Quantization of subgroups of the affine group Sergey Neshveyev - PowerPoint PPT Presentation

Quantization of subgroups of the affine group Sergey Neshveyev (Joint work with Pierre Bieliavsky, Victor Gayral and Lars Tuset) UiO July 3, 2019 S. Neshveyev (UiO) Bialowieza July 3, 2019 1 / 21

Quantization of Poisson–Lie groups and Lie bialgebras Recall that a Poisson–Lie group is a Lie group G with a Poisson bracket {· , ·} such that the mutliplication map m : G × G → G is a Poisson map. In the formal deformation setting, a quantization of G is a Hopf algebra structure on C ∞ ( G )[[ h ]] coinciding with the classical one modulo h and such that mod h 2 . m h ( f , g ) − m h ( g , f ) = { f , g } h Typically we assume that the coproduct, mapping a function f into ( g , h ) �→ f ( gh ), does not deform. This is not a restriction for connected reductive Lie groups. S. Neshveyev (UiO) Bialowieza July 3, 2019 2 / 21

On the dual side, a Poisson–Lie structure on G corresponds to a Lie bialgebra structure on g , namely, a skew-symmetric map δ : g → g ⊗ g such that δ is a 1-cocycle ( δ ([ X , Y ]) = X .δ ( Y ) − Y .δ ( X )) and δ ∗ : g ∗ ⊗ g ∗ → g ∗ is a Lie bracket. A quantization of a Lie bialgebra ( g , δ ) is a Hopf algebra structure on U g [[ h ]] coinciding with the classical one modulo h and such that ∆ h ( X ) − ∆ o p mod h 2 . h ( X ) = δ ( X ) h Typically we assume that the coproduct ∆( X ) = X ⊗ 1 + 1 ⊗ X (for X ∈ g ) does not change. Again, this is not a restriction if g is reductive. S. Neshveyev (UiO) Bialowieza July 3, 2019 3 / 21

Etingof–Kazhdan quantization theorem Theorem (Etingof–Kazhdan) Any Lie bialgebra can be quantized. There is nothing remotely close to this in the analytic setting. Some problems: - Arguments/formulas are difficult to make sense of when h is not a formal parameter. - There are real obstacles in the analytic setting. S. Neshveyev (UiO) Bialowieza July 3, 2019 4 / 21

Example: quantum SU (1 , 1) group � α � ¯ γ The classical SU (1 , 1) group consists of complex matrices of γ α ¯ determinant one. The quantized algebra of functions is generated by two elements α , γ such that αγ ∗ = q γ ∗ α, γγ ∗ = γ ∗ γ, αγ = q αγ, αα ∗ − q 2 γ ∗ γ = 1 . α ∗ α − γ ∗ γ = 1 , The coproduct is defined by � α � q γ ∗ � ∆( u ij ) = u ik ⊗ u kj for ( u ij ) ij = . α ∗ γ k S. Neshveyev (UiO) Bialowieza July 3, 2019 5 / 21

Woronowicz’s no-go theorem Theorem (Woronowicz) Given two irreducible representation π 1 and π 2 of the relations for α and γ on Hilbert spaces H 1 and H 2 , there is no way to define the tensor product representation, that is, a representation π such that π ( u ij ) extends � π 1 ( u ik ) ⊗ π 2 ( u kj ) . k As was realized by Korogodskii and later completed by Kustermans–Koelink, the right group to quantize in this case is SU (1 , 1) ⋊ Z / 2 Z , the normalizer of SU (1 , 1) in SL (2 , C ) S. Neshveyev (UiO) Bialowieza July 3, 2019 6 / 21

Coboundary Lie bialgebras In many cases, given a Lie bialgebra ( g , δ ), the cobracket δ is a coboundary, that is, δ ( X ) = − X . r for some r ∈ g ⊗ g . The axioms for the cobracket are equivalent to g -invariance of r + r 21 ∈ g ⊗ g and of [ r 12 , r 13 ] + [ r 12 , r 23 ] + [ r 13 , r 23 ] ∈ g ⊗ g ⊗ g . S. Neshveyev (UiO) Bialowieza July 3, 2019 7 / 21

Twisting of coproduct Theorem (Drinfeld) Assume r ∈ g ⊗ g is such that r 21 = − r and [ r 12 , r 13 ] + [ r 12 , r 23 ] + [ r 13 , r 23 ] = 0 . Then there exists an element J = 1 + 1 2 rh + · · · ∈ ( U g ⊗ U g )[[ h ]] such that ( J ⊗ 1)(∆ ⊗ ι )( J ) = (1 ⊗ J )( ι ⊗ ∆)( J ) . A quantization of ( g , δ ) can then be defined by letting ∆ h = J ∆( · ) J − 1 . Such a J is called a twist for U g or a dual 2-cocycle on G . S. Neshveyev (UiO) Bialowieza July 3, 2019 8 / 21

As was shown by Belavin–Drinfeld, all r -matrices r as above are obtained in the following way, up to passing to a Lie subalgebra: Assume B ( X , Y ) is a nondegenerate (as a bilinear form) skew-symmetric 2-cocycle on g . Take a basis ( X i ) i in g , put B ij = B ( X i , X j ), and then define � ( B − 1 ) ij X i ⊗ X j . r = i , j When there exists B which is a coboundary, so that B ( X , Y ) = f ([ X , Y ]) for some f ∈ g ∗ , then g is called a Frobenius Lie algebra. Note that the assumption of nondegeneracy of B in this case is equivalent to openness of the coadjoint orbit of f . S. Neshveyev (UiO) Bialowieza July 3, 2019 9 / 21

Examples: ax + b group � a � b Consider the group of real invertible matrices . Its Lie algebra is 0 1 spanned by two elements X , Y such that [ X , Y ] = Y . Consider the r -matrix r = X ⊗ Y − Y ⊗ X . As was shown by Coll–Gerstenhaber–Giaquinto and Ogievetsky the corresponding Lie bialgebra can be quantized using the twist J = exp { X ⊗ log(1 + hY ) } . An analogue of the above twist in the analytic setting was found by Stachura: � � � � J = exp X ⊗ log | 1 + iY | 1 ⊗ sgn (1 + iY ) , I ⊗ 1 , Ch � − 1 � 0 where I = (acting in the regular representation) and 0 1 Ch : {− 1 , 1 } × {− 1 , 1 } → {− 1 , 1 } is the unique nontrivial bicharacter. S. Neshveyev (UiO) Bialowieza July 3, 2019 10 / 21

Dual cocycle in analytic setting In operator algebras, a locally compact quantum group G is represented by a von Neumann algebra M = L ∞ ( G ) together with a (strongly operator continuous and ∗ -preserving) coproduct ∆: M → M ¯ ⊗ M satisfying a number of axioms. We also have a group von Neumann algebra M = W ∗ ( G ) with coproduct ˆ ˆ ∆, and L ∞ (ˆ G ) = ˆ M . A dual unitary 2-cocycle on G is a unitary element Ω ∈ W ∗ ( G )¯ ⊗ W ∗ ( G ) such that (Ω ⊗ 1)( ˆ ∆ ⊗ ι )(Ω) = (1 ⊗ Ω)( ι ⊗ ˆ ∆)(Ω) . By a result of De Commer, we then have a new locally compact quantum G Ω such that its group von Neumann algebra is W ∗ ( G ) equipped with the new coproduct Ω ˆ ∆( · )Ω ∗ . S. Neshveyev (UiO) Bialowieza July 3, 2019 11 / 21

Galois objects Every dual unitary cocycle Ω gives rise to a G -Galois object L ∞ ( G ) Ω , a deformation of the function algebra L ∞ ( G ) by Ω. It is equipped with an action of G (coaction of L ∞ ( G )). This action is free and transitive in an appropriate sense. Algebraically, given a finite dimensional Hopf algebra H and a twist J for the dual Hopf algebra H ∗ , we can deform the algebra H by defining a new product m J by m J ( x ⊗ y )( a ) = m ( ˆ ∆( a ) J − 1 ) . The coproduct ∆ defines a right coaction α of H n H J with trivial coinvariants and such that the map H J ⊗ H J → H J ⊗ H , a ⊗ b �→ α ( a )( b ⊗ 1) , is a linear isomorphism. S. Neshveyev (UiO) Bialowieza July 3, 2019 12 / 21

Existence of dual cocycles Theorem Let G be a second countable locally compact group. For a unitary representation π of G on a Hilbert space H, TFAE: ( B ( H ) , Ad π ) is a G-Galois object; π is irreducible and the regular representation of G is a multiple of π . Furthermore, if these conditions are satisfied, then there exists a unique up to coboundary dual unitary 2 -cocycle Ω on G such that B ( H ) ∼ = L ∞ ( G ) Ω as G-algebras. If G is nontrivial, the cocycle Ω is not a coboundary, ∆( · )Ω ∗ is not cocommutative. moreover, the coproduct Ω ˆ S. Neshveyev (UiO) Bialowieza July 3, 2019 13 / 21

Frobenius type subgroups of the affine group From now on we consider semidirect products Q ⋉ V , where Q and V are locally compact second countable groups, V is abelian, and there exists an element ξ 0 ∈ ˆ V such that the map φ : Q → ˆ q �→ q ♭ ξ 0 , V , is a measure class isomorphism, where ♭ denotes the dual action. Theorem (Ooms) Assume Q is a Lie group, ρ is a representation of Q on a vector space V of the same dimension as Q. Then the Lie algebra of Q ⋉ V is Frobenius if and only if the action of Q defined by the contragredient representation ρ c has an open orbit in V ∗ . S. Neshveyev (UiO) Bialowieza July 3, 2019 14 / 21

Some examples 1) Let K be a locally compact field, τ be an order-two ring automorphism of Mat n ( K ). Consider the quaternionic type group H ± n ( K , τ ) given by the subgroup of GL 2 n ( K ) of elements of the form � � A B , A , B ∈ Mat n ( K ) . ± τ ( B ) τ ( A ) Set Q = H ± V = Mat n ( K ) ⊕ Mat n ( K ) and n ( K , τ ) . Here both ( V , Q ) and ( ˆ V , Q ) satisfy our assumptions. S. Neshveyev (UiO) Bialowieza July 3, 2019 15 / 21

2) For n ≥ 1 and m ≥ 2, let   1 · · · 0 Mat n ( K ) . . . ...  . . .  . . . ˆ   V = Mat n ( K ) ⊕ · · · ⊕ Mat n ( K ) and Q =  .   0 · · · 1 Mat n ( K ) � �� �  m 0 · · · 0 GL n ( K ) Then, the dual pair ( V , Q ) satisfies our assumptions but the pair ( ˆ V , Q ) does not. 3) Let A be a nondiscrete second countable locally compact ring such that the set A × of invertible elements is of full Haar measure. Then, the pair (ˆ A , A × ) satisfies our assumptions. As a concrete example, choose a sequence { p n } n of prime numbers such that 1 � < ∞ p n n Then we can take A = � ′ n ( Q p n , Z p n ). S. Neshveyev (UiO) Bialowieza July 3, 2019 16 / 21