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
Recommend
More recommend
