Unexpected facets of the Yang-Baxter equation

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1 Unexpected facets of the Yang-Baxter equation Victoria LEBED University of Nantes Utrecht, September 29, 2015

2 1 Yang-Baxter equation Avectorspace V(oranobjectinany monoidal category) σ: V 2 V 2 Yang-Baxter equation(ybe): σ 1 σ 2 σ 1 =σ 2 σ 1 σ 2 : V 3 V 3 where σ i =Id i 1 V σ Id V. Amap σsatisfyingybeisabraiding.

3 1 Yang-Baxter equation Avectorspace V(oranobjectinany monoidal category) σ: V 2 V 2 Yang-Baxter equation(ybe): σ V V V V σ 1 σ 2 σ 1 =σ 2 σ 1 σ 2 : V 3 V 3 where σ i =Id i 1 V σ Id V. Amap σsatisfyingybeisabraiding. = (Reidemeister III)

4 1 Yang-Baxter equation Avectorspace V(oranobjectinany monoidal category) σ: V 2 V 2 Yang-Baxter equation(ybe): σ V V V V σ 1 σ 2 σ 1 =σ 2 σ 1 σ 2 : V 3 V 3 where σ i =Id i 1 V σ Id V. Amap σsatisfyingybeisabraiding. (Reidemeister III) History: Particle physics: factorization cond. for the dispersion matrix in the 1-dim. n-body problem(mcguire, Yang, 60 ). = collisions Statistical mechanics: partition function for exactly solvable lattice models(baxter, 70 ).

5 History: 1 Yang-Baxter equation Particle physics: factorization cond. for the dispersion matrix in the 1-dim. n-body problem(mcguire, Yang, 60 ). Statistical mechanics: partition function for exactly solvable lattice models(baxter, 70 ), quantum inverse scattering method for completely integrable systems (Faddeevetal.,1979). Field theories: factorizable S-matrices in 2-dim. quantum field theory(zamolodchikov, 1979), conformal field theory. Quantumgroups(Drinfel d,80 ). C algebras(woronowicz,80 ). Low-dimensional topology....

6 2 AhomologytheoryfortheYBE Aim: Unify homology theories for basic algebraic structures.

7 2 AhomologytheoryfortheYBE Aim: Unify homology theories for basic algebraic structures. Tools: Graphical calculus.

8 2 AhomologytheoryfortheYBE Aim: Unify homology theories for basic algebraic structures. Tools: Graphical calculus. Ingredients: A braided vector space (V, σ); aleftbraided V-module: (M,ρ: M V M)s.t. ρ ρ 1 =ρ ρ 1 σ 2 : M V V M ρ ρ = ρ ρ σ M V V M V V arightbraided V-module (N,λ: V N N).

9 2 AhomologytheoryfortheYBE Aim: Unify homology theories for basic algebraic structures. Tools: Graphical calculus. Ingredients: A braided vector space (V, σ); aleftbraided V-module: (M,ρ: M V M)s.t. ρ ρ 1 =ρ ρ 1 σ 2 : M V V M ρ ρ = ρ ρ σ M V V M V V arightbraided V-module (N,λ: V N N). Theorem(L.2013): M T(V) Ncarriesafamilyof differentials δ (α,β) =α δ+βδ, α,β k. (I.e., δ (α,β) δ (α,β) =0.)

10 2 AhomologytheoryfortheYBE Theorem(L.2013): M T(V) Ncarriesafamilyof differentials δ (α,β) =α δ+βδ, α,β k. δ= ( 1) i 1 ρ M V... V σ σ N δ = ( 1) i 1 M V... V N σ σ λ 1... i...n 1... i... n

11 2 AhomologytheoryfortheYBE Theorem(L.2013): M T(V) Ncarriesafamilyof differentials δ (α,β) =α δ+βδ, α,β k. δ= ( 1) i 1 ρ M V... V σ σ N δ = ( 1) i 1 M V... V σ σ N λ Proof: 1... i...n 1... i... n YBE = br. mod. = & sign = ( 1) #cross.

12 2 AhomologytheoryfortheYBE Theorem(L.2013): M T(V) Ncarriesafamilyof differentials δ (α,β) =α δ+βδ, α,β k. δ= ( 1) i ρ M V... V σ σ N δ = ( 1) i M V... V σ σ N λ 1... i...n 1... i... n Remarks: Functoriality. Interpretation in terms of quantum shuffles(rosso, 1995). Duality acohomologytheory. Pre-cubical structure.

13 2 AhomologytheoryfortheYBE Theorem(L.2013): M T(V) Ncarriesafamilyof differentials δ (α,β) =α δ+βδ, α,β k. Remarks: Functoriality. Interpretation in terms of quantum shuffles(rosso, 1995). Duality acohomologytheory. Pre-cubical structure. Degeneracies: Braidedcoalgebra: br. v. sp. (V,σ) & : V V Vs.t. = = = (Cf. Reidemeister moves for knotted 3-valent graphs!)

14 2 AhomologytheoryfortheYBE Theorem(L.2013): M T(V) Ncarriesafamilyof differentials δ (α,β) =α δ+βδ, α,β k. Remarks: Functoriality. Interpretation in terms of quantum shuffles(rosso, 1995). Duality acohomologytheory. Pre-cubical structure. Degeneracies: Braidedcoalgebra: br. v. sp. (V,σ) & : V V Vs.t. = = = (Cf. Reidemeister moves for knotted 3-valent graphs!) Theorem(L.2013): All δ (α,β) restrictto i Im( i). normalization

15 3 Alg. structures via braidings A Associative algebras

16 3 Alg. structures via braidings A Associative algebras (V,,1)s.t. Associativity: (u v) w=u (v w) Unit axiom: 1 v=v 1=v = = =

17 3 Alg. structures via braidings A Associative algebras (V,,1)s.t. Associativity: (u v) w=u (v w) = Unit axiom: 1 v=v 1=v Associativitybraiding σ Ass : v w 1 v w = = YBEfor σ Ass (un. ax.) associativity for

18 3 Alg. structures via braidings A Associative algebras (V,,1)s.t. Associativity: (u v) w=u (v w) = Unit axiom: 1 v=v 1=v Associativitybraiding σ Ass : v w 1 v w = = YBEfor σ Ass (un. ax.) associativity for Proof: =

19 3 Alg. structures via braidings A Associative algebras (V,,1) σ Ass : v w 1 v w YBEfor σ Ass (un.ax.) associativity for Afullyfaithfulfunctor Alg Br.

20 3 Alg. structures via braidings A Associative algebras (V,,1) σ Ass : v w 1 v w YBEfor σ Ass (un.ax.) associativity for Afullyfaithfulfunctor Alg Br. Duality: Alg Br coalg.

21 3 Alg. structures via braidings A Associative algebras (V,,1) σ Ass : v w 1 v w YBEfor σ Ass (un.ax.) associativity for Afullyfaithfulfunctor Alg Br. Duality: Alg Br coalg. σ Ass σ Ass =σ Ass = highlynon-invertible. Braidedmodulesfor (V,σ Ass ) modulesfor (V,,1).

22 3 Alg. structures via braidings A Associative algebras (V,,1) σ Ass : v w 1 v w YBEfor σ Ass (un.ax.) associativity for Afullyfaithfulfunctor Alg Br. Duality: Alg Br coalg. σ Ass σ Ass =σ Ass = highlynon-invertible. Braidedmodulesfor (V,σ Ass ) modulesfor (V,,1). Ass : v 1 v braidedcoalgebra.

23 3 Alg. structures via braidings A Associative algebras (V,,1) σ Ass : v w 1 v w YBEfor σ Ass (un.ax.) associativity for Afullyfaithfulfunctor Alg Br. Duality: Alg Br coalg. σ Ass σ Ass =σ Ass = highlynon-invertible. Braidedmodulesfor (V,σ Ass ) modulesfor (V,,1). Ass : v 1 v braidedcoalgebra. Braidedhomologiesfor (V,σ Ass )include bar; Hochschild; group.

24 3 Alg. structures via braidings B Leibniz algebras (V,[],1)s.t. Leibnizidentity: [v,[w,u]]=[[v,w],u] [[v,u],w]; Lieunitaxiom: [1,v]=[v,1]=0. (Bloh 1965, Loday& Cuvier 1991: a non-commutative generalization of Lie algebras.)

25 3 Alg. structures via braidings B Leibniz algebras (V,[],1)s.t. Leibnizidentity: [v,[w,u]]=[[v,w],u] [[v,u],w]; Lieunitaxiom: [1,v]=[v,1]=0. (Bloh 1965, Loday& Cuvier 1991: a non-commutative generalization of Lie algebras.) Leibnizbraiding σ Lei : v w w v+1 [v,w] YBEfor σ Lei (Lie un. ax.) Leibnizidentityfor [] Afullyfaithfulfunctor Lei Br.

26 3 Alg. structures via braidings B Leibniz algebras (V,[],1)s.t. Leibnizidentity: [v,[w,u]]=[[v,w],u] [[v,u],w]; Lieunitaxiom: [1,v]=[v,1]=0. (Bloh 1965, Loday& Cuvier 1991: a non-commutative generalization of Lie algebras.) Leibnizbraiding σ Lei : v w w v+1 [v,w] YBEfor σ Lei (Lie un. ax.) Leibnizidentityfor [] Afullyfaithfulfunctor Lei Br. σ Lei isinvertible. Braidedmod. for (V,σ Lei ) anti-symmetricleibniz mod. for (V,[],1).

27 3 Alg. structures via braidings B Leibniz algebras (V,[],1)s.t. [v,[w,u]]=[[v,w],u] [[v,u],w], [1,v]=[v,1]=0. Leibnizbraiding σ Lei : v w w v+1 [v,w] Supposethat V =V k1, [V,V ] V. Then Lei : v 1 v+v 1, v V ; turns (V,σ Lei )intoa braided coalgebra.

28 3 Alg. structures via braidings B Leibniz algebras (V,[],1)s.t. [v,[w,u]]=[[v,w],u] [[v,u],w], [1,v]=[v,1]=0. Leibnizbraiding σ Lei : v w w v+1 [v,w] Supposethat V =V k1, [V,V ] V. Then Lei : v 1 v+v 1, v V ; turns (V,σ Lei )intoa braided coalgebra. Braidedhomologiesfor (V,σ Lei )includeleibnizhomology. (M T(V),d Lei ) antisymm. Cuvier-Loday Lie V (M Λ(V),d CE ) Chevalley-Eilenberg

29 3 Alg. structures via braidings B Leibniz algebras (V,[],1)s.t. [v,[w,u]]=[[v,w],u] [[v,u],w], [1,v]=[v,1]=0. Leibnizbraiding σ Lei : v w w v+1 [v,w] Supposethat V =V k1, [V,V ] V. Then Lei : v 1 v+v 1, v V ; turns (V,σ Lei )intoa braided coalgebra. Braidedhomologiesfor (V,σ Lei )includeleibnizhomology. Lei Lie V (M T(V),d Lei ) antisymm. antisymm. Cuvier-Loday V (M Λ(V),d CE ) Chevalley-Eilenberg

30 3 Alg. structures via braidings B Leibniz algebras (V,[],1)s.t. [v,[w,u]]=[[v,w],u] [[v,u],w], [1,v]=[v,1]=0. Leibnizbraiding σ Lei : v w w v+1 [v,w] Supposethat V =V k1, [V,V ] V. Then Lei : v 1 v+v 1, v V ; turns (V,σ Lei )intoa braided coalgebra. Braidedhomologiesfor (V,σ Lei )includeleibnizhomology. Lei Lie V (M T(V),d Lei ) antisymm. antisymm. Cuvier-Loday V (M Λ(V),d CE ) Chevalley-Eilenberg Explains the choice of the lift of the Jacobi identity.

31 3 Alg. structures via braidings C Self-distributive structures

32 3 Alg. structures via braidings C Self-distributive structures b a b colorings by (S, ) a b SD braiding σ SD : (a,b) (b,a b) c b c (a b) c a b a b c RIII c b c (a c) (b c) a c a b c YBE RIII (a b) c=(a c) (b c) (SD)

33 3 Alg. structures via braidings C Self-distributive structures b a b colorings by (S, ) a b SD braiding σ SD : (a,b) (b,a b) c b c (a b) c a b a b c RIII c b c (a c) (b c) a c a b c YBE RIII (a b) c=(a c) (b c) (SD)

34 3 Alg. structures via braidings C Self-distributive structures b a b colorings by (S, ) a b SD braiding σ SD : (a,b) (b,a b) c b c (a b) c a b a b c RIII c b c (a c) (b c) a c a b c YBE RIII (a b) c=(a c) (b c) RII a a bbijective (Inv) σ 1 SD RI a a=a (SD) (Idem) RII RI

35 3 Alg. structures via braidings C Self-distributive structures b a b colorings by (S, ) a b SD braiding σ SD : (a,b) (b,a b) c b c (a b) c a b a b c RIII c b c (a c) (b c) a c a b c YBE RIII (a b) c=(a c) (b c) (SD) σ 1 SD RII a a bbijective (Inv) = RI a a=a (Idem) RII RI SD : a (a,a)

36 3 Alg. structures via braidings C Self-distributive structures b a b colorings by (S, ) Joyce, Matveev 1982: knotinvariants colorings quandle a b pos. braids RIII (a b) c=(a c) (b c) shelf braids RII a a bbijective rack knots RI a a=a quandle RII RI

37 3 Alg. structures via braidings C Self-distributive structures b a b colorings by (S, ) Joyce, Matveev 1982: knotinvariants colorings quandle a b pos. braids RIII (a b) c=(a c) (b c) shelf braids RII a a bbijective rack knots RI a a=a quandle Ex.: Conjugationquandles: (group G,g h=h 1 gh) coloring rule Wirtinger presentation rule, colorings Rep(π 1 (R 3 \K),G).

38 3 Alg. structures via braidings C Self-distributive structures b a b colorings by (S, ) Joyce, Matveev 1982: knotinvariants colorings quandle a b pos. braids RIII (a b) c=(a c) (b c) shelf braids RII a a bbijective rack knots RI a a=a quandle Ex.: Conjugationquandles: (group G,g h=h 1 gh) coloring rule Wirtinger presentation rule, colorings Rep(π 1 (R 3 \K),G). Ex.: Dihedralquandles: (Z n,a b=2b a) colorings n-fox colorings.

39 3 Alg. structures via braidings C Self-distributive structures b a b colorings by (S, ) Joyce, Matveev 1982: knotinvariants colorings quandle a b pos. braids RIII (a b) c=(a c) (b c) shelf braids RII a a bbijective rack knots RI a a=a quandle Ex.: Conjugationquandles: (group G,g h=h 1 gh) coloring rule Wirtinger presentation rule, colorings Rep(π 1 (R 3 \K),G). Ex.: Dihedralquandles: (Z n,a b=2b a) colorings n-fox colorings. n=3

40 3 Alg. structures via braidings C Self-distributive structures diagrams: D R-move D colorings: C C coloringsets: Col S (D) bij. Col S (D ) countinginvariants: #Col S (D) = #Col S (D )

41 3 Alg. structures via braidings C Self-distributive structures diagrams: D R-move D colorings: C C coloringsets: Col S (D) bij. Col S (D ) countinginvariants: #Col S (D) = #Col S (D ) Question: Extract more information? Idea: Some weight ωs.t. ω(c)=ω(c ) = {ω(c) C Col S (D)}={ω(C ) C Col S (D )}.

42 3 Alg. structures via braidings C Self-distributive structures diagrams: D R-move D colorings: C C coloringsets: Col S (D) bij. Col S (D ) countinginvariants: #Col S (D) = #Col S (D ) Question: Extract more information? Idea: Some weight ωs.t. ω(c)=ω(c ) = {ω(c) C Col S (D)}={ω(C ) C Col S (D )}. Answer: quandle cocycle invariants(carter-jelsovsky- Kamada-Langford-Saito 1999). φ:s S A Boltzmann weight: ω φ (C) = a b ±φ(a, b)

43 3 Alg. structures via braidings C Self-distributive structures Rack& quandle cohomology theories (Fenn-Rourke-Sanderson 1995, Carter et al. 1999) Motivation: {ω φ (C) C Col S (D)}yieldsabraid/knotinvariantwhen φ isarack/quandle 2-cocycle; this invariant is trivial when φ is a coboundary;

44 3 Alg. structures via braidings C Self-distributive structures Rack& quandle cohomology theories (Fenn-Rourke-Sanderson 1995, Carter et al. 1999) Motivation: {ω φ (C) C Col S (D)}yieldsabraid/knotinvariantwhen φ isarack/quandle 2-cocycle; this invariant is trivial when φ is a coboundary; adding coefficients allows to color diagram regions: a rack module M m m a ω φ (C) = ±φ(m,a,b) am b everythinggeneralizesto K n 1 R n+1.

45 3 Alg. structures via braidings C Self-distributive structures Rack& quandle cohomology theories (Fenn-Rourke-Sanderson 1995, Carter et al. 1999) Motivation: {ω φ (C) C Col S (D)}yieldsabraid/knotinvariantwhen φ isarack/quandle 2-cocycle; this invariant is trivial when φ is a coboundary; adding coefficients allows to color diagram regions: a rack module M m m a ω φ (C) = ±φ(m,a,b) am b everythinggeneralizesto K n 1 R n+1. Inspiration: (Co-)homology theories for associativity. Question(Przytycki): Explain the parallels between the associative and the SD worlds?

46 3 Alg. structures via braidings C Self-distributive structures Rack& quandle cohomology theories (Fenn-Rourke-Sanderson 1995, Carter et al. 1999) Motivation: {ω φ (C) C Col S (D)}yieldsabraid/knotinvariantwhen φ isarack/quandle 2-cocycle; this invariant is trivial when φ is a coboundary; adding coefficients allows to color diagram regions: a rack module M m m a ω φ (C) = ±φ(m,a,b) am b everythinggeneralizesto K n 1 R n+1. Inspiration: (Co-)homology theories for associativity. Question(Przytycki): Explain the parallels between the associative and the SD worlds? Answer: Common braided interpretation.

47 3 Alg. structures via braidings C Self-distributive structures Shelf (S, ) σ SD : (a,b) (b,a b) YBEfor σ SD SDfor Afullyfaithfulfunctor Shelf Br. σ SD isinvertible (S, )isarack. Braidedmodulesfor (V,σ SD ) rackmodulesfor (S, ). SD : a (a,a) weakbraidedcoalgebraif (S, )isa quandle. Braidedhomologiesfor (V,σ SD )includerack,quandle, and other SD homologies.

48 4 Multi-component braidings Question: How to treat more complicated structures?

49 4 Multi-component braidings Question: How to treat more complicated structures? Braidedsystem: V 1,V 2,...,V r and σ i,j :V i V j V j V i, i j, satisfying the colored Yang-Baxter equation(cybe): σ j,k 1 σi,k 2 σi,j 1 =σi,j 2 σi,k 1 σj,k 2 V i V j V k V k V j V i,i j k V i V j V k V i V j V k Thecollection (σ i,j )satisfyingcybeisamulti-braiding.

50 4 Multi-component braidings Question: How to treat more complicated structures? Braidedsystem: V 1,V 2,...,V r and σ i,j :V i V j V j V i, i j, satisfying the colored Yang-Baxter equation(cybe): σ j,k 1 σi,k 2 σi,j 1 =σi,j 2 σi,k 1 σj,k 2 V i V j V k V k V j V i,i j k V i V j V k V i V j V k Thecollection (σ i,j )satisfyingcybeisamulti-braiding. Left braided V-module: (M,(ρ i : M V i M))s.t. ρ j ρ i ρ = ρ j im Vi Vj σ i,j i j M V i V j

51 4 Multi-component braidings Question: How to treat more complicated structures? Braidedsystem: V 1,V 2,...,V r and σ i,j :V i V j V j V i, i j, satisfying the colored Yang-Baxter equation(cybe): σ j,k 1 σi,k 2 σi,j 1 =σi,j 2 σi,k 1 σj,k 2 V i V j V k V k V j V i,i j k V i V j V k V i V j V k Thecollection (σ i,j )satisfyingcybeisamulti-braiding. Left braided V-module: (M,(ρ i : M V i M))s.t. ρ j ρ i ρ = ρ j im Vi Vj σ i,j i j M V i V j Theorem(L.,2013): M T(V 1 ) T(V r ) Ncarriesa familyofdifferentials δ (α,β) =α δ+βδ, α,β k.

52 4 Multi-component braidings Finite-dim. bialgebra H (H,H ;σ H,H =σ r Ass (H),σ H,H =σ Ass(H ),σ H,H =σ YD ) σ H,H = σ H,H = σ H,H = ev h l l (1),h (2) l (2) h (1) YBEon H H H (un. ax.) bialgebra compatibility

53 4 Multi-component braidings Finite-dim. bialgebra H (H,H ;σ H,H =σ r Ass (H),σ H,H =σ Ass(H ),σ H,H =σ YD ) σ H,H = σ H,H = σ H,H = ev h l l (1),h (2) l (2) h (1) YBEon H H H (un. ax.) bialgebra compatibility Afullyfaithfulfunctor Bialg 2 BrSyst.

54 4 Multi-component braidings Finite-dim. bialgebra H (H,H ;σ H,H =σ r Ass (H),σ H,H =σ Ass(H ),σ H,H =σ YD ) σ H,H = σ H,H = σ H,H = ev h l l (1),h (2) l (2) h (1) YBEon H H H (un. ax.) bialgebra compatibility Afullyfaithfulfunctor Bialg 2 BrSyst. σ H,H isinvertible HisaHopfalgebra.

55 4 Multi-component braidings Finite-dim. bialgebra H (H,H ;σ H,H =σ r Ass (H),σ H,H =σ Ass(H ),σ H,H =σ YD ) σ H,H = σ H,H = σ H,H = ev h l l (1),h (2) l (2) h (1) YBEon H H H (un. ax.) bialgebra compatibility Afullyfaithfulfunctor Bialg 2 BrSyst. σ H,H isinvertible HisaHopfalgebra. Braided modules Hopf modules over H.

56 4 Multi-component braidings Finite-dim. bialgebra H (H,H ;σ H,H =σ r Ass (H),σ H,H =σ Ass(H ),σ H,H =σ YD ) σ H,H = σ H,H = σ H,H = ev h l l (1),h (2) l (2) h (1) YBEon H H H (un. ax.) bialgebra compatibility Afullyfaithfulfunctor Bialg 2 BrSyst. σ H,H isinvertible HisaHopfalgebra. Braided modules Hopf modules over H. Braided homologies include Gerstenhaber-Schack; Panaite-Ştefan.

57 4 Multi-component braidings Finite-dim. bialgebra H (H,H op,h,(h ) op ;...). Afullyfaithfulfunctor Bialg 4 BrSyst.

58 4 Multi-component braidings Finite-dim. bialgebra H (H,H op,h,(h ) op ;...). Afullyfaithfulfunctor Bialg 4 BrSyst. Braided modules Hopf bimodules over H.

59 4 Multi-component braidings Finite-dim. bialgebra H (H,H op,h,(h ) op ;...). Afullyfaithfulfunctor Bialg 4 BrSyst. Braided modules Hopf bimodules over H. Application: Hopf bimodules are modules over the Heisenberg double H (H)=H H Cibils-Rosso 1998: Hopf bimodules are modules over X (H)=(H H op ) (H (H ) op )

60 4 Multi-component braidings Finite-dim. bialgebra H (H,H op,h,(h ) op ;...). Afullyfaithfulfunctor Bialg 4 BrSyst. Braided modules Hopf bimodules over H. Application: Hopf bimodules are modules over the Heisenberg double H (H)=H H Cibils-Rosso 1998: Hopf bimodules are modules over X (H)=(H H op ) (H (H ) op ) Panaite 2002: Hopf bimodules are modules over... Y (H)=H #(H op H)#(H ) op & Z(H)=(H (H ) op ) (H op H)

61 4 Multi-component braidings Finite-dim. bialgebra H (H,H op,h,(h ) op ;...). Afullyfaithfulfunctor Bialg 4 BrSyst. Braided modules Hopf bimodules over H. Application: Hopf bimodules are modules over the Heisenberg double H (H)=H H Cibils-Rosso 1998: Hopf bimodules are modules over X (H)=(H H op ) (H (H ) op ) Panaite 2002: Hopf bimodules are modules over... Y (H)=H #(H op H)#(H ) op & Z(H)=(H (H ) op ) (H op H) Theorem(L. 2013): Hopf bimodules are modules over 4! = 24 pairwise isomorphic algebras.

62 4 Multi-component braidings Finite-dim. bialgebra H (H,H op,h,(h ) op ;...). Afullyfaithfulfunctor Bialg 4 BrSyst. Braided modules Hopf bimodules over H. Application: Hopf bimodules are modules over the Heisenberg double H (H)=H H Cibils-Rosso 1998: Hopf bimodules are modules over X (H)=(H H op ) (H (H ) op ) Panaite 2002: Hopf bimodules are modules over... Y (H)=H #(H op H)#(H ) op & Z(H)=(H (H ) op ) (H op H) Theorem(L. 2013): Hopf bimodules are modules over 4! = 24 pairwise isomorphic algebras. Braided homologies include the Ospel-Taillefer theory.

63 5 TheunifyingroleoftheYBE (multi-)braiding algebraic structure rbrsyst Struc (c)ybe the defining relations invertibility algebraic properties braided modules usual modules braided hom. theory usual hom. theories

64 5 TheunifyingroleoftheYBE (multi-)braiding algebraic structure rbrsyst Struc (c)ybe the defining relations invertibility algebraic properties braided modules usual modules braided hom. theory usual hom. theories Other braidable structures: module algebras(yau); Yetter-Drinfel dmodules(panaite-ştefan);

65 5 TheunifyingroleoftheYBE (multi-)braiding algebraic structure rbrsyst Struc (c)ybe the defining relations invertibility algebraic properties braided modules usual modules braided hom. theory usual hom. theories Other braidable structures: module algebras(yau); Yetter-Drinfel dmodules(panaite-ştefan); (non-commutative) Poisson algebras(fresse);

66 5 TheunifyingroleoftheYBE (multi-)braiding algebraic structure rbrsyst Struc (c)ybe the defining relations invertibility algebraic properties braided modules usual modules braided hom. theory usual hom. theories Other braidable structures: module algebras(yau); Yetter-Drinfel dmodules(panaite-ştefan); (non-commutative) Poisson algebras(fresse); multiple conjugation quandles(ishii) knotted handle-bodies

67 6 AbraidedversionofYDmodules Yetter-Drinfel dmoduleoverahopfalgebra H: vector space M; right H-action ρ; right H-coaction δ; compatibility.

68 6 AbraidedversionofYDmodules Yetter-Drinfel dmoduleoverahopfalgebra H: braiding σ vector space M; YD : ρ right H-action ρ; right H-coaction δ; τ compatibility. δ

69 6 AbraidedversionofYDmodules Yetter-Drinfel dmoduleoverahopfalgebra H: braiding σ vector space M; YD : ρ right H-action ρ; right H-coaction δ; τ compatibility. δ This construction yields all invertible f.-d. braidings!

70 M A M A 6 AbraidedversionofYDmodules Yetter-Drinfel dmoduleoverahopfalgebra H: braiding σ vector space M; YD : ρ right H-action ρ; right H-coaction δ; τ compatibility. δ This construction yields all invertible f.-d. braidings! L.-Wagemann2015: YDmoduleoverabr. system (C,A;σ): vector space M; rightaction ρof (A;σ A,A ); rightcoaction δof (C;σ C,C ); compatibility: M C M C δ ρ = ρ δ σ C,A

71 M A M A 6 AbraidedversionofYDmodules Yetter-Drinfel dmoduleoverahopfalgebra H: braiding σ vector space M; YD : ρ right H-action ρ; right H-coaction δ; τ compatibility. δ This construction yields all invertible f.-d. braidings! L.-Wagemann2015: YDmoduleoverabr. system (C,A;σ): vector space M; rightaction ρof (A;σ A,A ); rightcoaction δof (C;σ C,C ); compatibility: M C M C δ ρ = ρ δ σ C,A &a nice π: C A braiding σ GYD : ρ τ π δ

72 6 AbraidedversionofYDmodules YDmoduleoverabraidedsystem (C,A;σ): vector space M; rightaction ρof (A;σ A,A ); &a nice π: C A rightcoaction δof (C;σ C,C ); compatibility: braiding σ GYD : M C M C ρ δ ρ M A = ρ δ M σ C,A Examples: YDmodulesoveraHopfalgebra σ YD ; A τ π δ

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