Quantum symmetric spaces from refmection equation and module - - PowerPoint PPT Presentation

Quantum symmetric spaces from refmection equation and module categories

Makoto Yamashita (山下 真) joint works with Kenny De Commer, Sergey Neshveyev, Lars Tuset

University of Oslo

XXXVIII Workshop on Geometric Methods in Physics Białowieża, 2019 July

Introduction

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Yang–Baxter equation

Braided tensor categories in mathematical physics:

• quantum integrable systems (scattering in quantized setting)
• quantum fjeld theory (intrinsic symmetry of quantum fjelds)
• quantum groups (quantized spaces with group law)

= Yang–Baxter equation braided tensor category Hopf algebra universal R-matrix Knizhnik–Zamolodchikov equation Often: deformation of simple Lie groups ⇝ matrix solution to Yang–Baxter equation

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 1 / 23

Introduction

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Next to quantize: symmetric spaces

Defjnition (Symmetric space)

• Riemannian manifold M
• can model (x1, … , xn) ↦ (−x1, … , −xn) by isometry

1 U = Iso+(M): orientation preserving isometries 2 M ≅ U/ Stab(p) for any p ∈ M

Example (2-sphere S2 ⊂ ℝ3)

• symmetry around (1, 0, 0): (x, y, z) ↦ (x, −y, −z)
• S2 ≅ SU(2)/SO(2)

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 2 / 23

Introduction

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Spoiler: how it will go

Quantization will be ribbon twist braided module category: add refmection operator to Yang–Baxter equation X U X U “(quasi)particle bouncing off the boundary wall” U: quasiparticle following braid statistics X: quasiparticle on “boundary” Mathematically:

• U: an object of a braided monoidal category 𝒟
• X: an object of a module category 𝒠, X ⊗ U makes sense in 𝒠
• refmection operator: natural isomorphism X ⊗ U → X ⊗ U
• r better…

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 3 / 23

Introduction

• verview

Spoiler: how it will go

Quantization will be ribbon twist braided module category: add ribbon twist-braid operator to Yang–Baxter equation X X σ(U) U such that = with braided automorphism σ For irreducible compact symmetric spaces:

• purely algebraic quantization: Letzter–Kolb coideals of 𝒱q(𝔳)
• transcendental quantization: cyclotomic

Knizhnik–Zamolodchikov equations

• classifjcation through co-Hochschild cohomology

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 3 / 23

Introduction Poisson geometry

Poisson manifolds

Defjnition (Poisson manifold) Poisson bracket {f1, f2} ∈ C∞(M) for fi ∈ C∞(M):

• (C∞(M), {⋅, ⋅}): Lie algebra
• Leibniz rule {f1, f2f3} = {f1, f2}f3 + {f1, f3}f2

Example (Sklyanin bracket on SU(2)) C∞(SU(2)) = ⟨xi,j ∣ 1 ≤ i, j ≤ 2 coordinate functions⟩ {xij, xkl} = δ2iδ1kx1jx2l − δ1iδ2kx2jx1l − δj1δl2xi2xk1 + δj2δl1xi1xk2 Problem (deformation quantization) Can we fjnd associative products ⋆ℏ on (large subspace of) C∞(M) such that {f1, f2} = limℏ→0

1 ℏ (f1 ⋆ℏ f2 − f2 ⋆ℏ f1)?

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 4 / 23

Introduction Poisson geometry

Poisson–Lie groups and homogeneous spaces

With Poisson structure on U, M, N, …

• f ∶ M → N is a Poisson map if f # ∶ C∞(N) → C∞(M) respects the

brackets

• U is a Poisson–Lie group if the product map U × U → U is Poisson
• action U ↷ M is Poisson if U × M → M is Poisson

Lie algebraic characterizations (Drinfeld) Poisson–Lie group structure on U ↔ Lie bialgebra δ∶ 𝔳 → ⋀2 𝔳 ↔ classical Yang–Baxter equation for r s.t. δ(x) = [r, Δ(x)] Poisson action U ↷ U/K ↔ coisotropic subalg 𝔩 ⊂ 𝔳: δ(𝔩) ⊂ 𝔩 ⊗ 𝔳 + 𝔳 ⊗ 𝔩 Foth–Lu: possible for U = Iso+(M) ↷ M compact symmetric space

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 5 / 23

Introduction Poisson geometry

Poisson–Lie groups and homogeneous spaces

With Poisson structure on U, M, N, …

• f ∶ M → N is a Poisson map if f # ∶ C∞(N) → C∞(M) respects the

brackets

• U is a Poisson–Lie group if the product map U × U → U is Poisson
• action U ↷ M is Poisson if U × M → M is Poisson

Lie algebraic characterizations (Drinfeld) Poisson–Lie group structure on U ↔ Lie bialgebra δ∶ 𝔳 → ⋀2 𝔳 ↔ classical Yang–Baxter equation for r s.t. δ(x) = [r, Δ(x)] Poisson action U ↷ U/K ↔ coisotropic subalg 𝔩 ⊂ 𝔳: δ(𝔩) ⊂ 𝔩 ⊗ 𝔳 + 𝔳 ⊗ 𝔩 Example (Poisson homogeneous structures on S2) (SU(2), Sklyanin bracket) acts on (S2, {xi, xi+1} = (x1 + t)xi+2) (index mod3) (xi)3

i=1: coordinate on ℝ3 ⊃ S2, t ∈ ℝ

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 5 / 23

Quantization Hopf algebraic

Quantum groups

Drinfeld–Jimbo deformation of universal enveloping algebra U(𝔳) can be deformed as a Hopf algebra 𝒱ℏ(𝔳); ‘just’ deform coproduct: Δℏ(x) − fmip Δℏ(x) = 2ℏδ(x) + higher order in ℏ for x ∈ 𝔳 Example (𝒱ℏ(𝔱𝔳(2)))

• generators: Eℏ, Fℏ, Kℏ = eπ√−1ℏH
• relations: KℏEℏK−1

ℏ

= e2π√−1ℏEℏ, KℏFℏK−1

ℏ

= e−2π√−1ℏFℏ, EℏFℏ − FℏEℏ = (Kℏ − K−1

ℏ )/(eπ√−1ℏ − e−π√−1ℏ)

• coproduct: Δℏ(Kℏ) = Kℏ ⊗ Kℏ

Δℏ(Eℏ) = Eℏ ⊗ Kℏ + 1 ⊗ Eℏ, Δℏ(Fℏ) = Fℏ ⊗ 1 + K−1

ℏ

⊗ Fℏ Unitary structure (when ℏ ∈ √−1ℝ): K∗

ℏ = Kℏ, E∗ ℏ = FℏKℏ, F∗ ℏ = K−1 ℏ Eℏ

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 6 / 23

Quantization Hopf algebraic

Deformation quantization from matrix coeffjcients

dual Hopf algebra of matrix coeffjcients 𝒫ℏ(U) = ⟨𝒱ℏ(𝔳) → ℂ, T ↦ φ(Tξ) ∣ ξ ∈ V ∶ (admissible) fjnite dimensional 𝒱ℏ(𝔳)-module, φ ∈ V∗⟩ 𝒫ℏ(U) is a Hopf algebra by:

• linear combination from direct sum modules
• product, coproduct by transpose of Δℏ, product in 𝒱ℏ(𝔳)
• unitary structure from unitary structure and antipode of 𝒱ℏ(𝔳)

⟨Ki,ℏ ∣ i∶ vertex of Dynkin diagram⟩ ⊂ 𝒱ℏ(𝔳) ⇝ highest weight theory

• ‘same classifjcation’ of irreducible fjnite dimensional modules
• 𝒫ℏ(U) ≅ ⨁π∶ Irr 𝒱ℏ(𝔳) V∗

π ⊗ Vπ: ‘same’ coalgebra as 𝒫(U)

⇝ Coalgebra identifjcation 𝒫ℏ(U) = 𝒫(U) solves deformation quantization problem

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 7 / 23

Quantization Hopf algebraic

Equivariant quantization

Given Poisson action U ↷ M: Problem (Quantization as Uℏ-algebras) ∃ deformation quantization 𝒫ℏ(M) as 𝒫ℏ(U)-comodule algebra? Example (Podleś spheres, c ≥ 0) 𝒫(S2

ℏ,c) = ⟨A = A∗, B, B∗ ∣ BA = q2AB, B∗B = A−A2 +c, BB∗ = q2 −q4 +c⟩

If c = c(ℏ) depends on ℏ, 𝒫(S2

ℏ,c(ℏ)) is a deformation quantization for

{xi, xi+1} = 􏿶x1 +

1 √4c(0)+1􏿹 xi+2 (i mod 3)

Problem (Quantization of module categories) ∃ deformation of module category {equivariant vector bundle over M} ↶ Rep U?

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 8 / 23

Quantization categorical

Categorical framework

Defjnition (module category) A (right) module category over a tensor category 𝒟 is given by:

• linear category 𝒠
• bifunctor 𝒠 × 𝒟 → 𝒠, (X, V) ↦ X ⊗ V
• natural isomorphisms X ⊗ 1 → X, Ψ∶ (X ⊗ V) ⊗ W → X ⊗ (V ⊗ W)

satisfying pentagon equation, … Ostrik, De Commer–Y., Neshveyev: (in fjn. dim. or unitary setting) module category 𝒠 over Rep 𝒱ℏ(𝔳) & X ∈ 𝒠 ↔ 𝒫ℏ(U)-comodule algebra A s.t. 𝒠 ={equivariant A-modules}

• deformation of Rep 𝔩 ↶ Rep 𝔳 gives a quantization of U/K
• action of Ψ and Drinfeld twist on matrix coeffjcients

= coeffjcients of ⋆ℏ-product

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 9 / 23

Quantization categorical

Braided category from quantum groups

Universal R-matrix ℛ = 1 + 2ℏr + ⋯ ∈ 𝒱ℏ(𝔳) ⊗ 𝒱ℏ(𝔳) ℛΔℏ(x)ℛ−1 = fmip Δℏ(x) (Δℏ ⊗ id)(ℛ) = ℛ13ℛ23 = ∑i,j xi ⊗ xj ⊗ yiyj for ℛ = ∑i xi ⊗ yi, (id ⊗ Δℏ)(ℛ) = ℛ12ℛ13 = ∑i,j xixj ⊗ yi ⊗ yj β(ξ ⊗ η) = fmip (π ⊗ π′)(ℛ)(ξ ⊗ η) for ξ ∈ Vπ, η ∈ Vπ′:

• is an intertwiner of (fjnite dimensional) 𝒱ℏ(𝔳)-modules
• satisfjes the Yang-Baxter equation

The category of fjnite-dimensional 𝒱ℏ(𝔳)-modules is:

1 a tensor category, with tensor product module by Δℏ 2 with braiding by the action of β

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 10 / 23

Quantization rigidity of braided categories

Two viewpoints on deformation

Rep 𝒱ℏ(𝔳) is a deformation of Rep U by:

• keeping the structure of ℂ-linear category
• but change how π1 ⊗ π2 decomposes ↔ deformation of

coproduct Drinfeld: we could instead:

• keep how π1 ⊗ π2 decomposes

≡ use the same bifunctor ⊗∶ Rep U × Rep U → Rep U

• but change the associator Φ∶ (π1 ⊗ π2) ⊗ π3 ≅ π1 ⊗ (π2 ⊗ π3)

⇝ should consider quasi-Hopf algebra (𝒱(𝔳), Δ, Φ)

1 associator Φ: pentagon equation 2 braiding from R-matrix ℛ: hexagon equation with Φ

⇝ quasi-triangular quasi-Hopf algebra

3 deformation of 𝒱(𝔳) is controlled by co-Hochschild cohomology

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 11 / 23

Quantization rigidity of braided categories

How to use co-Hochschild cohomology

If Φ = ∑∞

k=0 ℏkΦ(k) ∈ (𝒱(𝔳)⊗3)𝔳ℏ is an associator in Rep U:

((V1 ⊗ V2) ⊗ V3) ⊗ V4 (V1 ⊗ (V2 ⊗ V3)) ⊗ V4 V1 ⊗ ((V2 ⊗ V3) ⊗ V4) V1 ⊗ (V2 ⊗ (V3 ⊗ V4)) (V1 ⊗ V2) ⊗ (V3 ⊗ V4)

• Φ1,2,3

Φ1,23,4 Φ2,3,4 Φ1,2,34 Φ12,3,4

and if Φ′ is another such, with Φ(k) = Φ′(k) for 0 ≤ k < n ⇒ linearized condition ψ1,2,3 + ψ1,23,4 + ψ2,3,4 = ψ12,3,4 + ψ1,2,34 for ψ ∈ Φ′(n) − Φ(n) ↔ 3-cocycle in the complex: ((𝒱(𝔳)⊗k)𝔳 → (𝒱(𝔳)⊗k+1)𝔳, alt. sum of Δ on different legs) ⟶

chm (⋀∗ ℂ 𝔳)𝔳

dim(⋀3 𝔳)𝔳 = 1 for simple 𝔳 (essential parameter space)

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 12 / 23

Quantization rigidity of braided categories

Braided category from confjguration spaces

Knizhnik–Zamolodchikov equations (π1, V1), … , (πn, Vn): representations of U tij: action of invariant 2-tensor t = − ∑k xk ⊗ xk ∈ 𝔳 ⊗ 𝔳 on Vi ⊗ Vj ⇝ KZn-equation 𝜖v 𝜖zi (z) = ℏ 􏾝

j≠i

tij zi − zj v(z) (1 ≤ i ≤ n)

• n Yn = {z = (z1, … , zn) ∈ ℂn ∣ zi ≠ zj}, v∶ Yn → V1 ⊗ ⋯ ⊗ Vn
• n = 3: associator Φℏ by monodromy from ‘z1 = z2’ to ‘z2 = z3’
• n = 2: braiding by 180° monodromy eπ√−1ℏt12 around ‘z1 = z2’

Drinfeld: Rep 𝒱ℏ(𝔳) ≅ (Rep U, Φℏ) as braided tensor categories β2 ∼ e2π√−1ℏt12 ∼ ℛ21ℛ contains all information

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 13 / 23

Quantization coideals

Involution

Understanding K = Stab(p) as fjxed point subalgebra Involution σ ∈ Aut(𝔳): σ ≠ id, σ2 = id ⇝ 𝔩 = Lie(K) is 𝔳σ = Lie(U)σ; K is Uσ Irreducible compact symmetric space is either: type I above construction with compact simple Lie group U type II M ≃ U ≃ (U × U)/U (Poisson torsor; we don’t care) Type I case:

• 𝔩 = 𝔳 ∩ 𝔥 for some real form 𝔥 ⊂ 𝔳 ⊗ ℂ
• M has Hermitian structure ⇔ M ⊂ U.μ for some μ ∈ 𝔳∗

Example (2-sphere)

• 𝔱𝔭(2; ℝ) = 𝔱𝔳(2) ∩ 𝔱𝔪(2; ℝ) ⊂ 𝔱𝔪(2; ℂ) = 𝔱𝔳(2) ⊗ ℂ
• SU(2) → SO(3) ↷ S2 ⊂ ℝ3 ≃ 𝔱𝔳(2)∗

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 14 / 23

Quantization coideals

Coideal quantization of symmetric spaces

Real form 𝔥 ⊂ 𝔳 ⊗ ℂ ⇝ classifjcation by Satake diagrams ⇝ try to mimic the ‘Satake involution’ θ ∈ Aut(𝔳) by θℏ ∈ Aut(𝒱ℏ(𝔳))

• replace Weyl group action W ↷ 𝔳 ⊗ ℂ by Lusztig braid

automorphisms

• use ⟨Ki,ℏ ∣ i⟩ ⊂ 𝒱ℏ(𝔳) as substitute of maximal torus

⇝ Letzter, Kolb: coideal subalgebra 𝒱θ

ℏ(𝔩) ⊂ 𝒱ℏ(𝔳)

• Specializes to 𝒱(𝔩) ⊂ 𝒱(𝔳)
• Dual coideal 𝒫ℏ(U/K) = {f ∈ 𝒫ℏ(U) ∣ T ⊳ f = ε(T)f for T ∈ 𝒱θ

ℏ(𝔩)} is

a deformation quantization of U/K

• Extra parameter for Hermitian case: 𝒱θ,t

ℏ (𝔩)

Example (Dual coideal of Podleś spheres) ∗-coideals 𝒱θ,t

ℏ (𝔱𝔭(2)) = ⟨Fℏ − e−2π√−1ℏEℏK−1 ℏ

+ √−1tK−1

ℏ ⟩ ⊂ 𝒱ℏ(𝔱𝔳(2)) for

t ∈ ℝ

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 15 / 23

Quantization refmection equation

Refmection equation

Gurevich, Donin, Mudrov …: refmection equation βV,VK1βV,VK1 = K1βV,VK1βV,V (β: braiding) for K ∈ Endℂ(V) ⇝ quantum homogeneous spaces Example (K-matrix for 𝒱ℏ(𝔱𝔳(2)), V = ℂ2 (defjning representation)) K = 􏿷√−1t(q−1 − q) −q−1/2 q−1/2 􏿺, β = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ q−1 1 1 q−1 − q q ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , q = eπ√−1ℏ Balagović, Kolb: universal K-matrix 𝒧 ∈ 𝒱θ(,t)

ℏ

(𝔩) ⊗ 𝒱ℏ(𝔳)

• we should think of 𝒧 as natural transformation

π1 ⊗ πσℏ

2 → π1 ⊗ π2 for π1 ∈ Rep 𝒱θ ℏ(𝔩), π2 ∈ Rep 𝒱ℏ(𝔳),

σℏ ∈ Aut(𝒱ℏ(𝔳)) quantization of involution σ

• (ε ⊗ id)(𝒧) = K solves the (modifjed) refmection equation

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 16 / 23

Quantization refmection equation

Braiding on module categories

Defjnition (Ribbon twist braid on module categories) Given

• (𝒟, β): braided tensor category
• σ∶ 𝒟 → 𝒟: braided autoequivalence
• 𝒠: 𝒟-module category

ribbon σ-braid on 𝒠 is: η∶ X ⊗ σ(V) → X ⊗ V satisfying

• refmection equation against braiding β
• multiplicativity in V

X X σ(U) U such that = =

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 17 / 23

Quantization refmection equation

Type B confjguration space

Original KZ equation: on Yn = ℂn⧵ (type A hyperplanes “zi = zj”) Cherednik, Leibman, Golubeva–Leksin, Eniquez–Etingof Adding ribbon twist η∶ X ⊗ σ(V) → X ⊗ V: adding “z1 = 0” Cyclotomic KZn-equation

• (πi, Vi)n

i=1: representations of U, (π0, X): representation of K

• t𝔩

0i: invariant 2-tensor of 𝔩 on X ⊗ Vi

• C𝔩

i: Casimir of 𝔩 on Vi

𝜖v 𝜖zi (z) = ℏ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 2t𝔩

0i + C𝔩 i

zi + 􏾝

j≠i

t𝔩

ij ± t𝔳⊖𝔩 ij

zi ∓ zj ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ v(z) (1 ≤ i ≤ n)

• n Y ′

n = {z = (z1, … , zn) ∈ ℂn ∣ zi ≠ zj, zi ≠ 0}, v∶ Y ′ n → X ⊗ V1 ⊗ ⋯ ⊗ Vn

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 18 / 23

Quantization refmection equation

Type B confjguration space

Cyclotomic KZn-equation

• (πi, Vi)n

i=1: representations of U, (π0, X): representation of K

• t𝔩

0i: invariant 2-tensor of 𝔩 on X ⊗ Vi

• C𝔩

i: Casimir of 𝔩 on Vi

𝜖v 𝜖zi (z) = ℏ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 2t𝔩

0i + C𝔩 i

zi + 􏾝

j≠i

t𝔩

ij ± t𝔳⊖𝔩 ij

zi ∓ zj ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ v(z) (1 ≤ i ≤ n)

• n Y ′

n = {z = (z1, … , zn) ∈ ℂn ∣ zi ≠ zj, zi ≠ 0}, v∶ Y ′ n → X ⊗ V1 ⊗ ⋯ ⊗ Vn

⇝ module category (Rep K, Ψℏ) ↶ (Rep U, Φℏ), with ribbon σ-twist braid

• n = 2: associator Ψℏ by monodromy from ‘z1 = 0’ to ‘z1 = z2’
• n = 1: η = eπ√−1ℏ(2t𝔩

01+C𝔩 1) by 180° monodromy around ‘z1 = 0’

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 18 / 23

Quantization refmection equation

Picard’s approximation

Associator: monodromy of cyclotomic KZ2 equation from ‘w = z1/z2 = 0’ to ‘w = 1’: Ψℏ = lima→0 a−ℏA1Ha(1 − a)aℏA0; Ha(w) = 1 +

∞

􏾝

n=1

ℏn 􏾝

i∗∈{−1,0,1}n

􏾚

w a

ωi1 ⋯ ωinAi1 ⋯ Ain with

• A−1 = t𝔩

12 − t𝔳⊖𝔩 12 , A1 = t𝔳 12, A0 = 2t𝔩 01 + C𝔩 1 (residues)

• ωk =

dx x−k

• ∫

w a ωi1 ⋯ ωin: iterated integral

⇒ Ψℏ = 1 + ℏ(log 2)A−1 + ℏ2 􏿶T +

π2 12[A0, A−1] + π2 6 [A1, A0]􏿹 + O(ℏ3)

for symmetric element T ∈ ℂ1𝒱(𝔩) ⊗ 𝒱(𝔳)⊗2

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 19 / 23

Quantization tying together

Conjectural correspondence

Conjecture (DC.–N.–T.–Y.) For ℏ ∈ √−1ℝ: ∃ unitary module category equivalence Rep 𝒱θ,t

ℏ (𝔩) ↶ Rep 𝒱ℏ(𝔳) ≡ (ℂχ. Rep U, Ψℏ) ↶ (Rep U, Φℏ)

up to Drinfeld’s equivalence Rep 𝒱ℏ(𝔳) ≡ (Rep U, Φ) Hermitian case: parameter correspondence t (for Poisson structures) ↔ character χ ∶ 𝔸(𝔩) → √−1ℝ (for KZ equation) Why care about unitary setting?

• Good duality between 𝒫ℏ(U)-coactions and module categories
• Hermitian case: 𝒱θ,t

ℏ (𝔩) for different t are isomorphic comodule

algebras, but correspond to different Poisson structures ⇝ they have different ∗-structures (and different unitary reps) ⇝ will give “correct” basepoint of module category

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 20 / 23

Quantization tying together

Sanity check: 2-spheres

Theorem (DC.–Y.; cf. Etingof–Ostrik) Unitary module categories over Rep 𝒱ℏ(𝔱𝔳(2)) are classifjed by certain weighted graphs

• Rep 𝒱ℏ(𝔱𝔳(2)): universal rigid unitary category generated by

U1/2 = (πnat, ℂ2) (Temperley–Lieb category)

• weighted graph describing how U1/2 and duality morphism

R∶ U0 → U1/2 ⊗ U1/2

• Podleś spheres correspond to the graphs

⋯ •

m−1

• m
• m+1 ⋯

qx+m+q−x−m qx+m−1+q−x−m+1 qx+m−1+q−x−m+1 qx+m+q−x−m qx+m+1+q−x−m−1 qx+m+q−x−m qx+m+q−x−m qx+m+1+q−x−m−1

⇝ brute force classifjcation

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 21 / 23

Quantization tying together

Getting relative co-Hochschild cohomology

For general (U, K): better to work over formal Laurent series ℂ[h−1, h

• work over quasi-Hopf algebra (𝒱(𝔳), Φ), quasi-coactions

(𝒱(𝔩), Ψ′)

• degree-wise comparison of associators Ψ, Ψ′ for same Φ

⇝ difference is a 2-cocycle in the co-Hochschild complex 𝒱(𝔩)𝔩 → (𝒱(𝔩) ⊗ 𝒱(𝔳))𝔩 → (𝒱(𝔩) ⊗ 𝒱(𝔳)⊗2)𝔩 → ⋯ ∼qis (Symc

ℂ(𝔩) ⊗tw Ω(Symc ℂ(𝔳)))𝔩 ∼qis (⋀∗ ℂ(𝔳 ⊖ 𝔩))𝔩 (Koszul duality)

Lemma For any symmetric pair (𝔳, 𝔩), (⋀2

ℂ(𝔳 ⊖ 𝔩))𝔩 ≅ H1(𝔩; ℂ) ≅ 𝔸(𝔩) ⊗ ℂ

• U/K irred. ⇒ (𝔳 ⊖ 𝔩)𝔩 = 0 ⇒ (⋀2(𝔳 ⊖ 𝔩))𝔩 ≃ H2(𝔳, 𝔩) up to linear dual
• look at Serre spectral sequence for K → U → U/K

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 22 / 23

Quantization tying together

Classifjcation at formal setting

Theorem (DC.–N.–Y., in prep.; classifjcation for non-Hermitian (𝔳, 𝔩)) all quasi-coaction of (𝒱(𝔳)ℏ, Φ) on 𝒱(𝔩)ℏ (with possibly deformed product) are equivalent non-Hermitian ⇔ 𝔩 semisimple implies (via Whitehead’s lemma):

• H2(𝒱(𝔩); 𝒱(𝔩)) ≅ H2(𝔩; ad𝒱(𝔩)) = 0 ⇒ 𝒱(𝔩)ℏ does not deform
• H1(𝒱(𝔩); M) ≅ H1(𝔩; adM) = 0 ⇒ coaction does not deform

Theorem (DC.–N.–Y., in prep.; classifjcation for Hermitian (𝔳, 𝔩)) Quasi-coaction (𝒱(𝔩)ℏ, ΨKZ ⊳ χ) for χ ∈ 𝔸(𝔩)∗ℏ exhaust the quasi-coactions (Δ, Ψ) by (𝒱(𝔳)ℏ, ΦKZ) with ‘Ψ(1) = 0’ up to equivalence So we should in fact take χ = χ(−1)ℏ−1 + χ(0) + χ(1)ℏ + ⋯ ∈ ℏ−1𝔸(𝔩)∗ℏ?

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 23 / 23

Quantization tying together

Classifjcation at formal setting

Theorem (DC.–N.–Y., in prep.; classifjcation for non-Hermitian (𝔳, 𝔩)) all quasi-coaction of (𝒱(𝔳)ℏ, Φ) on 𝒱(𝔩)ℏ (with possibly deformed product) are equivalent Theorem (DC.–N.–Y., in prep.; classifjcation for Hermitian (𝔳, 𝔩)) Quasi-coaction (𝒱(𝔩)ℏ, ΨKZ ⊳ χ) for χ ∈ 𝔸(𝔩)∗ℏ exhaust the quasi-coactions (Δ, Ψ) by (𝒱(𝔳)ℏ, ΦKZ) with ‘Ψ(1) = 0’ up to equivalence Need to check ΨKZ ⇝ ΨKZ ⊳ χ can ‘move’ cohomology class in H2(𝒱(𝔩)𝔩 → (𝒱(𝔩) ⊗ 𝒱(𝔳))𝔩 → (𝒱(𝔩) ⊗ 𝒱(𝔳)⊗2)𝔩 → ⋯)

• (𝒱(𝔩) ⊗ 𝒱(𝔳)⊗∗)𝔩 ∼qis (Symc

ℂ(𝔳 ⊖ 𝔩))𝔩 ≅ D∗ pol(U/K)U ∼qis T∗ pol(U/K)U

• pair T2

pol(U/K)U with the canonical 2-form on U/K ≃ U.μ for μ ∈ 𝔳∗

• ribbon σ-braid η = eπ√−1ℏ(2t𝔩

01+2(χ⊗id)(t)1+C𝔩 1) is a complete invariant

Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 23 / 23