l−j+m2+2−ǫ) ||l + 1, m1, m2 + 1; j ± , (ǫ := 1

2(−1)l+ 1

2 −j−m2) the first two satisfying the comm. rel. of SUq(2), the others

satisfying the comm. rel. of the equatorial Podle´ s sphere S2

• q. The map

x0 → −(αβ + β∗α∗)A , x1 →

• −α2 + q (β∗)2✁

A , x2 → B , leads to a ∗-rep. ˜ π of A on ˆ H, and Lemma a − W˜ π(a)W∗ ∈ J2 ∀ a ∈ A, where W is the orth. proj. ˆ H → H. Now, ζa(z) = ζW˜

π(a)W∗(z) for Re z > 2, and ζW˜ π(a)W∗ is easy to compute. We get,

Proposition Σ ∩ {Re z > 2} = {3, 4} consists of simple poles.

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 16 / 23

Reality and the first order condition

For the spectral triple associated to a G-homogeneous spin structure, charge conjugation C intertwines the natural representation U of G with the representation g → U(g−1)∗ . By dualization, the inverse becomes the antipode and we get the definition: an antilinear operator T is equivariant iff Th = S(h)∗T , ∀ h ∈ U . Similarly to other examples (cf. SUq(2)) we find an even (dim = 4) antilinear isometry J, J |l, m1, m2; j± = i2l+1(−1)j+m1 |l, −m1, −m2; j± , which is the antiunitary part of an equivariant T, has square −1 and commutes with D. Moreover, for all a, b ∈ A(S4

q),

[a, Jb∗J−1] ∈ OP−∞ , [[D, a], Jb∗J−1] ∈ OP−∞ .

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 17 / 23

The orientation condition

We call orientation an Hochschild cycle c = c(0) ⊗ c(1) ⊗ . . . ⊗ c(n) ∈ An , n = dim A , such that (in the even-dim. case) πD(c) := c(0)[D, c(1)] . . . [D, c(n)] = γ . This implies [c] = 0 in HHn(A). Since HH4(A) = 0 for S4

q, such a cycle c cannot exist.

On the other hand, if e = 1

2(1 + P γixi) , the tensor

ω4 := TrC4(η(e − 1

2) ˙ ⊗5)

is an element of Z4(A, κA) , with κ = σmod . Indeed its boundary is

1 8

❳ TrC4(ηγiγjγk) ·

• 2 ⊗ xi ⊗ xj ⊗ xk − xi ⊗ 1 ⊗ xj ⊗ xk + xi ⊗ xj ⊗ 1 ⊗ xk − xi ⊗ xj ⊗ xk ⊗ 1

✁ and vanishes since TrC4(ηγiγjγk) = 0 for all i, j, k ∈ {0, ±1, ±2}. Using index theory ⇒ [ω4] = 0 and then HHκ

4 (A) := H4(A, κA) = 0 .

What about a ‘twisted’ orientation?

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 18 / 23

Outline

1

Preliminary notions of NCG Introduction Spectral triples with symmetries Equivariant projective modules

2

Spectral triples for q-spaces The quantum 4-sphere and its symmetries The isospectral Dirac operator for S4

q 3

The remaining conditions for an NC-manifold Regularity condition Reality and the first order condition The orientation condition

4

Additional examples

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 19 / 23

The family of Podle´ s spheres

Let U := Uq(su(2)). It has generators K = K∗, K−1, E and F = E∗ and relations [E, F] = K2 − K−2 q − q−1 , KEK−1 = qE , KFK−1 = q−1F . The (left) U-module ∗-algebra A := A(S2

qs) has 3 generators {A = A∗, B, B∗} and can

be defined as follows. Let e := 1 1 + s2 ✥ s2 + A qB q−1B∗ 1 − q2A ✦ , then, with η := diag(q, q−1) and σ 1

2 the spin 1/2 irrep of U in matrix form,

Relations defining A

• e2 = e

∗-structure of A

• e∗ = η−1e η

Action of U on A

• h ⊲ e := σ 1

2 (h(1))t e σ 1 2 (S−1(h(2)))t

Remark: 0 ≤ s ≤ 1 (s = 0 is the ‘standard’ and s = 1 the ‘equatorial’ sphere).

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 20 / 23

A family of Dirac operators

Reps of A ⋊ U classified in [Schmüdgen-Wagner, 2003]: for N ∈ 1

2Z, let HN be the

Hilbert space with orth. basis |l, mN labelled by l ∈ N + |N| and m = −l, . . . , l. For any fixed N, if H := H−N ⊕ H+N , γ is the natural grading, and D |l, m±N := (l − |N| + 1) |l, m∓N , gives a 2-dimensional equivariant regular even spectral triple (A, H, D, γ). Index(e D+e) = chF

0 ([e]) = 2N ⇒ for different N we have inequivalent spectral triples.

An equivariant real structure is given by the antilinear isometry J |l, m±N := (−1)m+N |l, −m∓N . It commutes with D, anticommutes with γ, and the commutant and 1st order conditions hold modulo smoothing operators. Since J2 = (−1)2N , the KO-dim. is 2 mod 8 (resp. 6 mod 8) if 2N is odd (resp. even).

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 21 / 23

The dimension spectrum

Let ˆ H ≃ ℓ2(Z × N)2 be an Hilbert space with ortho. basis ||n, k ±, n ∈ 1

2Z and

n + k ∈ N. Define an inclusion W : H → ˆ H by: W |l, m±N := ||l − |N|, m + |N| ± . The following operators α, β ∈ B( ˆ H) satisfy the comm. relations of A(SUq(2)): α ||n, k ± = ♣ 1 − q2(n+k+1) ||n + 1

2, k + 1 2

± , β ||n, k ± = qn+k ||n + 1

2, k − 1 2

± . The embedding A(S2

qs) ֒

→ A(SUq(2)) leads to a ∗-rep. ˜ π : A(S2

qs) → B( ˆ

H) . Lemma For all a ∈ A , we have a − W∗˜ π(a)W ∈ OP−∞. Call C ⊂ B( ˆ H) the ∗-algebra gen. by α, β, α∗, β∗ and F. Then, Ψ0 ⊂ P CQ + OP−∞ . We determine Σ by looking at singularities of ζ-functions associated to P CQ. Only monomials T := P(ββ∗)kQ contribute to Σ. For such monomials ζT(z) = ✚ 4ζ(z − 1) + (4|N| − 2)ζ(z) if k = 0 ,

2 1−q2k ζ(z) + holomorphic function

if k = 0 , where ζ(z) is the Riemann zeta-function. ⇒ The dimension spectrum is Σ = {1, 2}.

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 22 / 23

Twisted orientation?

Let ω2 := TrC2 ✟ η(e − 1

2) ˙

⊗ e ˙ ⊗ (1 − e) ✠ , with η = diag(q, q−1) and e the idempotent defining A. This gives a class in HHκ

2 (A) := H2(A, κA), with κ(a) := σmod(a) = K2 ⊲ a. In fact, its boundary is 1 2TrC2(ηe ⊗ 1 − 1 ⊗ ηe)

which is zero since TrC2(ηe) is a number. With the twisted cocycles chF,κ

2

(a0, a1, a2) := 1

2 TrH(K−2γF[F, a0][F, a1][F, a2]) ,

• ne gets

chF,κ

2

(ω2) = [2N] . ⇒ a simple proof that HHκ

2 (A) = 0.

A ‘twisted’ version of the orientation axiom is satisfied by the 0+-dimensional D over the standard quantum sphere [Wagner, 2007].

Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 23 / 23