EQUIVARIANT ALGEBRAIC KK-THEORY 1. Kasparov's KK

EQUIVARIANT ALGEBRAIC KK-THEORY

EUGENIA ELLIS

1. Kasparov’s KK-theory Kasparov’s KK-theory is the major tool in noncommutative topology, [10]. The KK-theory of separable C∗-algebras is a common generalization both of topological K-homology and tolopological K-theory as an additive bivariant functor Let A and B separable C∗-algebras then a group KK(A, B) is deﬁned such that

top ∗ ∗ KK∗(C,B) ' K∗ (B) KK (A, C) ' Khom(A).

An important property of KK-theory is the so-called Kasparov product,

KK(A, B) × KK(B,C) → KK(A, C) which is bilinear with respect to the additive group structures. The Kasparov groups KK(A, B) for A, B ∈ C∗-Alg form a morphisms sets A → B of a category KK. The composition in KK is given by the Kasparov product and the category KK admits a triangulated category structure. There is a canonical functor k : C∗-Alg → KK that acts identically on ob- jects and every *-homomorphism f : A → B is represented by an element [f] ∈ KK(A, B). The functor k : C∗-Alg → KK

• ... is homotopy invariant: f0 ∼ f1 implies k(f0) = k(f1). • ... is C∗-stable: any corner embedding A → A ⊗ K(`2N) induces an isomorphism k(A) = k(A ⊗ K(`2N)). f g • ... is split-exact: for every split-extension I −→ A −→ A/I (i.e. there exists k(f) a *-homomorpfhism s : A/I → A such that g ◦ s = id) then k(I) −−−→ k(g) k(A) −−−→ k(A/I) is part of a distinguished triangle. The functor k : C∗-Alg → KK is the universal homotopy invariant, C∗-stable and split exact functor. Main authors who worked in the previous results: J. Cuntz, N. Higson, G. Kasparov, R. Meyer.

2. Algebraic kk-theory Algebraic kk-theory was introduced by G. Corti˜nasand A. Thom in order to show how methods from K-theory of operator algebras can be applied in completely algebraic setting. Let ` a commutative ring with unit and Alg the category of `- algebras (with or without unit). 1 2 EUGENIA ELLIS

Kasparov’s KK-theory ↔ Algebraic kk-theory [10] [2] bivariant K-theory on C∗-Alg bivariant K-theory on Alg k : C∗-Alg → KK j : Alg → KK k is stable w.r.t. compact operators j is stable w.r.t. matrices 2 [ A 'KK A ⊗ K(` (N)) A 'KK M∞(A) = Mn(A) n∈N k is continuous homotopy invariant j is polynomial homotopy invariant f f * ) A 4 B A 5 B g H V g H V ev0 ev1 ev0 ev1 H ' H C([0, 1],B) & B[t]

B 'KK C([0, 1],B) B 'KK B[t] k is split exact j is excisive k is universal for these properties j is universal for these properties

top KK∗(C,A) ' K∗ (A) kk∗(`, A) ' KH∗(A)

KH is Weibel’s homotopy K-theory deﬁned in [19]. Theorem 2.1. [2] The functor j : Alg → KK is an excisive, homotopy invariant, and M∞-stable functor and it is the universal functor for these properties. Let X be a inﬁnity set. Consider

MX := {a : X × X → ` : sopp(a) < ∞}.

Let A be an algebra, then MX A := MX ⊗` A.

Theorem 2.2. [13] The functor j : Alg → RX is an excisive, homotopy invariant, and MX -stable functor and it is the universal functor for these properties. If X = N both theorems are the same. 3. Equivariant algebraic kk-theory We introduce in [5] an algebraic bivariant K-theory for the category of G-algebras where G is a group. Equivariant Kasparov’s KK-theory ↔ Equivariant algebraic kk-theory [10] [5] bivariant K-theory on G-C∗-Alg bivariant K-theory on G-Alg k : G-C∗-Alg → KKG j : G-Alg → KKG k is stable w.r.t. compact operators j is G-stable 2 A 'KKG A ⊗ K(` (G × N)) A 'KKG M∞MG(A) k is continuous homotopy invariant j is polynomial homotopy invariant B 'KKG C([0, 1],B) B 'KKG B[t] k is split exact j is excisive k is universal for these properties j is universal for these properties

1 G compact G ﬁnite and |G| ∈ ` G top G KK∗ (C,A) ' K∗ (A o G) kk∗ (`, A) ' KH∗(A o G) EQUIVARIANT ALGEBRAIC KK-THEORY 3

Let A be a G-algebra and

MG := {a : G × G → ` : sopp(a) < ∞}.

Consider in MG ⊗ A the following action of G

g · (es,t ⊗ a) = egs,gt ⊗ g · a

A G-stable functor identiﬁes any G-algebra A with MG ⊗ A. Theorem 3.1 ([5]). The functor j : G-Alg → KKG is an excisive, homotopy invariant, and G-stable functor and it is the universal functor for these properties. 3.1. Green-Julg Theorem. The functors

o , G-Alg l Alg τ A o G = A ⊗ `G (a n g)(b n h) = a[g · b] n gh can be extended to

o , G-Alg l Alg o not adjoint functors τ jG j o KKG + KK adjoint functors if G is ﬁnite and 1 ∈ ` l o |G| τ

1 Theorem 3.2. [5] Let G be a finite group of n elements such that n ∈ `. Let A be a G-algebra and B an algebra. There is an isomorphism G τ ψGJ : kk (B ,A) → kk(B,A o G) 1 Corollary 3.3. [5] Let G be a finite group of n elementrs such that n ∈ `. Then G G kk (`, A) ' KH(A o G) kk (`, `) ' KH(`G) G H 3.2. Adjointness between IndH and ResG . Let H be a subgroup of G and A an H-algebra. Define • A(G) = {α : G → A : α is a function with finite support} G (G) • IndH (A)= {α ∈ A : α(s) = h · α(sh) ∀h ∈ H, s ∈ G} • (g · α)(s) = α(g−1s) The functors

H ResG G-Alg , H-Alg IndG is NOT left adjoint to ResH l o H G G IndH can be extended to

H ResG KKG , KKH IndG is a left adjoint to ResH k o H G G IndH Theorem 3.4. [5] Let G be a group, H a subgroup of G, B an H-algebra and A a G-algebra. Then there is an isomorphism G G H H ψIR : kk (IndH (B),A) → kk (B, ResG (A)) 4 EUGENIA ELLIS

G (G/H) H H Corollary 3.5. • kk (` ,A) ' kk (`, ResG (A)). • If H is ﬁnite then kkG(`(G/H),A) ' KH(A o H). • kkG(`(G),A) ' KH(A). 3.3. Baaj-Skandalis duality. A G-graduation on an algebra A is a decomposition on submodules M A = As AsAt ⊆ Ast ∀s, t ∈ G s∈G The functors o - G-Alg l Ggr-Alg ˆ o can be extended to

o - G-Alg l Ggr-Alg o not an equivalence ˆ o jG ˆjG o G , ˆ G KK l KK o an equivalence ˆ o

4. Algebraic quantum kk-theory 4.1. Van Daele’s algebraic quantum groups. Let ` = C and (G, ∆, ϕ) an algebraic quantum group. That means, (G, ∆) is a multiplier Hopf algebra: •G associative algebra over C with non-degenerate product

• M(G) multiplier algebra of G:(ρ1, ρ2) ∈ M(G) if – ρi : G → G is a linear map (i = 1, 2) – ρ1(hk) = ρ1(h)k ρ2(hk) = hρ2(k) ρ2(h)k = hρ1(k) ∀h, k ∈ G – (ρ1, ρ2)(˜ρ1, ρ˜2) = (ρ1ρ˜1, ρ˜2ρ2) • An homomorphism∆: G → M(G ⊗ G) is a comultiplication if – ∆(h)(1 ⊗ k) ∈ G ⊗ G (h ⊗ 1)∆(k) ∈ G ⊗ G ∀h, k ∈ G – The coassociativity property is satisﬁed:

(h ⊗ 1 ⊗ 1)(∆ ⊗ idG)(∆(k)(1 ⊗ r)) = (idG ⊗∆)((h ⊗ 1)∆(k))(1 ⊗ 1 ⊗ r) ∀h, k, r ∈ G • The following maps are bijective:

Ti : G ⊗ G → G ⊗ G T1(h ⊗ k) = ∆(h)(1 ⊗ k) T2(h ⊗ k) = (h ⊗ 1)∆(k) Proposition 4.1. If (G, ∆) is a multiplier Hopf algebra there is a unique homomorphism : G → C, called counit, such that

( ⊗ idG )(∆(h)(1 ⊗ k)) = hk (idG ⊗)((h ⊗ 1)∆(k)) = hk ∀h, k ∈ G. There is also a unique anti-homomorphism S : G → M(G), called antipode, such that

m(S ⊗ idG )(∆(h)(1 ⊗ k)) = (h)k m(idG ⊗S)((h ⊗ 1)∆(k)) = (k)h ∀h, k ∈ G here m is the multiplication map. EQUIVARIANT ALGEBRAIC KK-THEORY 5

(G, ∆) is a regular multiplier Hopf algebra if S(G) ⊆ G and S is invertible. There is a natural embedding ιG : G → M(G) which is an homomorphism

h 7→ (Lh,Rh) Lh(k) = hk Rh(k) = kh Moreover ρh ∈ G and hρ ∈ G for all h ∈ G and ρ ∈ M(G),

ρh = (Lρ1(h),Rρ1(h)) hρ = (Lρ2(h),Rρ2(h))

We write ρh = ρ1(h) and hρ = ρ2(h). A right invariant functional on G is a non-zero linear map ψ : G → C such that

(ψ ⊗ idG)∆(h) = ψ(h)1

Here (ψ ⊗ idG)∆(h) denotes the element ρ ∈ M(G) such that

ρk = (ψ ⊗ idG)(∆(h)(1 ⊗ k)) kρ = (ψ ⊗ idG)((1 ⊗ k)∆(h))

Similarly, a left invariant functional on G is a non-zero linear map ϕ : G → C such that

(idG ⊗ϕ)∆(h) = ϕ(h)1. Invariant functionals do not always exist. If ϕ is a left invariant functional on G then it is unique up to scalar multiplication and ψ = ϕ ◦ S is a right invariant functional. algebraic quantum group = regular multiplier Hopf algebra with invariants The dual of (G, ∆) is (Gˆ, ∆):ˆ • The elements of Gˆ are the linear functionals of the form ϕ(h·) ˆ G = {ξh : G → C : ξh(x) = ϕ(hx)} The elements of Gˆ can also be written as ϕ(·h), ψ(h·), ψ(·h). • The product on Gˆ is deﬁned as follows

(ξh · ξk)(x) = (ϕ ⊗ ϕ)(∆(x)(h ⊗ k)) • The coproduct ∆ˆ : Gˆ → M(Gˆ ⊗ Gˆ) is deﬁned by deﬁning the elements ∆(ˆ ξ1)(1 ⊗ ξ2) and (ξ1 ⊗ 1)∆(ˆ ξ2) in Gˆ ⊗ Gˆ as follows

((ξ1 ⊗ 1)∆(ˆ ξ2))(h ⊗ k) = (ξ1 ⊗ ξ2)(∆(h)(1 ⊗ k))

(∆(ˆ ξ1)(1 ⊗ ξ2))(h ⊗ k) = (ξ1 ⊗ ξ2)((h ⊗ 1)(∆(k)) • (Gˆ, ∆)ˆ is isomorphic to (G, ∆) as algebraic quantum group

4.2. Examples. •G = CG with the usual Hopf algebra structure.

1 e = h ϕ = ψ = χ : G → χ (h) = e C e 0 e 6= h G is compact type because 1 ∈ G. 6 EUGENIA ELLIS

ˆ X •G = CG = { agχg : ag ∈ C ag 6= 0 for a ﬁnite amount of g} g∈G χ g = h χ χ = g g h 0 g 6= h X ∆ : G → M(G ⊗ G) ∆(χg) = χgt−1 ⊗ χt t∈G

The integral is ϕ = ψ : G → C ϕ(χh) = ψ(χh) = 1 G is discrete type because exists k ∈ G xk = (x)k . •G = H a ﬁnite dimensional Hopf algebra. G is compact and discrete Let (G, ∆) be an algebraic quantum group and A be a G-module algebra. Aˆ(G) := G ⊗ Gˆ (g ⊗ f)(˜g ⊗ f˜) = gf(˜g) ⊗ f˜ ev

P t · (g ⊗ f) = t(1) · g ⊗ t(2) · f

(t · f)(g) = f(S(t)g)

Aˆ(G) ⊗ A (g ⊗ f ⊗ a)(˜g ⊗ f˜⊗ a˜) = gf(˜g) ⊗ f˜⊗ aa˜

P t · (g ⊗ f ⊗ a) = t(1) · g ⊗ t(3) · f ⊗ t(2) · a

A functor F : G-Alg → D is G-stable if F (ι1) and F (ι2) are isomorphism where Aˆ(G) ⊗ A 0 ι : A → ← Aˆ(G) ⊗ A : ι 2 0 A 1 are corner inclusions.

5. Algebraic quantum kk-theory

Theorem 5.1. Let X be a set such that card(X ) = N × dimC(G). Let F : G- Alg → D be a MX -stable functor. The functor Fˆ : G-Alg → D A 7→ F (Aˆ(G) ⊗ A) is G-stable. Theorem 5.2. Theorem The functor jG : G-Alg → KKG is an excisive, homotopy invariant, and G-stable functor. Moreover, it is the universal functor for these properties.

6. Adjointness theorems in algebraic quantum kk-theory 6.1. Green-Julg theorem. X A#H = A ⊗ H (a#h)(b#k) = a(h(1) · b)#h(2)k Theorem 6.1. Let H be a semisimple Hopf algebra and A be an H-module algebra then H kk (C,A) ' KH(A#H) EQUIVARIANT ALGEBRAIC KK-THEORY 7

6.1. Green-Julg theorem. Let G be an algebraic quantum group. • Gˆ is a G-module: (g * f)(k) = f(kg) ˆ P •G is a G-module: f * g = f(g(2))g(1) Theorem 6.2. (B. Drabant, A. Van Daele, Y. Zhang) Let A be a G-module algebra then (A#G)#G'ˆ Aˆ(G) ⊗ A

ˆ ˆ Theorem 6.3. The functors #G : KKG → KKG #Gˆ : KKG → KKG are equivalences and kkG(A, B) ' kkGˆ(A#G,B#G) where A, B are G-module algebras.

7. Conclusion G group G algebraic quantum group

equivariant KK KKG KKG ˆ stability B 'KKG MGB B 'KKG A(G) ⊗ B

o # KKG ) KK KKG ) KK j τ j τ Green-Julg adjoints functors adjoint functors if G = H 1 Theorem G is ﬁnite, |G| ∈ ` semisimple Hopf algebra G H kk∗ (`, A) ' KH∗(A o G) kk∗ (`, A) ' KH∗(A#H) H ResG Ind-Res KKG + KKH ? k G IndH adjoint functors G Imprimitivity IndH (B) o G 'KK B o H ?

o # ˆ Baaj-Skandalis KKG * KKˆ G KKG * KKG j j ˆ # o duality equivalences equivalences

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IMERL, Facultad de Ingenier´ıa, Universidad de la Republica,´ Julio Herrera y Reissig 565, 11.300, Montevideo, Uruguay