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 S 4 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 Preliminary notions of NCG 1 Introduction Spectral triples with symmetries Equivariant projective modules Spectral triples for q -spaces 2 The quantum 4 -sphere and its symmetries The isospectral Dirac operator for S 4 q The remaining conditions for an NC-manifold 3 Regularity condition Reality and the first order condition The orientation condition Additional examples 4 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 of 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 ) . . . . . . ✒ ✓ Connes, 1996 Compact spin manifolds Comm. real spectral triples 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. ( D 2 + 1 ) − 1 / 2 ∈ L ( p , ∞ ) ( H ) ; ⇒ the triple is even if ∃ γ = γ ∗ , γ 2 = 1 , such that γ D + D γ = 0 and a γ = γ a ∀ a ∈ A . Examples: The prototype: ( C ∞ ( M ) , L 2 ( M , S ) , D / ) . M = compact spin c manifold, S = spinor bundle, D / = Dirac operator. Baby example: ( C ∞ ( S 1 ) , L 2 ( S 1 ) , − i ∂ θ ) . A simple NC-example: ( A , ℓ 2 ( N ) , N ) with | n � can. ortho. basis of ℓ 2 ( 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 ∈ U ( g ) , a ∈ C ∞ ( M ) , ξ ∈ H ∞ . h ( 1 ) aS ( h ( 1 ) ) ξ = ( h ⊲ a ) ξ ( ∆ 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 A N := A ⊗ C N there is a natural rep. ❳ N ( a · v ) i := av i , ( h · v ) i := j = 1 ( h ( 1 ) ⊲ v j ) σ ij ( h ( 2 ) ) , with a ∈ A , v = ( v 1 , . . . , v N ) , h ∈ U and σ : U → Mat N ( 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 ❳ N i = 1 ϕ ( v ∗ v , w ∈ A N . � v , w � := i w i ) , 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 = σ mod is called modular automorphism . Let e = ( e ij ) ∈ Mat N ( A ) and define π : A N → A N by ❳ N v ∈ A N , j = 1 , . . . , N . π ( v ) j := i = 1 v i e ij , Then, π is an orthogonal projection (w.r.t. � , � ) iff e 2 = 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 Preliminary notions of NCG 1 Introduction Spectral triples with symmetries Equivariant projective modules Spectral triples for q -spaces 2 The quantum 4 -sphere and its symmetries The isospectral Dirac operator for S 4 q The remaining conditions for an NC-manifold 3 Regularity condition Reality and the first order condition The orientation condition Additional examples 4 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 := U q ( so ( 5 )) is generated, as a ∗ -algebra, by K i = K ∗ i , K − 1 , E i , F i := E ∗ i ( i = 1 , 2 ), with relations i K 2 j − K − 2 j K i E i K − 1 K i E j K − 1 = q − 1 E j if i � = j , = q i E i , [ E i , F j ] = δ ij , [ . . . ] q j − q − j i i For each l ∈ 1 2 N , we have a (finite-dim.) vector space V l with orthonormal basis | l , m 1 , m 2 ; j � (labels satisfying suitable constraints) and an irreducible ∗ -representation σ l : U → End ( V l ) given by [Chakrabarti, 1994] σ l ( K 1 ) | l , m 1 , m 2 ; j � = q m 1 | l , m 1 , m 2 ; j � , σ l ( K 2 ) | l , m 1 , m 2 ; j � = q m 2 − m 1 | l , m 1 , m 2 ; j � , ♣ σ l ( E 1 ) | l , m 1 , m 2 ; j � = [ j − m 1 ][ j + m 1 + 1 ] | l , m 1 + 1 , m 2 ; j � , ♣ σ l ( E 2 ) | l , m 1 , m 2 ; j � = [ j − m 1 + 1 ][ j − m 1 + 2 ] a l ( j , m 2 ) | l , m 1 − 1 , m 2 + 1 ; j + 1 � ♣ + [ j + m 1 ][ j − m 1 + 1 ] b l ( j , m 2 ) | l , m 1 − 1 , m 2 + 1 ; j � ♣ + [ j + m 1 ][ j + m 1 − 1 ] c l ( j , m 2 ) | l , m 1 − 1 , m 2 + 1 ; j − 1 � . [ x ] := ( q x − q − x ) / ( q − q − 1 ) Here is the q -analogue of x . Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 9 / 23

The quantum 4 -sphere q ) has 5 generators { x 0 = x ∗ 0 , x 1 , x ∗ 1 , x 2 , x ∗ The (left) U -module ∗ -algebra A := A ( S 4 2 } and can be defined as follows. Let γ i be the matrices ✵ ✶ 1 ✵ − q ✶ ✵ ✶ . . . . . . . . . . − q 2 q 3 ❇ ❈ γ 0 = ❆ , γ 1 = ❆ , γ − 1 = ❆ , ❇ ❈ ❇ ❈ ❇ . . . ❈ . . . . . . . ❇ ❈ ❇ ❈ ❇ − q 2 ❈ − q − 1 ❇ ❈ ❇ ❈ ❇ ❈ . . . ❅ . . . . ❅ . . . ❅ q 4 q . . . . . . . . . . ✵ q 3 ✶ ✵ ✶ . . . . . . . q − 3 γ 2 = ❇ ❆ , ❈ γ − 2 = ❇ ❆ , ❈ ❇ . . . . ❈ ❇ . . . ❈ ❇ q 3 ❈ ❇ ❈ ❇ ❈ ❇ ❈ . . . . . . . ❅ ❅ q − 3 . . . . . . . and call 2 ( 1 + γ 0 x 0 + γ 1 x 1 + γ 2 x 2 + γ − 1 x ∗ 1 + γ − 2 x ∗ e = 1 2 ) . Then, with η := diag ( q 4 , q − 2 , q 2 , q − 4 ) and σ 1 2 the spin 1 / 2 irrep of U in matrix form, e 2 = e Relations defining A � e ∗ = η − 1 e η ∗ -structure of A � 2 ( h ( 1 ) ) t e σ 1 2 ( S − 1 ( h ( 2 ) )) t Action of U on A h ⊲ e := σ 1 � Francesco D’Andrea (SISSA) NCG and QG 26th October 2007 10 / 23