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 1

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 ) a ∈ M + , ϕ(a) < ∞ , ω ∈ M + if ∗ , ψ((ι ⊗ ω) ∆ (a)) = ψ(a) ω( 1 ) a ∈ M + , ψ(a) < ∞ , ω ∈ M + if ∗ . Classical case: M = L ∞ (G) , ∆ (f)(p, q) = f(pq) and � ϕ(f) = f(p) dp . 2

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. 3

Multiplicative unitary We fix (M, ∆ ) and a left invariant weight ϕ . • GNS construction: H ϕ and Λ ϕ : N ϕ → H ϕ . • Partial isometry W on H ϕ ⊗ H ϕ : � W ∗ ( Λ ϕ (x) ⊗ Λ ϕ (y)) = ( Λ ϕ ⊗ Λ ϕ ) ∆ (y)(x ⊗ 1 )) . One can prove that W is a multiplicative unitary (in the sense of S.Baaj & G.Skandalis ): W 12 W 13 W 23 = W 23 W 12 . Classical case: W acts on L 2 (G × G) and (Wξ)(p, q) = ξ(p, p − 1 q) . 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 , � � = (ι ⊗ ω)(W ∗ ) . (ι ⊗ ω)(W) such that S 4

Associated C ∗ -algebras • The closure of the linear space { (ι ⊗ ω)(W) | ω ∈ B(H ϕ ) ∗ } 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(H ϕ ) ∗ } 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 W 12 W 13 W 23 = W 23 W 12 . Main possible axioms: Regularity: (S. Baaj and G. Skandalis) Closure A ˆ A = K . Semi-regularity: (S. Baaj) Closure A ˆ A contains K . 5

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 π 2 π 1 ˆ → ( ˆ M 1 , ˆ − − − − − − → (M, ∆ ) − − − − − − (M 2 , ∆ 2 ) ∆ 1 ) , are obtained through the cocycle bicrossed product construction . Method to construct examples by taking M 1 and M 2 ordinary l.c. groups. Remark: One can define closed normal quantum subgroups , construct the quotient and obtain a short exact sequence. 6

Matched pairs of l.c. groups G is a l.c. group. • i : G 1 ֒ → G and j : G 2 ֒ → G closed subgroups, but j anti-homomorphism. • Θ : G 1 × G 2 → G : Θ (g, s) = i(g) j(s) is a Borel isomorphism. So, L ∞ (G 1 ⊗ G 2 ) ≅ L ∞ (G) . Two actions α and β , by defining j(α g (s)) i(β s (g)) = i(g) j(s) . Cocycles U : G 1 × G 1 × G 2 → T V : G 1 × G 2 × G 2 → T satisfying cocycle equations . Locally compact quantum group as cocycle crossed product and ”cocycle crossed coproduct”. 7

Smooth examples: (with L. Vainerman) Extensions of Lie groups When G 1 , G 2 and G are Lie groups, we always have Θ (G 1 × G 2 ) open in G and Θ diffeomorphism. How to find examples? • g = g 1 ⊕ g 2 matched pair of Lie algebras. Exponentiation? • Cocycles on this matched pair of Lie algebras: u : g 1 ∧ g 1 → g ∗ v : g 2 ∧ g 2 → g ∗ and 2 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 . 8

Concrete example Lie algebra g = sl 2 ( R ) : [H, X] = 2 X , [H, Y ] = − 2 Y , [X, Y ] = H . Two Lie subalgebras g 1 = span { H, X } , g 2 = span { Y } , g = g 1 ⊕ g 2 . Exponentiation: G = PSL 2 ( R ) , � � � � a x 1 0 mod {± 1 } , mod {± 1 } . G 1 : G 2 : 1 s 1 0 a Mutual actions s α (a,x) (s) = a(a + xs) , β s (a, x) = ( | a + xs | , Sgn (a + xs)x) . Non-trivial cocycle on infinitesimal level: u λ : g 1 ∧ g 1 → g ∗ 2 : u λ (H, X) = λY . Exponentiation exists for λ = 4 n π . 9

Semi-regularity (joint work with S. Baaj and G. Skandalis) Recall the most general setting: Θ : G 1 × G 2 → G : Θ (g, s) = i(g) j(s) is a Borel isomorphism. So, L ∞ (G 1 ⊗ G 2 ) ≅ L ∞ (G) . Take trivial cocycles . C ∗ -algebras of bicrossed product l.c. quantum group ˆ A = G 1 r ⋉ C 0 (G/G 1 ) , A = C 0 (G 2 \ G) ⋊ r G 2 . A r ⋉ ˆ A = A ˆ A = (G 1 × G 2 ) r ⋉ C 0 (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 (G 1 × G 2 ) is not open in G . Is the last situation possible? No, if G is Lie. 10

Some more details We have Θ : G 1 × G 2 → G : Θ (g, s) = i(g) j(s) a Borel isomorphism. We investigate A r ⋉ ˆ A = (G 1 × G 2 ) r ⋉ C 0 (G) . Reduced crossed product is obtained in the covariant representation associated to the orbit of e ∈ G . This orbit is precisely Θ (G 1 × G 2 ) . 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 (G 1 × G 2 ) is dense, we have our three cases. 11

General example Let A be a locally compact ring . Let the complement of the group of units A ∗ have (additive) Haar measure zero. Define G = A ∗ × A (a, x) · (b, y) = (ab, x + ay) . with G 1 = { (a, a − 1 ) | a ∈ A ∗ } , G 2 = { (s, 0 ) | s ∈ A ∗ } . We have a matched pair of G 1 ≅ A ∗ and G 2 ≅ 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. 12

Concrete example Let P be a family of prime numbers with � 1 p < ∞ . p ∈P Define � A = restricted Q p . p ∈P (x p ) ∈ A if x p ∈ Z p for p large enough. (x p ) ∈ A ∗ iff x p ∈ Q ∗ p for all p and x p ∈ Z ∗ p for p large enough. A ∗ has empty interior. p ) = 1 With obvious normalization: λ( Z p \ Z ∗ p . λ( A\A ∗ ) = 0 by the Borel-Cantelli lemma because � 1 p < ∞ . p ∈P 13

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 ⋉ C 0 (H) = H f ⋉ C 0 (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 = (G 1 × G 2 ) f,r ⋉ C 0 (G) . Example with l.c. ring: i(G 1 ) = u j(G 2 ) u − 1 in G and A f,r ⋉ ˆ A = (G 1 × G 2 ) f,r ⋉ C 0 (G) ∼ M G 1 f,r ⋉ C 0 (G/G 2 ) ≅ G 1 f,r ⋉ C 0 (G/G 1 ) = A u,r . Hence, A f ⋉ ˆ A = A r ⋉ ˆ A iff (A, ∆ ) is amenable, so, iff G 1 = A ∗ is amenable. Non-amenable example: A = M 2 ( R ) . 14