Unexpected facets of the Yang Baxter equation

Transcription

1 Unexpected facets of the Yang Baxter equation Victoria LEBED (ab)c = a(bc) = z 1 (y 1 xy)z= (z 1 y 1 z)(z 1 xz)(z 1 yz) TCD, Octobre 2016

2 1 Yang Baxter equation: basics Data: vectorspacev, ff:v 2 V 2. Yang-Baxter equation(ybe) ff 1 ff 2 ff 1 =ff 2 ff 1 ff 2 :V 3 V 3 ff 1 =ff Id V,ff 2 =Id V ff

3 1 Yang Baxter equation: basics Data: vectorspacev, ff:v 2 V 2. Yang-Baxter equation(ybe) ff 1 ff 2 ff 1 =ff 2 ff 1 ff 2 :V 3 V 3 ff 1 =ff Id V,ff 2 =Id V ff Avatars: factorization condition for the dispersion matrix in the 1-dim. n-body problem(mcguire& Yang 60 ); condition for the partition function in an exactly solvable lattice model(onsager 44; Baxter 70 );

4 1 Yang Baxter equation: basics Data: vectorspacev, ff:v 2 V 2. Yang-Baxter equation(ybe) ff 1 ff 2 ff 1 =ff 2 ff 1 ff 2 :V 3 V 3 ff 1 =ff Id V,ff 2 =Id V ff Avatars: factorization condition for the dispersion matrix in the 1-dim. n-body problem(mcguire& Yang 60 ); condition for the partition function in an exactly solvable lattice model(onsager 44; Baxter 70 ); quantum inverse scattering method for completely integrable systems(faddeev et al. 79); factorisable S-matrices in 2-dim. quantum field theory (Zamolodchikov 79); R-matricesinquantumgroups(Drinfel 0 d80 ); C algebras(woronowicz80 );...

5 1 Yang Baxter equation: basics Data: vectorspacev, ff:v 2 V 2. Yang-Baxter equation(ybe) ff 1 ff 2 ff 1 =ff 2 ff 1 ff 2 :V 3 V 3 ff Avatars: 1 =ff Id V,ff 2 =Id V ff braid equation in low-dimensional topology ff YBE = Reidemeister III move

6 2 Braided sets Data: sets, ff:sˆ2 Sˆ2. Set-theoreticYBE(Drinfel 0 d 90) ff 1 ff 2 ff 1 =ff 2 ff 1 ff 2 :Sˆ3 Sˆ3 ff Solutions are called braided sets. 1 =ffˆid S,ff 2 =Id Sˆff

7 2 Braided sets Data: sets, ff:sˆ2 Sˆ2. Set-theoreticYBE(Drinfel 0 d 90) ff 1 ff 2 ff 1 =ff 2 ff 1 ff 2 :Sˆ3 Sˆ3 ff Solutions are called braided sets. 1 =ffˆid S,ff 2 =Id Sˆff braided sets linearise deform general solutions

8 2 Braided sets Data: sets, ff:sˆ2 Sˆ2. Set-theoreticYBE(Drinfel 0 d 90) ff 1 ff 2 ff 1 =ff 2 ff 1 ff 2 :Sˆ3 Sˆ3 ff Solutions are called braided sets. 1 =ffˆid S,ff 2 =Id Sˆff linearise deform braided sets Examples: ff(x;y)=(x;y); ff(x;y)=(y;x) R-matrices; general solutions

9 2 Braided sets Data: sets, ff:sˆ2 Sˆ2. Set-theoreticYBE(Drinfel 0 d 90) ff 1 ff 2 ff 1 =ff 2 ff 1 ff 2 :Sˆ3 Sˆ3 ff Solutions are called braided sets. braided sets Examples: ff(x;y)=(x;y); ff(x;y)=(y;x) linearise deform R-matrices; Liealgebra (V; []),centralelement12v, ff(x y)=y x+ h1 [x;y]. 1 =ffˆid S,ff 2 =Id Sˆff YBEforff Jacobiidentityfor [] general solutions

10 3 Self-distributivity sets,binaryoperation, ff(x;y)=(y;x y) YBEforff self-distributivityfor Self-distributivity: (x y) z=(x z) (y z)

11 3 Self-distributivity sets,binaryoperation, ff(x;y)=(y;x y) YBEforff self-distributivityfor Self-distributivity: (x y) z=(x z) (y z) Examples: groupswithx y=y 1 xyyieldaquandle: (SD) & 8y;x7 x yisabijection & x x=x; z 1 (y 1 xy)z=(z 1 y 1 z)(z 1 xz)(z 1 yz)

12 3 Self-distributivity sets,binaryoperation, ff(x;y)=(y;x y) YBEforff self-distributivityfor Self-distributivity: (x y) z=(x z) (y z) Examples: groupswithx y=y 1 xyyieldaquandle: (SD) & 8y;x7 x yisabijection & x x=x; z 1 (y 1 xy)z=(z 1 y 1 z)(z 1 xz)(z 1 yz) abeliangroups,t:s S, a b=ta+(1 t)b.

13 3 Self-distributivity sets,binaryoperation, ff(x;y)=(y;x y) YBEforff self-distributivityfor Self-distributivity: (x y) z=(x z) (y z) Examples: groupswithx y=y 1 xyyieldaquandle: (SD) & 8y;x7 x yisabijection & x x=x; z 1 (y 1 xy)z=(z 1 y 1 z)(z 1 xz)(z 1 yz) abeliangroups,t:s S, a b=ta+(1 t)b. Applications: invariants of knots and knotted surfaces (Joyce&Matveev 82);

14 (S; )-colourings c y x y of knot diagrams: b a x y cf. Wirtinger presentation ofı 1 (R 3 \K): p

15 (S; )-colourings c y x y of knot diagrams: b a x y Proposition: (S; )isaquandle = #{(S; )-colouringsofdiagrams } c b c (a b) c isaknotinvariant. c b c (a c) (b c) a b RIII a c a b c a b c

16 (S; )-colourings c y x y of knot diagrams: b a x y Proposition: (S; )isaquandle = #{(S; )-colouringsofdiagrams } Example: (Z 3 ;a b=2b a) isaknotinvariant.

17 (S; )-colourings c y x y of knot diagrams: b a x y Proposition: (S; )isaquandle = #{(S; )-colouringsofdiagrams } Example: (Z 3 ;a b=2b a) isaknotinvariant. 3 colourings 9 colourings

18 3 Self-distributivity sets,binaryoperation, ff(x;y)=(y;x y) YBEforff self-distributivityfor Self-distributivity: (x y) z=(x z) (y z) Applications: invariants of knots and knotted surfaces (Joyce&Matveev 82); atotalorderonbraidgroups(dehornoy 91);

19 3 Self-distributivity sets,binaryoperation, ff(x;y)=(y;x y) YBEforff self-distributivityfor Self-distributivity: (x y) z=(x z) (y z) Applications: invariants of knots and knotted surfaces (Joyce&Matveev 82); atotalorderonbraidgroups(dehornoy 91); Hopf algebra classification(andruskiewitsch Graña 03).

20 4 More examples of braided sets monoid (S; ;1), ff(x;y)=(1;x y); YBEforff associativityfor

21 4 More examples of braided sets monoid (S; ;1), ff(x;y)=(1;x y); YBEforff associativityfor monoidfactorizationg=hk, S=H[K, ff(x;y)=(h;k),h2h,k2k,hk=xy;

22 4 More examples of braided sets monoid (S; ;1), ff(x;y)=(1;x y); YBEforff associativityfor monoidfactorizationg=hk, S=H[K, ff(x;y)=(h;k),h2h,k2k,hk=xy; lattice (S; V ; W ), ff(x;y)=(x V y;x W y).

23 4 More examples of braided sets monoid (S; ;1), ff(x;y)=(1;x y); YBEforff associativityfor monoidfactorizationg=hk, S=H[K, ff(x;y)=(h;k),h2h,k2k,hk=xy; lattice (S; V ; W ), ff(x;y)=(x V y;x W y). All these braidings are idempotent: ffff = ff.

24 5 Universal enveloping constructions Universal enveloping monoids: Mon(S;ff)=hS xy=y 0 x 0 wheneverff(x;y)=(y 0 ;x 0 )i U. e. (semi)groups and algebras are defined similarly. methods braided sets groups& algebras

25 5 Universal enveloping constructions Universal enveloping monoids: Mon(S;ff)=hS xy=y 0 x 0 wheneverff(x;y)=(y 0 ;x 0 )i U. e. (semi)groups and algebras are defined similarly. braided sets methods examples groups& algebras Theorem: (S;ff)a nice finitebraidedset,ff 2 =Id = Mon(S; ff) is of I-type, cancellative, Ore; Grp(S; ff) is solvable, Garside; k Mon(S; ff) is Koszul, noetherian, Cohen Macaulay, Artin Schelter regular (Manin, Gateva-Ivanova& Van den Bergh, Etingof Schedler Soloviev, Jespers Okniński, Chouraqui ).

26 Example: S={a;b}, aa ff bb; ab ff; ba ff; Grp(S;ff)=ha;b a 2 =b 2 i=:g.

27 Example: S={a;b}, aa ff bb; ab ff; ba ff; Grp(S;ff)=ha;b a 2 =b 2 i=:g. RealisationbyEuclideantransformationsofR 2 : b a 0 b=a 0 ba 0 a=a0 b a 2 =b 2

28 Example: S={a;b}, aa ff bb; ab ff; ba ff; Grp(S;ff)=ha;b a 2 =b 2 i=:g. RealisationbyEuclideantransformationsofR 2 : b a 0 b=a 0 ba 0 a=a0 b a 2 =b 2 R 2 =G =Kleinbottle:

29 5 Universal enveloping constructions Universal enveloping monoids: Mon(S;ff)=hS xy=y 0 x 0 wheneverff(x;y)=(y 0 ;x 0 )i Examples: monoid (S; ;1), ff(x;y)=(1;x y), S Mon(S;ff) = 1S =1 Mon ;

30 5 Universal enveloping constructions Universal enveloping monoids: Mon(S;ff)=hS xy=y 0 x 0 wheneverff(x;y)=(y 0 ;x 0 )i Examples: monoid (S; ;1), ff(x;y)=(1;x y), S Mon(S;ff) = 1S =1 Mon ; Liealgebra (V; [];1), ff(x y)=y x+ h1 [x;y], UEA(V; []) kmon(s;ff) = 1=1Mon.

31 6 Representations Mon(S;ff)=hS xy=y 0 x 0 wheneverff(x;y)=(y 0 ;x 0 )i Representations of (S; ff) := representations of k Mon(S; ff), i.e. vectorspacesmwithmˆs M s.t. (m x) y = (m y 0 ) x 0 = ff m x y m x y

32 6 Representations Mon(S;ff)=hS xy=y 0 x 0 wheneverff(x;y)=(y 0 ;x 0 )i Representations of (S; ff) := representations of k Mon(S; ff), i.e. vectorspacesmwithmˆs M s.t. (m x) y = (m y 0 ) x 0 = ff m x y m x y Examples: trivialrep.: M=k;m x=m; M=kMon(S;ff);m x=mx; usual reps for monoids, Lie algebras, self-distributive structures.

33 7 A cohomology theory? A cohomology theory for YBE solutions should: 1)Describedeformations: ff 0 ff 0 + hff 1 + h 2 ff 2 +.

34 7 A cohomology theory? A cohomology theory for YBE solutions should: 1)Describedeformations: ff 0 ff 0 + hff 1 + h 2 ff 2 +. Difficult! Pioneers: Freyd Yetter 89, Eisermann 05.

35 7 A cohomology theory? A cohomology theory for YBE solutions should: 1)Describedeformations: ff 0 ff 0 + hff 1 + h 2 ff 2 +. Difficult! Pioneers: Freyd Yetter 89, Eisermann 05. First approximation: diagonal deformations ff q (x;y)=q!(x;y) ff(x;y);!:sˆs Z:!a2-cocycle = ff q aybesolution.

36 7 A cohomology theory? A cohomology theory for YBE solutions should: 1)Describedeformations: ff 0 ff 0 + hff 1 + h 2 ff 2 +. Difficult! Pioneers: Freyd Yetter 89, Eisermann 05. First approximation: diagonal deformations ff q (x;y)=q!(x;y) ff(x;y);!:sˆs Z:!a2-cocycle = ff q aybesolution. 2) Yield knot and knotted surface invariants: (S;ff)-coloureddiagram (D;C) &!:SˆS Z Boltzmannweight B! (C) =!(x;y)!(x;y): y 0 x x 0 y x y y 0 x 0!a2-cocycle = aknotinvariantgivenby {B! (C) Cisa(S;ff)-colouringofD}:

37 7 A cohomology theory? A cohomology theory for YBE solutions should: 3) Unify cohomology theories for associative structures, Lie algebras, self-distributive structures etc. + explain parallels between them, + suggest theories for new structures.

38 7 A cohomology theory? A cohomology theory for YBE solutions should: 3) Unify cohomology theories for associative structures, Lie algebras, self-distributive structures etc. + explain parallels between them, + suggest theories for new structures. 4)ComputethecohomologyofkMon(S;ff).

39 8 Braided cohomology Data: braidedset (S;ff) & bimodulemoverit.

40 8 Braided cohomology Data: braidedset (S;ff) & bimodulemoverit. Construction: C n (S;ff;M)=Maps(Sˆn ;M), d n = n+1 i=1 ( 1)i 1 (d n;i l d n;i r ):C n C n+1, d n;i l f: x 0 i f(x0 1:::x 0 i 1 x i+1:::x n+1 ) x 0 i x0 1:::x 0 i 1 x i+1:::x n+1 ff 1 ff i 1 x 1 :::x n+1 f ff ff x 1 ::: x i :::x n+1

41 8 Braided cohomology Data: braidedset (S;ff) & bimodulemoverit. Construction: C n (S;ff;M)=Maps(Sˆn ;M), d n = n+1 i=1 ( 1)i 1 (d n;i l d n;i r ):C n C n+1, d n;i l f: x 0 i f(x0 1:::x 0 i 1 x i+1:::x n+1 ) x 0 i x0 1:::x 0 i 1 x i+1:::x n+1 ff 1 ff i 1 x 1 :::x n+1 f ff ff x 1 ::: x i :::x n+1 Theorem: d n+1 d n =0;

42 8 Braided cohomology Data: braidedset (S;ff) & bimodulemoverit. Construction: C n (S;ff;M)=Maps(Sˆn ;M), d n = n+1 i=1 ( 1)i 1 (d n;i l d n;i r ):C n C n+1, d n;i l f: x 0 i f(x0 1:::x 0 i 1 x i+1:::x n+1 ) x 0 i x0 1:::x 0 i 1 x i+1:::x n+1 ff 1 ff i 1 x 1 :::x n+1 f ff ff x 1 ::: x i :::x n+1 Theorem: d n+1 d n =0; H n (S;ff;M)=Kerd n =Imd n 1 isthenthcohomologygroupof (S;ff)withcoefficientsinM;

43 8 Braided cohomology Data: braidedset (S;ff) & bimodulemoverit. Construction: C n (S;ff;M)=Maps(Sˆn ;M), d n = n+1 i=1 ( 1)i 1 (d n;i l d n;i r ):C n C n+1, d n;i l f: x 0 i f(x0 1:::x 0 i 1 x i+1:::x n+1 ) x 0 i x0 1:::x 0 i 1 x i+1:::x n+1 ff 1 ff i 1 x 1 :::x n+1 f ff ff x 1 ::: x i :::x n+1 Theorem: d n+1 d n =0; H n (S;ff;M)=Kerd n =Imd n 1 isthenthcohomologygroupof (S;ff)withcoefficientsinM; for nice M,acupproduct :H n H m H n+m ;

44 8 Braided cohomology Data: braidedset (S;ff) & bimodulemoverit. Construction: C n (S;ff;M)=Maps(Sˆn ;M), d n = n+1 i=1 ( 1)i 1 (d n;i l d n;i r ):C n C n+1, d n;i l f: x 0 i f(x0 1:::x 0 i 1 x i+1:::x n+1 ) x 0 i x0 1:::x 0 i 1 x i+1:::x n+1 ff 1 ff i 1 x 1 :::x n+1 f ff ff x 1 ::: x i :::x n+1 Theorem: d n+1 d n =0; H n (S;ff;M)=Kerd n =Imd n 1 isthenthcohomologygroupof (S;ff)withcoefficientsinM; for nice M,acupproduct :H n H m H n+m ; other good properties.

45 9 Agoodtheory? 1)&2) For!2C 2 (S;ff;Z), d 2!=0 =!yieldsboltzmannweights & diagonal deformations,!=d 1 =!yieldstrivial...

46 9 Agoodtheory? 1)&2) For!2C 2 (S;ff;Z), d 2!=0 =!yieldsboltzmannweights & diagonal deformations,!=d 1 =!yieldstrivial... 3) Unifies classical cohomology theories. Example: monoid (S; ;1), ff(x;y)=(1;x y), d n;i l f: :::x i 2 x i 1 x i x i+1 ::: ffi 1

47 9 Agoodtheory? 1)&2) For!2C 2 (S;ff;Z), d 2!=0 =!yieldsboltzmannweights & diagonal deformations,!=d 1 =!yieldstrivial... 3) Unifies classical cohomology theories. Example: monoid (S; ;1), ff(x;y)=(1;x y), d n;i l f: :::x i 2 x i 1 x i x i+1 ::: ffi 1 :::x i 2 1(x i 1 x i )x i+1 ::: ffi 2

48 9 Agoodtheory? 1)&2) For!2C 2 (S;ff;Z), d 2!=0 =!yieldsboltzmannweights & diagonal deformations,!=d 1 =!yieldstrivial... 3) Unifies classical cohomology theories. Example: monoid (S; ;1), ff(x;y)=(1;x y), d n;i l f: :::x i 2 x i 1 x i x i+1 ::: ffi 1 :::x i 2 1(x i 1 x i )x i+1 ::: ffi 2 :::1x i 2 (x i 1 x i )x i+1 ::: :::

49 9 Agoodtheory? 1)&2) For!2C 2 (S;ff;Z), d 2!=0 =!yieldsboltzmannweights & diagonal deformations,!=d 1 =!yieldstrivial... 3) Unifies classical cohomology theories. Example: monoid (S; ;1), ff(x;y)=(1;x y), d n;i l f: :::x i 2 x i 1 x i x i+1 ::: ffi 1 :::x i 2 1(x i 1 x i )x i+1 ::: ffi 2 :::1x i 2 (x i 1 x i )x i+1 ::: ::: 1x 1 :::x i 2 (x i 1 x i )x i+1 :::

50 9 Agoodtheory? 1)&2) For!2C 2 (S;ff;Z), d 2!=0 =!yieldsboltzmannweights & diagonal deformations,!=d 1 =!yieldstrivial... 3) Unifies classical cohomology theories. Example: monoid (S; ;1), ff(x;y)=(1;x y), d n;i l f: :::x i 2 x i 1 x i x i+1 ::: ffi 1 :::x i 2 1(x i 1 x i )x i+1 ::: ffi 2 :::1x i 2 (x i 1 x i )x i+1 ::: ::: 1x 1 :::x i 2 (x i 1 x i )x i+1 ::: f(:::x i 2 (x i 1 x i )x i+1 :::).

51 9 Agoodtheory? 4) Quantum symmetriser QS: braided cohomology of (S;ff)withcoefsinM QS Hochschildcohomologyof kmon(s;ff)withcoefsinm

52 9 Agoodtheory? 4) Quantum symmetriser QS: braided cohomology of (S;ff)withcoefsinM cup product QS Hochschildcohomologyof kmon(s;ff)withcoefsinm cup product

53 9 Agoodtheory? 4) Quantum symmetriser QS: braided cohomology of (S;ff)withcoefsinM cup product smaller complexes QS Hochschildcohomologyof kmon(s;ff)withcoefsinm cup product tools

54 9 Agoodtheory? 4) Quantum symmetriser QS: braided cohomology of (S;ff)withcoefsinM cup product smaller complexes QS Hochschildcohomologyof kmon(s;ff)withcoefsinm cup product tools QS is an isomorphism when ffff=idandchark=0(farinati&garcía-galofre 16); ffff=ff(l. 16).

55 9 Agoodtheory? 4) Quantum symmetriser QS: braided cohomology of (S;ff)withcoefsinM cup product smaller complexes QS Hochschildcohomologyof kmon(s;ff)withcoefsinm cup product tools QS is an isomorphism when ffff=idandchark=0(farinati&garcía-galofre 16); ffff=ff(l. 16). Applications: factorizable groups, Young tableaux.

56 9 Agoodtheory? 4) Quantum symmetriser QS: braided cohomology of (S;ff)withcoefsinM cup product smaller complexes QS Hochschildcohomologyof kmon(s;ff)withcoefsinm cup product tools QS is an isomorphism when ffff=idandchark=0(farinati&garcía-galofre 16); ffff=ff(l. 16). Applications: factorizable groups, Young tableaux. Openproblem: HowfarisQSfrombeinganisoingeneral?