On set-theoretic solutions to the Yang-Baxter equation. Victoria LEBED (Nantes) with Leandro VENDRAMIN (Buenos Aires)

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1 On set-theoretic solutions to the Yang-Baxter equation Victoria LEBED (Nantes) with Leandro VENDRAMIN (Buenos Aires) Turin, January 2016

2 1 Yang-Baxter equation X: vector space, ff : X 2 X 2. Yang-Baxter equation (YBE) ff 1 ff 2 ff 1 = ff 2 ff 1 ff 2 : X 3 X 3 where ff 1 = ff Id X, ff 2 = Id X ff. Origins: factorization condition for the dispersion matrix in the 1-dim. n-body problem (McGuire & Yang 60 ); partition function for exactly solvable lattice models (Baxter 70 ).

3 X: set, ff : Xˆ2 Xˆ2 1 Set-theoretic Yang-Baxter equation Yang-Baxter equation (YBE) ff 1 ff 2 ff 1 = ff 2 ff 1 ff 2 : Xˆ3 Xˆ3 where ff 1 = ffˆ Id X, ff 2 = Id X ˆff. Origins: Drinfel 0 d 1990.

4 1 Set-theoretic Yang-Baxter equation X: set, ff : Xˆ2 Xˆ2, (x; y) 7 ( x y; x y ) Yang-Baxter equation (YBE) ff 1 ff 2 ff 1 = ff 2 ff 1 ff 2 : Xˆ3 Xˆ3 where ff 1 = ffˆ Id X, ff 2 = Id X ˆff. Origins: Drinfel 0 d (X; ff) is called a braided set, with a braiding ff. x y x y ff x y = (Reidemeister III)

5 1 Set-theoretic Yang-Baxter equation X: set, ff : Xˆ2 Xˆ2, (x; y) 7 ( x y; x y ) Yang-Baxter equation (YBE) ff 1 ff 2 ff 1 = ff 2 ff 1 ff 2 : Xˆ3 Xˆ3 where ff 1 = ffˆ Id X, ff 2 = Id X ˆff. Origins: Drinfel 0 d (X; ff) is called a braided set, with a braiding ff. Left non-degenerate braided set: x 7 x y is a bijection X X for all y. x y x y ff x y = (Reidemeister III) x y x y x y

6 1 Set-theoretic Yang-Baxter equation X: set, ff : Xˆ2 Xˆ2, (x; y) 7 ( x y; x y ) Yang-Baxter equation (YBE) ff 1 ff 2 ff 1 = ff 2 ff 1 ff 2 : Xˆ3 Xˆ3 where ff 1 = ffˆ Id X, ff 2 = Id X ˆff. Origins: Drinfel 0 d (X; ff) is called a braided set, with a braiding ff. Left non-degenerate braided set: x 7 x y is a bijection X X for all y. Birack: left and right non-degenerate braided set with invertible ff. x y x y ff x y = (Reidemeister III) x y x y x y

7 2 Structure (semi)group X: set, ff : Xˆ2 Xˆ2, (x; y) 7 ( x y; x y ). Structure (semi)group of (X; ff): (S)G X;ff = h X xy = x yx y i

8 2 Structure (semi)group X: set, ff : Xˆ2 Xˆ2, (x; y) 7 ( x y; x y ). Structure (semi)group of (X; ff): Captures properties of ff. (S)G X;ff = h X xy = x yx y i

9 2 Structure (semi)group X: set, ff : Xˆ2 Xˆ2, (x; y) 7 ( x y; x y ). Structure (semi)group of (X; ff): Captures properties of ff. (S)G X;ff = h X xy = x yx y i A source of interesting groups and algebras: Theorem: If (X; ff) is a finite RI-compatible birack with ff 2 = Id, then SG X;ff is of I-type, cancellative, Öre; G X;ff is solvable, Garside; ksg X;ff is Koszul, noetherian, Cohen-Macaulay, Artin-Schelter regular (Manin, Gateva-Ivanova & Van den Bergh, Etingof-Schedler-Soloviev, Jespers-Okniński, Chouraqui ).

10 3 Self-distributive structures Shelf: set X & : X ˆ X X s.t. (x y) z = (x z) (y z) ff = y x x y y is a braiding on X

11 3 Self-distributive structures Shelf: set X & : X ˆ X X s.t. (x y) z = (x z) (y z) Rack: & x 7 x y bijective 8y. ff = y x x y y is a braiding on X ff is LND (X; ff ) is a birack

12 3 Self-distributive structures Shelf: set X & : X ˆ X X s.t. (x y) z = (x z) (y z) Rack: & x 7 x y bijective 8y. Quandle: & x x = x. ff = y x x y y is a braiding on X ff is LND (X; ff ) is a birack (x; x) ff 7 (x; x).

13 3 Self-distributive structures Shelf: set X & : X ˆ X X s.t. (x y) z = (x z) (y z) Rack: & x 7 x y bijective 8y. Quandle: & x x = x. Examples: x y = f(x) for any f : X X; ff = y x x y y is a braiding on X ff is LND (X; ff ) is a birack (x; x) ff 7 (x; x).

14 3 Self-distributive structures Shelf: set X & : X ˆ X X s.t. (x y) z = (x z) (y z) Rack: & x 7 x y bijective 8y. Quandle: & x x = x. ff = y x x y y is a braiding on X ff is LND (X; ff ) is a birack (x; x) ff 7 (x; x). Examples: x y = f(x) for any f : X X; group X & x y = y 1 xy; G X;ff = h X x y = y 1 xy i.

15 3 Self-distributive structures Shelf: set X & : X ˆ X X s.t. (x y) z = (x z) (y z) Rack: & x 7 x y bijective 8y. Quandle: & x x = x. ff = y x x y y is a braiding on X ff is LND (X; ff ) is a birack (x; x) ff 7 (x; x). Examples: x y = f(x) for any f : X X; group X & x y = y 1 xy; G X;ff = h X x y = y 1 xy i. Applications: invariants of knots and knotted surfaces (Joyce & Matveev 1982); study of large cardinals (Laver 1980s); Hopf algebra classification (Andruskiewitsch-Graña 2003).

16 3 Self-distributive structures Shelf: set X & : X ˆ X X s.t. (x y) z = (x z) (y z) Rack: & x 7 x y bijective 8y. Quandle: & x x = x. ff = y x x y y is a braiding on X ff is LND (X; ff ) is a birack (x; x) ff 7 (x; x). Examples: x y = f(x) for any f : X X; group X & x y = y 1 xy; G X;ff = h X x y = y 1 xy i. Applications: invariants of knots and knotted surfaces (Joyce & Matveev 1982); study of large cardinals (Laver 1980s); Hopf algebra classification (Andruskiewitsch-Graña 2003). Remark: Hopf algebras are linearized quandles.

17 3 Self-distributive structures Shelf: set X & : X ˆ X X s.t. (x y) z = (x z) (y z) Rack: & x 7 x y bijective 8y. Quandle: & x x = x. y x x is a ff 0 = braiding x y on X ff is RND (X; ff ) is a birack (x; x) ff 7 (x; x). Examples: x y = f(x) for any f : X X; group X & x y = y 1 xy; G X;ff = h X x y = y 1 xy i. Applications: invariants of knots and knotted surfaces (Joyce & Matveev 1982); study of large cardinals (Laver 1980s); Hopf algebra classification (Andruskiewitsch-Graña 2003). Remark: Hopf algebras are linearized quandles.

18 4 Cycle sets Cycle set: set X & : X ˆ X X s.t. (x y) (x z) = (y x) (y z) & y 7 x y bijective 8x. x y x ff = y x y is a LND braiding on X with ff 2 = Id

19 4 Cycle sets Cycle set: set X & : X ˆ X X s.t. (x y) (x z) = (y x) (y z) & y 7 x y bijective 8x. Non-degenerate CS: & x 7 x x bijective. x y x ff = y x y is a LND braiding on X with ff 2 = Id ff is RND (X; ff ) is a birack

20 4 Cycle sets Cycle set: set X & : X ˆ X X s.t. (x y) (x z) = (y x) (y z) & y 7 x y bijective 8x. Non-degenerate CS: & x 7 x x bijective. x y x ff = y x y is a LND braiding on X with ff 2 = Id ff is RND (X; ff ) is a birack (Etingof-Schedler- Soloviev 1999, Rump 2005) Applications: (semi)group theory; construction of f.-d. complex semisimple triangular Hopf algebras (Etingof-Gelaki 1998).

21 4 Cycle sets Cycle set: set X & : X ˆ X X s.t. (x y) (x z) = (y x) (y z) & y 7 x y bijective 8x. Non-degenerate CS: & x 7 x x bijective. x y x ff = y x y is a LND braiding on X with ff 2 = Id ff is RND (X; ff ) is a birack (Etingof-Schedler- Soloviev 1999, Rump 2005) Applications: (semi)group theory; construction of f.-d. complex semisimple triangular Hopf algebras (Etingof-Gelaki 1998). Examples: x y = f(y) for any f : X X.

22 4 Cycle sets Cycle set: set X & : X ˆ X X s.t. (x y) (x z) = (y x) (y z) & y 7 x y bijective 8x. Non-degenerate CS: & x 7 x x bijective. x y x ff = y x y is a LND braiding on X with ff 2 = Id ff is RND (X; ff ) is a birack (Etingof-Schedler- Soloviev 1999, Rump 2005) Applications: (semi)group theory; construction of f.-d. complex semisimple triangular Hopf algebras (Etingof-Gelaki 1998). Examples: x y = f(y) for any f : X X. G X;ff = h X (x y)x = (y x)y i.

23 5 Monoids For a monoid (X; ; 1), the associativity of ff = 1 x y is a braiding x y on X (with ff 2 = ff )

24 5 Monoids For a monoid (X; ; 1), the associativity of ff = 1 x y is a braiding x y on X (with ff 2 = ff ) X is a group ff is LND (L. 2013)

25 5 Monoids For a monoid (X; ; 1), the associativity of ff = 1 x y is a braiding x y on X (with ff 2 = ff ) X is a group ff is LND (L. 2013) One has a semigroup isomorphism SG X;ff X; Xˆk 3 x 1 x k 7 x 1 x k :

26 6 Associated shelf Fix a LND braided set (X; ff). x y x y ff x y x y y x x y Proposition (L.-V. 2015): one has a shelf (X; ff ), where (y x) y =: x ff y y x y x

27 6 Associated shelf Fix a LND braided set (X; ff). Proposition (L.-V. 2015): one has a shelf (X; ff ), where (y x) y =: x ff y y x y x Proof:

28 6 Associated shelf Fix a LND braided set (X; ff). Proposition (L.-V. 2015): one has a shelf (X; ff ), where (y x) y =: x ff y y x y x Proof:

29 6 Associated shelf Fix a LND braided set (X; ff). Proposition (L.-V. 2015): one has a shelf (X; ff ), where (y x) y =: x ff y y x y x Proof:

30 6 Associated shelf Fix a LND braided set (X; ff). Proposition (L.-V. 2015): one has a shelf (X; ff ), where (y x) y =: x ff y y x y x Proof:

31 6 Associated shelf Fix a LND braided set (X; ff). Proposition (L.-V. 2015): one has a shelf (X; ff ), where y x (y x) y =: x ff y y x Motto: ff is often simpler than ff, and encodes its important properties.

32 6 Associated shelf Fix a LND braided set (X; ff). Proposition (L.-V. 2015): one has a shelf (X; ff ), where y x (y x) y =: x ff y y x Motto: ff is often simpler than ff, and encodes its important properties. Proposition (L.-V. 2015): (X; ff ) is a rack ff is invertible; (X; ff ) is a trivial (x ff y = x) ff 2 = Id; x ff x = x ff(x x; x) = (x x; x).

33 7 Guitar map J (n) : Xˆn Xˆn ; (x 1 ; : : : ; x n ) 7 (x x2 xn 1 ; : : : ; x xn n 1; x n ); where x xi+1 xn i = (: : : (x xi+1 i ) : : :) xn. J 1(x) J 2(x) J n(x) x 1 xn.

34 7 Guitar map J (n) : Xˆn Xˆn ; (x 1 ; : : : ; x n ) 7 (x x2 xn 1 ; : : : ; x xn n 1; x n ); where x xi+1 xn i = (: : : (x xi+1 i ) : : :) xn. J 1(x) J 2(x). J n(x) J 2(x)= x x3x4x5 2 x 1 xn x 2 x 3 x 4 x 5

35 7 Guitar map J (n) : Xˆn Xˆn ; (x 1 ; : : : ; x n ) 7 (x x2 xn 1 ; : : : ; x xn n 1; x n ); where x xi+1 xn i = (: : : (x xi+1 i ) : : :) xn. J 1(x) J 2(x). J n(x) J 2(x)= x x3x4x5 2 x 1 xn x 2 x 3 x 4 x 5 Proposition (L.-V. 2015): Jff i = ff 0 i J, where ff 0 = ff 0 ff. ff = x y x y y ff x x ff 0 = x y x y

36 7 Guitar map J (n) : Xˆn Xˆn ; (x 1 ; : : : ; x n ) 7 (x x2 xn 1 ; : : : ; x xn n 1; x n ); where x xi+1 xn i = (: : : (x xi+1 i ) : : :) xn. J 1(x) J 2(x). J n(x) J 2(x)= x x3x4x5 2 x 1 xn x 2 x 3 x 4 x 5 Proposition (L.-V. 2015): Jff i = ff 0 i J, where ff 0 = ff 0 ff. Corollary: ff and ff 0 yield isomorphic B n -actions on Xˆn.

37 7 Guitar map J (n) : Xˆn Xˆn ; (x 1 ; : : : ; x n ) 7 (x x2 xn 1 ; : : : ; x xn n 1; x n ); where x xi+1 xn i = (: : : (x xi+1 i ) : : :) xn. J 1(x) J 2(x). J n(x) J 2(x)= x x3x4x5 2 x 1 xn x 2 x 3 x 4 x 5 Proposition (L.-V. 2015): Jff i = ff 0 i J, where ff 0 = ff 0 ff. Corollary: ff and ff 0 yield isomorphic B n -actions on Xˆn. Warning: In general, (X; ff) (X; ff 0 ) as braided sets!

38 8 RI-compatibility RI-compatible braiding: 9t : X X s.t. ff(t(x); x) = (t(x); x). t(x) x = x = x x x x (Reidemeister I) t 1 (x)

39 8 RI-compatibility RI-compatible braiding: 9t : X X s.t. ff(t(x); x) = (t(x); x). t(x) x = x = x x x x (Reidemeister I) t 1 (x) Examples: for a rack, it means x x = x (here t(x) = x);

40 8 RI-compatibility RI-compatible braiding: 9t : X X s.t. ff(t(x); x) = (t(x); x). t(x) x = x = x x x x (Reidemeister I) t 1 (x) Examples: for a rack, it means x x = x (here t(x) = x); for a cycle set, it means the non-degeneracy (here t(x) = x x).

41 9 Structure group via associated shelf Theorem (L.-V. 2015): (1) The guitar maps induce a bijective 1-cocycle J : SG X;ff SGX;ff 0, where ff 0 = ff 0 ff. Reminder: SG X;ff = h X xy = x yx y i.

42 9 Structure group via associated shelf Theorem (L.-V. 2015): (1) The guitar maps induce a bijective 1-cocycle }{{} J : SG X;ff SG X;ff 0, where ff 0 = ff 0 ff. J(xy) = J(x) y J(y)

43 9 Structure group via associated shelf Theorem (L.-V. 2015): (1) The guitar maps induce a bijective 1-cocycle }{{} J : SG X;ff SG X;ff 0, where ff 0 = ff 0 ff. J(xy) = J(x) y J(y) (x 1 ; : : : ; x n ) y = x y (x y 1; : : : ; x y n) x y

44 9 Structure group via associated shelf Theorem (L.-V. 2015): (1) The guitar maps induce a bijective 1-cocycle }{{} J : SG X;ff SG X;ff 0, where ff 0 = ff 0 ff. J(xy) = J(x) y J(y) (x 1 ; : : : ; x n ) y = (x y 1; : : : ; x y n) (2) If (X; ff) is an RI-compatible birack, then the maps Kˆn J (n) induce a bijective 1-cocycle G X;ff GX;ff 0, x x y y

45 9 Structure group via associated shelf Theorem (L.-V. 2015): (1) The guitar maps induce a bijective 1-cocycle }{{} J : SG X;ff SG X;ff 0, where ff 0 = ff 0 ff. J(xy) = J(x) y J(y) (x 1 ; : : : ; x n ) y = x y (x y 1; : : : ; x y n) x y (2) If (X; ff) is an RI-compatible birack, then the maps Kˆn J (n) induce a bijective 1-cocycle G X;ff GX;ff 0, where J (n) is extended to (X t X 1 )ˆn by x 1 x 2 1 x 3 1 y 1 y 2 1 y 3 1

46 9 Structure group via associated shelf Theorem (L.-V. 2015): (1) The guitar maps induce a bijective 1-cocycle }{{} J : SG X;ff SG X;ff 0, where ff 0 = ff 0 ff. J(xy) = J(x) y J(y) (x 1 ; : : : ; x n ) y = x y (x y 1; : : : ; x y n) x y (2) If (X; ff) is an RI-compatible birack, then the maps Kˆn J (n) induce a bijective 1-cocycle G X;ff GX;ff 0, where J (n) is extended to (X t X 1 )ˆn by K(x) = x, K(x 1 ) = t(x) 1. x 1 x 2 1 x 3 1 y 1 y 2 1 y 3 1

47 10 Associated shelf: examples For a rack (X; ) ff =, J : ff ff 0.

48 10 Associated shelf: examples For a rack (X; ) ff =, J : ff ff 0. For a cycle set (X; ) x ff y = x,

49 10 Associated shelf: examples For a rack (X; ) ff =, J : ff ff 0. For a cycle set (X; ) x ff y = x, y x J : ff flip, x y (S)G X;ff 0 is the free abelian (semi)group on X.

50 10 Associated shelf: examples For a rack (X; ) ff =, J : ff ff 0. For a cycle set (X; ) x ff y = x, y x J : ff flip, x y (S)G X;ff 0 is the free abelian (semi)group on X. For a group (X; ; 1) x ff y = y,

51 10 Associated shelf: examples For a rack (X; ) ff =, J : ff ff 0. For a cycle set (X; ) x ff y = x, y x J : ff flip, x y (S)G X;ff 0 is the free abelian (semi)group on X. For a group (X; ; 1) x ff y = y, x x J : ff, x y SG X;ff 0 X, x1 x k 7 x 1.

52 11 Braided modules Right braided module over (X; ff): set M & j : M ˆ X M s.t. j j = j j ff M X X M X X

53 11 Braided modules Right braided module over (X; ff): set M & j : M ˆ X M s.t. j j = j j ff M X X M X X Examples: usual modules for the structures above;

54 11 Braided modules Right braided module over (X; ff): set M & j : M ˆ X M s.t. j j = j j ff M X X M X X Examples: usual modules for the structures above; trivial module: M = { };

55 11 Braided modules Right braided module over (X; ff): set M & j : M ˆ X M s.t. j j = j j ff M X X M X X Examples: usual modules for the structures above; trivial module: M = { }; adjoint modules: M = Xˆk, j(x; y) = x y. x y x y

56 12 Braided homology Ingredients: a braided set (X; ff); right braided module (M; j) over (X; ff); left braided module (N; ) over (X; ff); abelian group A.

57 12 Braided homology Ingredients: a braided set (X; ff); right braided module (M; j) over (X; ff); left braided module (N; ) over (X; ff); abelian group A. Put C k = A (MˆXˆkˆN) = A Z ZM ˆ Xˆk ˆ N.

58 12 Braided homology Ingredients: a braided set (X; ff); right braided module (M; j) over (X; ff); left braided module (N; ) over (X; ff); abelian group A. Put C k = A (MˆXˆkˆN) = A Z ZM ˆ Xˆk ˆ N. Theorem (Carter-Elhamdadi-Saito 2004, L. 2013): C k carry a family of differentials ( ; ) = +, ; 2 Z. M X ::: X = j ff ( 1) i 1 ff N M X ::: X N = ff ( 1) i 1 ff 1 ::: i ::: n 1 ::: i ::: n

59 13 Braided homology.v2 Theorem (Fenn-Rourke-Sanderson 1992, L.-V. 2015): If moreover ff is LND, then C k carry a second family of differentials b ( ; ) = c + c, ; 2 Z, where c (m; y; n) = ( 1) i 1 (j(m; ffl i (y)); : : : ; cy i ; : : : ; n), ffl i (y) = x1:::xi 1 x i ; x = J 1 (y);

60 13 Braided homology.v2 Theorem (Fenn-Rourke-Sanderson 1992, L.-V. 2015): If moreover ff is LND, then C k carry a second family of differentials b ( ; ) = c + c, ; 2 Z, where c (m; y; n) = ( 1) i 1 (j(m; ffl i (y)); : : : ; cy i ; : : : ; n), ffl i (y) = x1:::xi 1 x i ; x = J 1 (y); c (m; y; n) = ( 1) i 1 (m; y i y 1 ; : : : ; y i y i 1 ; y i y i+1 ; : : : ; y i y k ; (y i ; n)).

61 (m; y 1 ; y 2 ; y 3 ; y 4 ; n) 7 (m; y 3 y 1 ; y 3 y 2 ; y 3 y 4 ; (y 3 ; n)). y 1 y 2 y 3 y 1 y 3 y 3 y 4 y 1 y 3 y 2 y 3 y 2 y 3 y 3 y 1 y 3 y 2 y 4 y 3 y 4 y 3 y 4 y 3 y 3

62 14 Braided homology: v1 = v2 Theorem (L.-V. 2015): If ff is LND, then J : (C ; ( ; ) ) (C ; b ( ; ) ) yields a chain complex iso.

63 14 Braided homology: v1 = v2 Theorem (L.-V. 2015): If ff is LND, then J : (C ; ( ; ) ) (C ; b ( ; ) ) yields a chain complex iso. Examples: group (X; ; 1) { (: : : ; x i 1 x i ; : : :); i > 1; i (m; x; n) = (m x 1 ; : : :) i = 1; { c (: : : ; cx i ; : : :); i > 1; i (m; x; n) = (m x 1 x 1 2 ; : : :) i = 1; J(m; x; n) = (m; x 1 x k ; : : : ; x k 1 x k ; x k ; n),

64 14 Braided homology: v1 = v2 Theorem (L.-V. 2015): If ff is LND, then J : (C ; ( ; ) ) (C ; b ( ; ) ) yields a chain complex iso. Examples: group (X; ; 1) { (: : : ; x i 1 x i ; : : :); i > 1; i (m; x; n) = (m x 1 ; : : :) i = 1; { c (: : : ; cx i ; : : :); i > 1; i (m; x; n) = (m x 1 x 1 2 ; : : :) i = 1; J(m; x; n) = (m; x 1 x k ; : : : ; x k 1 x k ; x k ; n), 2 forms of the bar complex

65 14 Braided homology: v1 = v2 Theorem (L.-V. 2015): If ff is LND, then J : (C ; ( ; ) ) (C ; b ( ; ) ) yields a chain complex iso. Examples: rack (X; ) rack homology 1-term self-distributive homology

66 14 Braided homology: v1 = v2 Theorem (L.-V. 2015): If ff is LND, then J : (C ; ( ; ) ) (C ; b ( ; ) ) yields a chain complex iso. Examples: rack (X; ) rack homology 1-term self-distributive homology c, c c their alternative versions (Przytycki 2011)

67 14 Braided homology: v1 = v2 Theorem (L.-V. 2015): If ff is LND, then J : (C ; ( ; ) ) (C ; b ( ; ) ) yields a chain complex iso. Examples: rack (X; ) rack homology 1-term self-distributive homology c, c c their alternative versions (Przytycki 2011) cycle set (X; ): new homology theory (L.-V. 2015)

68 14 Braided homology: v1 = v2 Theorem (L.-V. 2015): If ff is LND, then J : (C ; ( ; ) ) (C ; b ( ; ) ) yields a chain complex iso. Examples: rack (X; ) rack homology 1-term self-distributive homology c, c c their alternative versions (Przytycki 2011) cycle set (X; ): new homology theory (L.-V. 2015) c c, M = { }, N = X: H 1 central extensions