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

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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)

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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 mh(f , g) − mh(g, f ) = {f , g}h mod h2. 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.

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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 Ug[[h]] coinciding with the classical one modulo h and such that ∆h(X) − ∆op

h (X) = δ(X)h

mod h2. 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.

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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.
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Example: quantum SU(1, 1) group

The classical SU(1, 1) group consists of complex matrices α ¯ γ γ ¯ α

f

determinant one. The quantized algebra of functions is generated by two elements α, γ such that αγ = qαγ, αγ∗ = qγ∗α, γγ∗ = γ∗γ, α∗α − γ∗γ = 1, αα∗ − q2γ∗γ = 1. The coproduct is defined by ∆(uij) =

k

uik ⊗ ukj for (uij)ij = α qγ∗ γ α∗

.
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Woronowicz’s no-go theorem

Theorem (Woronowicz)

Given two irreducible representation π1 and π2 of the relations for α and γ

n Hilbert spaces H1 and H2, there is no way to define the tensor product

representation, that is, a representation π such that π(uij) extends

k

π1(uik) ⊗ π2(ukj). As was realized by Korogodskii and later completed by Kustermans–Koelink, the right group to quantize in this case is SU(1, 1) ⋊ Z/2Z, the normalizer of SU(1, 1) in SL(2, C)

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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 + r21 ∈ g ⊗ g and of [r12, r13] + [r12, r23] + [r13, r23] ∈ g ⊗ g ⊗ g.

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Twisting of coproduct

Theorem (Drinfeld)

Assume r ∈ g ⊗ g is such that r21 = −r and [r12, r13] + [r12, r23] + [r13, r23] = 0. Then there exists an element J = 1 + 1

2rh + · · · ∈ (Ug ⊗ Ug)[[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 Ug or a dual 2-cocycle on G.

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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 (Xi)i in g, put Bij = B(Xi, Xj), and then define r =

i,j

(B−1)ijXi ⊗ Xj. 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 .

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Examples: ax + b group

Consider the group of real invertible matrices a b 1

. Its Lie algebra is

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 |
Ch
1 ⊗ sgn(1 + iY ), I ⊗ 1
,

where I = −1 1

(acting in the regular representation) and

Ch : {−1, 1} × {−1, 1} → {−1, 1} is the unique nontrivial bicharacter.

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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 ˆ M = W ∗(G) with coproduct 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 GΩ such that its group von Neumann algebra is W ∗(G) equipped with the new coproduct Ω ˆ ∆(·)Ω∗.

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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 mJ by mJ(x ⊗ y)(a) = m( ˆ ∆(a)J−1). The coproduct ∆ defines a right coaction α of H n HJ with trivial coinvariants and such that the map HJ ⊗ HJ → HJ ⊗ H, a ⊗ b → α(a)(b ⊗ 1), is a linear isomorphism.

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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, moreover, the coproduct Ω ˆ ∆(·)Ω∗ is not cocommutative.

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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 ξ0 ∈ ˆ V such that the map φ: Q → ˆ V , q → q♭ξ0, 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

f 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 ρc has an open orbit in V ∗.

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Some examples

1) Let K be a locally compact field, τ be an order-two ring automorphism

f Matn(K). Consider the quaternionic type group H±

n (K, τ) given by the

subgroup of GL2n(K) of elements of the form

A

B ±τ(B) τ(A)

,

A, B ∈ Matn(K). Set V = Matn(K) ⊕ Matn(K) and Q = H±

n (K, τ).

Here both (V , Q) and ( ˆ V , Q) satisfy our assumptions.

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2) For n ≥ 1 and m ≥ 2, let ˆ V = Matn(K) ⊕ · · · ⊕ Matn(K)

m

and Q =      1 · · · Matn(K) . . . ... . . . . . . · · · 1 Matn(K) · · · GLn(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 {pn}n of prime numbers such that

n

1 pn < ∞ Then we can take A = ′

n(Qpn, Zpn).

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Kohn–Nirenberg quantization

The Kohn–Nirenberg quantization is initially defined as the continuous injective linear map OpKN : S′(V × ˆ V ) − → L

S(V ), S′(V )
,

from tempered Bruhat distributions on V × ˆ V to continuous linear

perators from the Bruhat-Schwartz space on V to tempered Bruhat

distributions on V by the formula OpKN(F)ϕ(v) :=

V × ˆ

V

eiξ,v−v′ F(v′, ξ) ϕ(v′) dv′dξ for F ∈ S′(V × ˆ V ), ϕ ∈ S(V ). The distributional kernel of the operator OpKN(F) is therefore given by (v, v′) →

(1 ⊗ F∗

V )F

(v′, v − v′),

where FV denotes the partial Fourier transfrom in coordinate V .

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From this we see that the Kohn–Nirenberg quantization map OpKN extends to a unitary isomorphism from L2(V × ˆ V ) onto HS(L2(V )). Hence, the Hilbert space L2(V × ˆ V ) can be endowed with an associative product f1 ⋆0 f2 := Op∗

KN

OpKN(f1) OpKN(f2)
.

By suitably normalizing the Haar measure on G = Q ⋉ V we get a unitary

perator operator U : L2(V × ˆ

V ) → L2(G) defined by (Uf )(q, v) = f

v, φ(q)
.

Using this unitary we can then transport the product ⋆0 to a product ⋆

n L2(G). This product is equivariant with respect to the action of G on

itself by left translations.

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Theorem

The distributional kernel of ⋆ defines a unitary operator in W ∗(G)¯ ⊗W ∗(G). It is a dual unitary 2-cocycle on G cohomologous to the dual cocycle defined in the previous theorem. Explicitly, this cocycle equals (F∗

V ⊗ 1) UΞ (FV ⊗ 1),

where UΞ is the unitary defined by the transfromation Ξ : Q × ˆ V × G → Q × ˆ V × G, (q, ξ, g) →

q, ξ, φ−1(ξ0 + ξ)g
.

For Q = R×, V = R and ξ0 = −1, the cohomologus cocycle (ˆ R ⊗ ˆ R)(Ω∗

21)

coincides with Stachura’s cocycle.

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Bicrossed products

A pair (G1, G2) of closed subgroups a locally compact second countable group G is called a matched pair if G1 ∩ G2 = {e} and G1G2 is a subset

f G of full measure.

Given such a pair, we have almost everywhere defined measurable left actions α of G1 and β of G2 on the measure spaces G2 and G1, resp., such that gs−1 = αg(s)−1βs(g) for g ∈ G1, s ∈ G2. We can then define a bicrossed product a locally compact quantum group with the function algebra G1 ⋉α L∞(G2).

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Twistings as bicrossed products

Theorem

With G = Q ⋉ V and Ω as before, the quantum group GΩ = (W ∗(G), Ω ˆ ∆(·)Ω∗) is isomorphic to the bicrossed product quantum group defined by the matched pair (Q, ξ0Qξ−1

0 ) of subgroups of Q ⋉ ˆ

V .

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