Semi-regularity of Locally Compact Quantum Groups Stefaan Vaes - - PDF document

Semi-regularity of Locally Compact Quantum Groups

Stefaan Vaes Institute of Mathematics Jussieu, Paris Department of Mathematics, K.U.Leuven

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L.c. quantum groups

(joint work with J. Kustermans) We call (M, ∆) a locally compact quantum group, if

• ∆ : M → M ⊗ M is a normal ∗-homomorphism, which

is co-associative: (∆ ⊗ ι)∆ = (ι ⊗ ∆)∆ ;

• there exist invariant n.s.f. weights ϕ and ψ on M:

ϕ((ω ⊗ ι)∆(a)) = ϕ(a) ω(1) if a ∈ M+, ϕ(a) < ∞, ω ∈ M+

∗,

ψ((ι ⊗ ω)∆(a)) = ψ(a) ω(1) if a ∈ M+, ψ(a) < ∞, ω ∈ M+

∗.

Classical case: M = L∞(G) , ∆(f)(p, q) = f(pq) and ϕ(f) =

• f(p) dp .

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Discussion

• Based on important work of: G.I. Kac & L. Vainerman,
• M. Enock & J.-M. Schwartz, S.L. Woronowicz, S. Baaj &
• G. Skandalis, E. Kirchberg and A. Van Daele.
• Existence of Haar measure is an axiom, but their

uniqueness is a theorem.

• No antipode or co-inverse in the definition: they will

be constructed.

• Difficulty in giving axioms without Haar measure:

characterization of the antipode in terms of the co- multiplication. The Hopf algebraic formula m(ι ⊗ S)(∆(x)) = ε(x)1 has no meaning.

• There exists an equivalent C∗-algebraic formulation.

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Multiplicative unitary

We fix (M, ∆) and a left invariant weight ϕ.

• GNS construction: Hϕ and Λϕ : Nϕ → Hϕ.
• Partial isometry W on Hϕ ⊗ Hϕ:

W ∗(Λϕ(x) ⊗ Λϕ(y)) = (Λϕ ⊗ Λϕ)

• ∆(y)(x ⊗ 1)) .

One can prove that W is a multiplicative unitary (in the sense of S.Baaj & G.Skandalis): W12W13W23 = W23W12 . Classical case: W acts on L2(G × G) and (Wξ)(p, q) = ξ(p, p−1q) . Remark: M is generated by (ι ⊗ ω)(W) and ∆(x) = W ∗(1 ⊗ x)W . Antipode: There exists a closed map S : D(S) ⊂ M → M , such that S

• (ι⊗ω)(W)
• = (ι⊗ω)(W ∗).

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Associated C∗-algebras

• The closure of the linear space

{(ι ⊗ ω)(W) | ω ∈ B(Hϕ)∗} is a C∗-algebra, denoted by A.

• ∆ : A → M(A ⊗ A) is non-degenerate.
• All objects can be restricted to A.
• The C∗-algebra of the dual is the closure of

{(ω ⊗ ι)(W) | ω ∈ B(Hϕ)∗} and denoted by ˆ A.

• S. Baaj and G. Skandalis take as the starting point a

multiplicative unitary, i.e. W ∈ B(H ⊗ H) with W12W13W23 = W23W12 . Main possible axioms: Regularity: (S. Baaj and G. Skandalis) Closure A ˆ A = K. Semi-regularity: (S. Baaj) Closure A ˆ A contains K.

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Extensions of l.c. quantum groups

(joint work with L. Vainerman) History: G.I. Kac, M. Takeuchi, S. Majid, S. Baaj & G. Skandalis. Precisely all short exact sequence with the cleftness property (M2, ∆2)

π2

− − − − − − → (M, ∆)

ˆ π1

− − − − − − → ( ˆ M1, ˆ ∆1) , are obtained through the cocycle bicrossed product construction. Method to construct examples by taking M1 and M2 ordinary l.c. groups. Remark: One can define closed normal quantum subgroups, construct the quotient and obtain a short exact sequence.

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Matched pairs of l.c. groups

G is a l.c. group.

• i : G1 ֒

→ G and j : G2 ֒ → G closed subgroups, but j anti-homomorphism.

• Θ : G1 × G2 → G : Θ(g, s) = i(g) j(s)

is a Borel isomorphism. So, L∞(G1 ⊗ G2) ≅ L∞(G). Two actions α and β, by defining j(αg(s)) i(βs(g)) = i(g) j(s) . Cocycles U : G1 × G1 × G2 → T V : G1 × G2 × G2 → T satisfying cocycle equations. Locally compact quantum group as cocycle crossed product and ”cocycle crossed coproduct”.

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Smooth examples:

(with L. Vainerman)

Extensions of Lie groups

When G1, G2 and G are Lie groups, we always have Θ(G1 × G2)

• pen in G

and Θ diffeomorphism. How to find examples?

• g = g1 ⊕ g2 matched pair of Lie algebras.

Exponentiation?

• Cocycles on this matched pair of Lie algebras:

u : g1 ∧ g1 → g∗

2

and v : g2 ∧ g2 → g∗

1

satisfying cocycle identities. Exponentiation? Exponentiation is always possible in dimension 1 + 1 and 2 + 1, but Lie groups have to be taken non- connected in some cases. Classification of all extensions in dimension 2 + 1.

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Concrete example

Lie algebra g = sl2(R): [H, X] = 2X , [H, Y ] = −2Y , [X, Y ] = H . Two Lie subalgebras g1 = span{H, X} , g2 = span{Y } , g = g1 ⊕ g2 . Exponentiation: G = PSL2(R) , G1 :

• a

x

1 a

• mod{±1} ,

G2 :

• 1

s 1

• mod{±1} .

Mutual actions α(a,x)(s) = s a(a + xs) , βs(a, x) = (|a + xs|, Sgn(a + xs)x) . Non-trivial cocycle on infinitesimal level: uλ : g1 ∧ g1 → g∗

2 : uλ(H, X) = λY .

Exponentiation exists for λ = 4n

π .

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Semi-regularity

(joint work with S. Baaj and G. Skandalis) Recall the most general setting: Θ : G1 × G2 → G : Θ(g, s) = i(g) j(s) is a Borel isomorphism. So, L∞(G1 ⊗ G2) ≅ L∞(G). Take trivial cocycles. C∗-algebras of bicrossed product l.c. quantum group A = G1 r⋉ C0(G/G1) , ˆ A = C0(G2\G) ⋊r G2 . A r⋉ ˆ A = A ˆ A = (G1 × G2) r⋉ C0(G) . Regular multiplicative unitary: K = A r⋉ ˆ A iff Θ homeomorphism onto G. Semi-regular multiplicative unitary: K ⊂ A r⋉ ˆ A iff Θ homeomorphism onto open Ω ⊂ G. Non-semi-regular multiplicative unitary: K ⊂ A r⋉ ˆ A iff Θ(G1 × G2) is not open in G. Is the last situation possible? No, if G is Lie.

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Some more details

We have Θ : G1 × G2 → G : Θ(g, s) = i(g) j(s) a Borel isomorphism. We investigate A r⋉ ˆ A = (G1 × G2) r⋉ C0(G) . Reduced crossed product is obtained in the covariant representation associated to the orbit of e ∈ G. This orbit is precisely Θ(G1 × G2). Every orbit gives repr. of the full crossed product. We know that, for a free orbit, the image is K iff the orbit is closed and homeomorphic to G; image contains K iff the orbit is locally closed and homeomorphic to G. Because Θ(G1 × G2) is dense, we have our three cases.

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General example

Let A be a locally compact ring. Let the complement

• f the group of units A∗ have (additive) Haar measure

zero. Define G = A∗ × A with (a, x) · (b, y) = (ab, x + ay) . G1 = {(a, a − 1) | a ∈ A∗} , G2 = {(s, 0) | s ∈ A∗} . We have a matched pair of G1 ≅ A∗ and G2 ≅ A∗. Bicrossed product l.c. quantum group. Semi-regular iff A∗ open in A. There exists a locally compact ring A such that the complement of A∗ is dense, with measure zero ! There exist l.c. quantum groups whose multiplicative unitary is not semi-regular.

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Concrete example

Let P be a family of prime numbers with

• p∈P

1 p < ∞ . Define A = restricted

• p∈P

Qp . (xp) ∈ A if xp ∈ Zp for p large enough. (xp) ∈ A∗ iff xp ∈ Q∗

p for all p and xp ∈ Z∗ p for p

large enough. A∗ has empty interior. With obvious normalization: λ(Zp \ Z∗

p) = 1 p.

λ(A\A∗) = 0 by the Borel-Cantelli lemma because

• p∈P

1 p < ∞ .

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Remark on A r⋉ ˆ A and A f⋉ ˆ A

Typical example of a proper action: l.c. group H acting on itself. Then, H r⋉ C0(H) = H f⋉ C0(H) (and this is K). There exist l.c. quantum groups (A, ∆) such that A f⋉ ˆ A ≠ A r⋉ ˆ A . Their action on themselves is, in a sense, non-proper ! Bicrossed products: A f,r⋉ ˆ A = (G1 × G2) f,r⋉ C0(G) . Example with l.c. ring: i(G1) = u j(G2) u−1 in G and A f,r⋉ ˆ A = (G1 × G2) f,r⋉ C0(G) ∼

M G1 f,r⋉ C0(G/G2)

≅ G1 f,r⋉ C0(G/G1) = Au,r . Hence, A f⋉ ˆ A = A r⋉ ˆ A iff (A, ∆) is amenable, so, iff G1 = A∗ is amenable. Non-amenable example: A = M2(R).

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