Noncommutative topology and quantum groups . Takeshi Katsura Faculty of Science and Technology, Keio University 27th / June / 2017 Boston University / Keio University Workshop 2017 Geometry and Mathematical Physics Boston University 26th-30th / June / 2017 Takeshi Katsura Noncommutative topology and quantum groups Plan PART I. Noncommutative topology x1: C*-algebras and Gelfand transform x2: Deformation quantization x3: Group C*-algebras x4: Crossed products x5: Warped crossed products PART II. Quantum groups Takeshi Katsura Noncommutative topology and quantum groups PART I. Noncommutative topology • A suitable algebra A of functions on a space X remembers the space X. • Transfer a theory of spaces X to a theory of the algebras A of functions on X. • Extend such theories from commutative algebras to noncommutative algebras. • Regard a noncommutative algebra as an algebra of “functions” on a “noncommutative space”. Takeshi Katsura Noncommutative topology and quantum groups x1: Definition of C*-algebras . .Definition *-algebra := algebra A over complex numbers C ∗ . w/ involution A 3 x 7! x 2 A (αx)∗ = αx∗ (x + y)∗ = x∗ + y∗ (xy)∗ = y∗x∗ (x∗)∗ = x . .Definition C*-algebra := ∗-algebra A w/ complete norm k · k ∗ 2 . satisfying kxyk ≤ kxkkyk, kx xk = kxk In this talk, every C*-algebra is unital i.e. has a multiplicative identity 1 Takeshi Katsura Noncommutative topology and quantum groups x1: Examples of C*-algebras . Example (C*-algebra) . • complex numbers C • the matrix algebras Mn(C) (n = 1; 2;:::) • the algebra B(H) of bounded operators . on a Hilbert space H . Theorem (Gelfand-Naimark) . Every C*-algebra is isomorphic H . to a C*-subalgebra of B( ). Takeshi Katsura Noncommutative topology and quantum groups x1: Gelfand transform . Example (commutative C*-algebra) . X: compact space f ! C j g .C(X) := f : X continuous T := fz 2 C j jzj = 1g n-torus Tn n-sphere Sn real projective n-space RPn . Theorem (Gelfand) . Every unital commutative C*-algebra is isomorphic . to C(X) for a unique compact space X. Takeshi Katsura Noncommutative topology and quantum groups x2: Deformation quantization “continuous” map [0; 1) 3 q 7! Aq: C*-algebra s.t. A0 C(X): commutative C*-algebra Recipe for deformation quantization of a compact space X: ∗ (i) C(X) C (f1;:::; fn j Rc [Ro) Rc: commutation relations fk fl = flfk , ∗ ∗ fk fl = flfk Ro: other relations among ffk g ∗ ;:::; j Rq [R (ii) Aq := C (a1 an c o) Rq iq c : relations ak al = e alak etc. Takeshi Katsura Noncommutative topology and quantum groups x2: Generators and relations X: compact space f1;:::; fn 2 C(X) f1;:::; fn genarates C(X) n () f := (f1;:::; fn): X ! C is injective other relations Ro among ffk g ! relations determining f(X) ⊂ Cn . Example . • C(T) C∗(u j u∗u = uu∗ = 1) u: unitary n ∗ • C(T ) C (u1;:::; un j uk ul = uluk uk : unitary) • n−1 ∗ ;:::; j C(S ) C (h1 hn hk hPl = hlhk ∗; n 2 . hk = hk k=1 hk = 1) Takeshi Katsura Noncommutative topology and quantum groups x2: Examples of deformation quantization • Noncommutative 2-torus 2 R 3 θ 7! Aθ (A0 = C(T )) ∗ 2πiθ Aθ := C (u; v j uv = e vu; u; v: unitary) −n 2 2 ♠ Kn(Aθ) = Kn(A0) = K (T ) = Z (n = 0; 1) • Noncommutative 3-sphere 3 R 3 θ 7! Bθ (B0 = C(S )) 3 ∗ C(S ) Bθ := C (u; v j uv = e2πiθvu; u∗u = uu∗ v∗v = vv∗; u∗u + v∗v = 1) ♠ U(1)-bundle over S2, Heegaard splitting, noncommutative lens space Takeshi Katsura Noncommutative topology and quantum groups x3: Group C*-algebras Γ: (discrete) group ∗ ∗ C (Γ) := C (fut gt2Γ j ut : unitary; ut us = uts) C∗(Γ) remembers all unitary representations of Γ C∗(Γ) is commutative () Γ is abelian in this case C∗(Γ) C(bΓ) where bΓ: the Pontryagin dual of Γ . Example . C∗(Z) C(T), C∗(Zn) C(Tn) F: finite abelian ) C∗(F) C(bF) C(F) CjFj ∗ C ⊕ C ⊕ C . C (S3) M2( ) Takeshi Katsura Noncommutative topology and quantum groups x3: Twisted Group C*-algebra !:Γ × Γ ! T: 2-cocycle of Γ −1 −1 (!(t0; t1)!(t0; t1t2) !(t0t1; t2)!(t1; t2) = 1) ∗ ∗ C!(Γ) := C (fut gt2Γ j ut : unitary; ut us = !(t; s)uts) . Example . 2 2 θ 2 R, !θ : Z × Z ! T by 2πimkθ !θ((n; m); (k; l)) = e ∗ 2 C (Z ) = Aθ: noncommutative 2-torus . !θ In part II, we discuss examples for the Klein 4-group K (Z=2Z) × (Z=2Z). Takeshi Katsura Noncommutative topology and quantum groups x4: Crossed products Γ: (discrete) group, X: compact space, α:Γ y X: action (i.e. Γ 3 t 7! αt 2 Homeo(X): group hom) C(X) oα Γ: crossed product C*-algebra generated by C(X) and unitaries fut gt2Γ satisfying ut us = uts and fut = ut (f ◦ αt ) If α: trivial (αt = id) and Γ is abelian, ∗ b C(X) oα Γ C(X) ⊗ C (Γ) C(X × Γ) Takeshi Katsura Noncommutative topology and quantum groups x4: noncommutative 2-torus revisited C(X) oα Γ “” C(X=Γ) ⊗ MjΓj(C) θ 2 R αθ : Z y T: action by Z 3 n 7! (z 7! e2πinθz) 2 Homeo(T) T o Z Then, C( ) αθ Aθ: noncommutative 2-torus In this way, one can get many deformation quantization of spaces like X × T. Noncommutative crossed product, twisted crossed product, ::: Takeshi Katsura Noncommutative topology and quantum groups x5: Warped crossed products X: compact space, α: Z y X: action ! α1 2 Homeo(X) L: Hermitian line bundle over X C(X) oα,L Z: warped crossed product (Søren Eilers-TK) 1 α: trivial (α1 = id), ) C(X) oα,L Z C(L ) where L1 := fv 2 L j kvk = 1g principal U(1)-bundle associated with L X = S2 and L varies (e.g. Hopf fibration), deformation quantization of S3 or the Lens spaces (Matsumoto-Tomiyama) Takeshi Katsura Noncommutative topology and quantum groups PART II. Quantum groups PART I: X: compact space ! C(X): C*-algebra PART II G: compact group i.e. G: compact space ! C(G): C*-algebra with product G × G ! G with coproduct ∆: C(G) ! C(G × G) by ∆(f)(t; s) = f(ts) ♠ C(G × G) C(G) ⊗ C(G) (c.f. Hopf algebra) Takeshi Katsura Noncommutative topology and quantum groups x1. Definition and Examples . .Definition compact quantum group := C*-algebra A with coproduct ∆: A ! A ⊗ A s.t. ⊗ ◦ ⊗ ◦ 9 . (∆ idA ) ∆ = (idA ∆) ∆ and “ inverse” . Example (compact quantum group) . • C(G) for a compact group G • C∗(Γ) for a discrete group Γ with ∆(ut ) = ut ⊗ ut • ::: . SUq(n), Takeshi Katsura Noncommutative topology and quantum groups x1: Duality Theorem ♠ Dual quantum group a compact quantum group (A; ∆) a discrete quantum group (Ab; ∆)b , a discrete quantum group (A; ∆) a compact quantum group (Ab; ∆)b , . Example . F: finite group (C(F); ∆) and (C∗(F); ∆) are . the duals of each others. Takeshi Katsura Noncommutative topology and quantum groups x2: Definition of free symmetric groups n2 Sn ,! U(n) ,! Mn(C) C ∗ f g n j R [R C(Sn) C ( pk;l k;l=1 c o) Rc: commutation relations pk;lpk 0;l0 = pk 0;l0 pk;l ∗ 2 R ; = = ; o: pPk l projection (i.e. pk;l P pk;l pk l) n n p ; = 1 (8k), p ; = 1 (8l) l=1P k l k=1 k l n ⊗ ∆(pk;l) = m=1 pk;m pm;l . Definition (Wang ’98) . (As(n); ∆): free symmetric group ∗ f g n j R As(n) := C ( pk;l k;l=1 o) . magic square C*-algebra Takeshi Katsura Noncommutative topology and quantum groups x2: Results on free symmetric groups . Proposition . n = 1; 2; 3 ) (As(n); ∆) (C(Sn); ∆) n ≥ 4 ) As(n): noncommutative, 1-dim. n ≥ 5 ) (As(n); ∆): not coamenable, . As(n): not exact . Proposition (Banica-Collins, Ogawa) . ,! R 3 ⊗ C R 3; C .As(4) C( P ) M4( ) C( P M4( )) Takeshi Katsura Noncommutative topology and quantum groups x3: Magic square C*-algebra of size 4 ∗ f g 4 j R As(4) = C ( pk;l k;l=1 o) ∗ 2 R : p ; projection (i.e. p = p = p ; ) o Pk l k;l P k;l k l 4 8 4 8 l=1 pk;l = 1 ( k), k=1 pk;l = 1 ( l) . .Definition K := ft1; t2; t3; t4g ⊂ S4: Klein 4-group . t2 = (12)(34), t3 = (13)(24), t4 = (14)(23) α: S4 × S4 y As(4): action ασ,τ(pk;l) = pσ(k),τ(l) Restrict α to K × K ⊂ S4 × S4. Takeshi Katsura Noncommutative topology and quantum groups x3: 2-cocycle of K k⧹l 1 2 3 4 1 1 1 1 1 "k;l := 2 1 −1 1 −1 3 1 −1 −1 1 4 1 1 −1 −1 !: K × K ! T: 2-cocycle by !(tk ; tl) := "k;l ∗ ∗ 4 C!(K) M2(C) (c.f. C (K) C ) c1; c2; c3; c4 2 M2(C): unitary tk tl = tm ck cl = "k;lcm Takeshi Katsura Noncommutative topology and quantum groups x3: Pauli matrices Pauli matrix c ; c ; c ; c 2 M (C) 1 ! 2 3 4 2 ! 1 0 ; i 0 ; c1 = c2 = − 0 1 ! 0 i ! 0 1 0 i c = ; c = 3 −1 0 4 i 0 fc1; c2; c3; c4g: orthonormal basis of the Hilbert space (M (C); trace) (C4; ⟨·|·⟩) P2 3 3 ; ; ; 7! 4 2 S (a1 a2 a3 a4) k=1 ak ck SU(2) SU(2)={±1g RP3 Takeshi Katsura Noncommutative topology and quantum groups x3: Real projective space n−1 n n−1 S ,! R h1;:::; hn 2 C(S ) n−1 n−1 hk hl factors through RP fk;l 2 C(RP ) .