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 overview 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) Hopf algebra universal R -matrix braided tensor category Knizhnik–Zamolodchikov equation Yang–Baxter 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 overview Next to quantize: symmetric spaces Defjnition (Symmetric space) • Riemannian manifold M • can model ( x 1 , … , x n ) ↦ (− x 1 , … , − x n ) by isometry 1 U = Iso + ( M ) : orientation preserving isometries 2 M ≅ U / Stab ( p ) for any p ∈ M Example ( 2 -sphere S 2 ⊂ ℝ 3 ) • symmetry around ( 1 , 0 , 0 ) : ( x , y , z ) ↦ ( x , − y , − z ) • S 2 ≅ SU ( 2 )/ SO ( 2 ) Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 2 / 23
Introduction overview Spoiler: how it will go Quantization will be ribbon twist braided module category : add refmection operator to Yang–Baxter equation X U “(quasi)particle bouncing off the boundary wall” U : quasiparticle following braid statistics X : quasiparticle on “boundary” X U 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 or better… Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 3 / 23
= Introduction overview Spoiler: how it will go Quantization will be ribbon twist braided module category : add ribbon twist-braid operator to Yang–Baxter equation U X such that X σ ( U ) with braided automorphism σ For irreducible compact symmetric spaces: • purely algebraic quantization: Letzter–Kolb coideals of 𝒱 q (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 { f 1 , f 2 } ∈ C ∞ ( M ) for f i ∈ C ∞ ( M ) : • ( C ∞ ( M ), {⋅, ⋅}) : Lie algebra • Leibniz rule { f 1 , f 2 f 3 } = { f 1 , f 2 } f 3 + { f 1 , f 3 } f 2 Example (Sklyanin bracket on SU ( 2 ) ) C ∞ ( SU ( 2 )) = ⟨ x i , j ∣ 1 ≤ i , j ≤ 2 coordinate functions ⟩ { x ij , x kl } = δ 2 i δ 1 k x 1 j x 2 l − δ 1 i δ 2 k x 2 j x 1 l − δ j 1 δ l 2 x i 2 x k 1 + δ j 2 δ l 1 x i 1 x k 2 Problem (deformation quantization) Can we fjnd associative products ⋆ ℏ on (large subspace of) C ∞ ( M ) 1 such that { f 1 , f 2 } = lim ℏ→ 0 ℏ ( f 1 ⋆ ℏ f 2 − f 2 ⋆ ℏ f 1 ) ? 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 S 2 ) (SU ( 2 ) , Sklyanin bracket) acts on ( S 2 , { x i , x i + 1 } = ( x 1 + t ) x i + 2 ) (index ( x i ) 3 i = 1 : coordinate on ℝ 3 ⊃ S 2 , t ∈ ℝ mod 3 ) Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 5 / 23
ℏ ℏ ℏ Quantization Hopf algebraic Quantum groups Drinfeld–Jimbo deformation of universal enveloping algebra U (𝔳) can be deformed as a Hopf algebra 𝒱 ℏ (𝔳) ; ‘just’ deform coproduct: Δ ℏ ( x ) − fmip Δ ℏ ( x ) = 2 ℏ δ ( x ) + higher order in ℏ for x ∈ 𝔳 Example ( 𝒱 ℏ (𝔱𝔳( 2 )) ) • generators: E ℏ , F ℏ , K ℏ = e π √− 1 ℏ H • relations: K ℏ E ℏ K − 1 = e 2 π √− 1 ℏ E ℏ , K ℏ F ℏ K − 1 = e − 2 π √− 1 ℏ F ℏ , E ℏ F ℏ − F ℏ E ℏ = ( K ℏ − K − 1 ℏ )/( e π √− 1 ℏ − e − π √− 1 ℏ ) • coproduct: Δ ℏ ( K ℏ ) = K ℏ ⊗ K ℏ Δ ℏ ( E ℏ ) = E ℏ ⊗ K ℏ + 1 ⊗ E ℏ , Δ ℏ ( F ℏ ) = F ℏ ⊗ 1 + K − 1 ⊗ F ℏ ℏ = K − 1 Unitary structure (when ℏ ∈ √− 1 ℝ ): K ∗ ℏ = K ℏ , E ∗ ℏ = F ℏ K ℏ , F ∗ ℏ E ℏ 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 𝒫 ℏ ( U ) = ⟨ 𝒱 ℏ (𝔳) → ℂ , T ↦ φ ( T ξ ) ∣ ξ ∈ V ∶ (admissible) fjnite dimensional 𝒱 ℏ (𝔳) -module , φ ∈ V ∗ ⟩ 𝒫 ℏ ( U ) is a Hopf algebra by: • linear combination from direct sum modules • product, coproduct by transpose of Δ ℏ , product in 𝒱 ℏ (𝔳) • unitary structure from unitary structure and antipode of 𝒱 ℏ (𝔳) ⟨ K i ,ℏ ∣ i ∶ vertex of Dynkin diagram ⟩ ⊂ 𝒱 ℏ (𝔳) ⇝ highest weight theory • ‘same classifjcation’ of irreducible fjnite dimensional modules • 𝒫 ℏ ( U ) ≅ ⨁ π ∶ Irr 𝒱 ℏ (𝔳) V ∗ π ⊗ V π : ‘same’ coalgebra as 𝒫 ( U ) ⇝ Coalgebra identifjcation 𝒫 ℏ ( 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 U ℏ -algebras) ∃ deformation quantization 𝒫 ℏ ( M ) as 𝒫 ℏ ( U ) -comodule algebra? Example (Podleś spheres, c ≥ 0 ) 𝒫 ( S 2 ℏ, c ) = ⟨ A = A ∗ , B , B ∗ ∣ BA = q 2 AB , B ∗ B = A − A 2 + c , BB ∗ = q 2 − q 4 + c ⟩ If c = c (ℏ) depends on ℏ , 𝒫 ( S 2 ℏ, c (ℏ) ) is a deformation quantization for 1 { x i , x i + 1 } = x 1 + √ 4 c ( 0 )+ 1 x i + 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 𝒱 ℏ (𝔳) & X ∈ ↔ 𝒫 ℏ ( 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 + 2 ℏ r + ⋯ ∈ 𝒱 ℏ (𝔳) ⊗ 𝒱 ℏ (𝔳) ℛ Δ ℏ ( x ) ℛ − 1 = fmip Δ ℏ ( x ) (Δ ℏ ⊗ id )( ℛ ) = ℛ 13 ℛ 23 = ∑ i , j x i ⊗ x j ⊗ y i y j for ℛ = ∑ i x i ⊗ y i , ( id ⊗ Δ ℏ )( ℛ ) = ℛ 12 ℛ 13 = ∑ i , j x i x j ⊗ y i ⊗ y j β ( ξ ⊗ η ) = fmip ( π ⊗ π ′ )( ℛ )( ξ ⊗ η ) for ξ ∈ V π , η ∈ V π ′ : • is an intertwiner of (fjnite dimensional) 𝒱 ℏ (𝔳) -modules • satisfjes the Yang-Baxter equation The category of fjnite-dimensional 𝒱 ℏ (𝔳) -modules is: 1 a tensor category, with tensor product module by Δ ℏ 2 with braiding by the action of β Makoto Yamashita (Oslo) Quantum Symmetric Spaces Białowieża, 2019 July 10 / 23
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