Universal K-matrices for quantum symmetric pairs Martina Balagovi - PowerPoint PPT Presentation

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Universal K-matrices for quantum symmetric pairs Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Quantum groups and their analysis, Oslo, August 2019 Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices If you like: ...then you should also like: 1. quantum enveloping algebras 1. quantum symmetric pairs 2. R matrices 2. K matrices 3. the quantum Yang Baxter 3. the reflection equation equation 4. braided module categories 4. braided tensor categories [Balagovi´ c, Kolb, The bar involution for quantum symmetric pairs, Represent. Theory 19 (2015), 186-210 ] [Balagovi´ c, Kolb, Universal K-matrix for quantum symmetric pairs, Journal f¨ ur die reine und angewandte Mathematik 747 (2019), 299–353 ] [Kolb, Braided module categories via quantum symmetric pairs ] Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices ◮ Particle on a line � V ◮ Two particles � V ⊗ W ◮ Scattering: ∼ � c V , W : V ⊗ W − → W ⊗ V ◮ Quantum Yang Baxter equation: � ( c W , U ⊗ 1) (1 ⊗ c V , U ) ( c V , W ⊗ 1) = = (1 ⊗ c V , W ) ( c V , U ⊗ 1) (1 ⊗ c W , U ) Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Braided tensor categories ◮ we choose V from a tensor category V ∼ ◮ commutativity constraint c V , W : V ⊗ W − → W ⊗ W ◮ QYBE = the action of the braid group of type on V ⊗ n Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Braided tensor categories ◮ we choose V from a tensor category V ∼ ◮ commutativity constraint c V , W : V ⊗ W − → W ⊗ W ◮ QYBE = the action of the braid group of type on V ⊗ n ◮ hexagon axiom (similar for c V , W ⊗ U ): c V ⊗ W , U = ( c V , U ⊗ 1) ◦ (1 ⊗ c W , U ) Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Quasitriangular Hopf algebras ◮ Hopf algebra H , V (some nice) category of representations ◮ quasitraingular = exists R ∈ H ⊗ H , ˇ R = flip ◦ R c V , W = ˇ R| V ⊗ W : V ⊗ W → W ⊗ V R ∆( a ) = ∆( a ) ˇ ˇ R ◮ The hexagon axiom becomes: (∆ ⊗ 1)( R ) = R 13 R 23 (1 ⊗ ∆)( R ) = R 13 R 12 ◮ QYBE R 12 R 13 R 23 = R 23 R 13 R 12 Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Quantum enveloping algebra ◮ g , U q g , O int ◮ The construction of the R-matrix [Lusztig]: Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Quantum enveloping algebra ◮ g , U q g , O int ◮ The construction of the R-matrix [Lusztig]: ◮ Define the bar involution on U q g : K i �→ K − 1 q �→ q − 1 E i �→ E i , F i �→ F i , , i ◮ Find the quasi R-matrix R 0 ∈ U q n − ⊗ U q n + such that R 0 ∆( a ) = ∆( a ) R 0 ◮ Set R = R 0 · q − H ⊗ H , R = R 0 ◦ q − H ⊗ H ◦ flip ˇ ◮ Prove (∆ ⊗ 1)( R ) = . . . (1 ⊗ ∆)( R ) = . . . ◮ ⇒ QYBE ◮ [Reshethikhin-Turaev] Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Reflection equation ◮ particle on a line + a wall: � t V : V → V ◮ Reflection Equation: c W , V ( t W ⊗ 1) c V , W ( t V ⊗ 1) = ( t V ⊗ 1) c W , V ( t W ⊗ 1) c V , W Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices ◮ braids with a fixed pole: c W , V ( t W ⊗ 1) c V , W ( t V ⊗ 1) = ( t V ⊗ 1) c W , V ( t W ⊗ 1) c V , W ◮ Naturality condition in ⊗ : t V ⊗ W = ( t V ⊗ 1) c W , V ( t W ⊗ 1) c V , W ◮ Naturality condition ⇒ RE Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Braided module categories ◮ V monoidal category, M module category ( ⊠ : M × V → M ) ◮ e M , V : M ⊠ V → M ⊠ V ◮ e M ⊠ V , W = ( id M ⊠ c V , W )( e M , W ⊠ id V )( id M ⊠ c W , V ) ◮ e M , V ⊗ W = ( id M ⊠ c W , V )( e M , W ⊠ id V )( id M ⊠ c V , W )( e M , V ⊠ id W ) Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Braided module categories ◮ V monoidal category, M module category ( ⊠ : M × V → M ) ◮ e M , V : M ⊠ V → M ⊠ V ◮ e M ⊠ V , W = ( id M ⊠ c V , W )( e M , W ⊠ id V )( id M ⊠ c W , V ) ◮ e M , V ⊗ W = ( id M ⊠ c W , V )( e M , W ⊠ id V )( id M ⊠ c V , W )( e M , V ⊠ id W ) ◮ Recover t V = e Triv , V ◮ Representation of the braid group of type B on M ⊠ V n [Kolb, Braided module categories via quantum symmetric pairs ] [Brochier, Cyclotomic associators and finite type invariants for tangles in the solid torus, Algebraic and Geometric Topology, 2013. ] Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Quasitriangular comodule algebras ◮ H quasitriangular Hopf algebra, B algebra, ∆ B : B → B ⊗ H ◮ V = Rep ( H ), M = Rep ( B ), ⊠ : M × V → M ◮ Want: element K ∈ B ⊗ H , e M , V = K| M ⊠ V ◮ Conditions: ◮ K ∆ B ( b ) = ∆ B ( b ) K ◮ (∆ B ⊗ id )( K ) = R 32 K 13 R 23 ◮ ( id ⊗ ∆)( K ) = R 32 K 13 R 23 K 12 Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Quasitriangular comodule algebras ◮ H quasitriangular Hopf algebra, B algebra, ∆ B : B → B ⊗ H ◮ V = Rep ( H ), M = Rep ( B ), ⊠ : M × V → M ◮ Want: element K ∈ B ⊗ H , e M , V = K| M ⊠ V ◮ Conditions: ◮ K ∆ B ( b ) = ∆ B ( b ) K ◮ (∆ B ⊗ id )( K ) = R 32 K 13 R 23 ◮ ( id ⊗ ∆)( K ) = R 32 K 13 R 23 K 12 ◮ K = ( ε ⊗ id )( K ) will then satisfy the reflection equation: ( K ⊗ 1) ˇ R ( K ⊗ 1) ˇ R = ˇ R ( K ⊗ 1) ˇ R ( K ⊗ 1) Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Main point: Theorem Quantum symmetric pairs provide examples of this structure. If you like: ...then you should also like: 1. quantum groups 1. quantum symmetric pairs 2. R matrices 2. K matrices 3. the quantum Yang Baxter 3. the reflection equation equation 4. braided module categories 4. braided tensor categories Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Classical symmetric pairs: ◮ g finite dimensional simple Lie algebra ◮ θ : g → g an involution ◮ k = g θ fixed points ◮ ( g , k ) is a symmetric pair Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Satake diagrams Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs

Motto U q g and universal R-matrices Quantum symmetric pairs Universal K-matrices Quantum symmetric pairs: ◮ ( g , k ) a symmetric pair ◮ ( U q g , U q k ) not compatible deformations Martina Balagovi´ c (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Universal K-matrices for quantum symmetric pairs