A Wells type exact sequence for non-degenerate unitary solutions of the Yang--Baxter equation
AA WELLS TYPE EXACT SEQUENCE FOR NON-DEGENERATE UNITARYSOLUTIONS OF THE YANG–BAXTER EQUATION
VALERIY BARDAKOV AND MAHENDER SINGH
Abstract.
Cycle sets are known to give non-degenerate unitary solutions of the Yang–Baxterequation and linear cycle sets are enriched versions of these algebraic systems. The paper exploresthe recently developed cohomology and extension theory for linear cycle sets. We derive a fourterm exact sequence relating 1-cocycles, second cohomology and certain groups of automorphismsarising from central extensions of linear cycle sets. This is an analogue of a similar exact sequencefor group extensions known due to Wells. We also compare the exact sequence for linear cycle setswith that for their underlying abelian groups and discuss generalities on dynamical 2-cocycles. Introduction
The quantum Yang–Baxter equation is a fundamental equation arising in theoretical physics andhas deep connections with mathematics specially braid groups and knot theory. A solution of thequantum Yang–Baxter equation is a linear map R : V ⊗ V → V ⊗ V satisfying R R R = R R R , where V is a vector space and R ij : V ⊗ V ⊗ V → V ⊗ V ⊗ V acts as R on the ( i, j ) tensorfactor and as the identity on the remaining factor. If F : V ⊗ V → V ⊗ V is the flip operator F ( v ⊗ w ) = w ⊗ v , then R : V ⊗ V → V ⊗ V is a solution of the quantum Yang–Baxter equation ifand only if R = F ◦ R satisfies the braid relation R R R = R R R . Topologically, the braid relation is simply the third Reidemeister move of diagrams of links on theplane as shown in Figure 1.
Figure 1.
The braid relationIf X is a basis of the vector space V , then a map r : X × X → X × X satisfying r r r = r r r induces a solution of the Yang–Baxter equation. In this case, we say that ( X, r ) is a set-theoretic solution of the Yang–Baxter equation. Writing r ( x, y ) = ( σ x ( y ) , τ y ( x )) for x, y ∈ X , wesay that the solution r is non-degenerate if σ x and τ x are invertible for all x ∈ X . The problem offinding these set-theoretic solutions was posed by Drinfeld [9] and has attracted a lot of attention. Mathematics Subject Classification.
Primary 16T25, 20N02; Secondary 55N35, 57M27, 20J05.
Key words and phrases.
Brace, cycle set cohomology, linear cycle set, extension, group cohomology, Yang–Baxterequation. a r X i v : . [ m a t h . G R ] F e b VALERIY BARDAKOV AND MAHENDER SINGH A (left) cycle set , as defined by Rump [24], is a non-empty set X with a binary operation · havingbijective left translations X → X , x (cid:55)→ y · x , and satisfying the relation( x · y ) · ( x · z ) = ( y · x ) · ( y · z )(1.0.1)for all x, y, z ∈ X . A cycle set is non-degenerate if the squaring map a (cid:55)→ a · a is invertible. It isknown that every finite cycle set is non-degenerate [24, Theorem 2]. Rump showed that cycle setsare in bijection with non-degenerate unitary set-theoretic solutions of the Yang–Baxter equation.These solutions give a rich class of structures and are connected with semigroups of special type,Bieberbach groups [14], biquandles [12, 16], colourings of plane curves [11], Hopf algebras [10]and Garside groups [7], to name a few. Special solutions, particularly, the ones possessing self-distributivity are intimately connected to invariants of knots and links in the 3-space and thickenedsurfaces [12, 16, 17]. Cycle sets have been proved to be very useful in understanding the structureof solutions of the Yang–Baxter equation and for obtaining general classification results. Cyclesets as braces give only unitary solutions whereas skew braces, racks, bi-quandles etc. give generalsolutions. The structure of cycle sets is still far from being completely understood, and manyimportant questions on the topic are yet not answered. The reader is referred to [4, 5, 7, 8, 11, 13,19, 20, 21, 25, 26, 27, 28] for some recent works.A (left) brace is an abelian group ( A, +) with an additional group operation ◦ such that a ◦ ( b + c ) + a = a ◦ b + a ◦ c (1.0.2)holds for all a, b, c ∈ A . Braces were introduced by Rump [25] in a slightly different but equivalentform where he showed that these algebraic systems give set-theoretic solutions of the Yang–Baxterequation. The preceding definition is due to Ced´o, Jespers and Okni´nski [5]. Each abelian group istrivially a brace with a + b = a ◦ b . In addition, regular rings give a large supply of braces. Relationsbetween the additive and the multiplicative groups of a brace have been explored in many recentworks, for example, [6, 15, 22].A (left) linear cycle set is a cycle set ( X, · ) with an abelian group operation + satisfying theconditions x · ( y + z ) = x · y + x · z, (1.0.3) ( x + y ) · z = ( x · y ) · ( x · z )(1.0.4)for all x, y, z ∈ X . This notion goes back to Rump [25], who showed it to be equivalent to the bracestructure via the relation x · y = x − ◦ ( x + y ) , where x − is inverse with respect to ◦ . An abelian group can be viewed as a linear cycle set bytaking x · y = y for all x, y ∈ X , and referred as a trivial linear cycle set. Rump [25] showed thatlinear cycle sets are closely related to radical rings.It was pointed out in [2] that an extension theory for cycle sets (equivalently braces) would becrucial for a classification of these objects. This led to development of an extension theory byLebed and Vendramin [19, 20]. A homology and cohomology theory for linear cycle sets (and hencefor braces) was developed recently in [19]. As in case of groups, Lie algebras, quandles or any nicealgebraic system, the second cohomology groups were shown to classify central cycle set extensions.A cohomology theory for general cycle sets was developed in [20]. We will follow the linear cycleset language of [19] since it gives a neat construction of cohomology and extension theory. It isworth noting that the right and the two-sided analogues of braces and cycle sets can be definedanalogously and have been considered in the literature. ELLS TYPE EXACT SEQUENCE FOR SOLUTIONS OF THE YANG–BAXTER EQUATION 3
In this paper, we derive an exact sequence relating 1-cocycles, certain group of automorphismsand second cohomology groups of linear cycle sets. This can be thought of as a cycle set analogueof a fundamental exact sequence due to Wells [31]. For notational convenience, sometimes, we willdenote the value of a map φ at a point x by φ x . We use the notation Aut, Z and H to denotegroup of automorphisms, group of 1-cocycles and second cohomology group of a linear cycle set,respectively. To distinguish linear cycle sets from groups, we use the bold notation Aut , Z and H to denote group of automorphisms, group of 1-cocycles and second cohomology of a group,respectively.We begin Section 2 by recalling basic definitions and results. We prove that there is a naturalgroup homomorphism from the second linear cycle set cohomology to the second symmetric coho-mology of the underlying abelian group (Proposition 2.6) and also examine this homomorphism fortrivial cycle sets (Proposition 2.7). Section 3 prepares the foundation for the main result. Givena linear cycle set ( X, · , +) and an abelian group A , we define an action of Aut( X ) × Aut ( A ) onH ( X ; A ). As a consequence, we obtain a lower bound on the size of H ( X ; A ) (Corollary 3.2). InSection 4, we prove our main theorem (Theorem 4.4) that associates to each central extension oflinear cycle sets a four term exact sequence relating group of 1-cocycles, certain group of automor-phisms and second cohomology groups. Section 5 explores properties of the important connectingmap in this exact sequence (Theorem 5.3). In Section 6, we compare the exact sequence with thecorresponding Wells exact sequence for the underlying extension of abelian groups (Theorem 6.1).Finally, in Section 7, we discuss some generalities on bi-groupoids and dynamical extensions of(linear) cycle sets. 2. Preliminaries and some basic results
We begin with the following observation.
Lemma 2.1.
Let ( X, · , ∗ ) be a bi-groupoid and S : X × X → X × X given by S ( x, y ) = ( x · y, y ∗ x ) for x, y ∈ X . Then the following hold: (1) The pair ( X, S ) is a set theoretic solution of the Yang–Baxter equation if and only if theequalities ( x · ( y · z )) = ( x · y ) · (( y ∗ x ) · z )(( y · z ) ∗ x ) · ( z ∗ y ) = (( y ∗ x ) · z ) ∗ ( x · y )( z ∗ y ) ∗ (( y · z ) ∗ x ) = z ∗ ( y ∗ x ) hold for all x, y, z ∈ X . (2) If x · y = y for all x, y ∈ X , then the pair ( X, S ) is a set theoretic solution of the Yang–Baxterequation if and only if the operation ∗ is left distributive, i.e. (2.0.1) z ∗ ( y ∗ x ) = ( z ∗ y ) ∗ ( z ∗ x ) for all x, y, z ∈ X . Recall that ( X, ∗ ) is a (left) rack if the map x (cid:55)→ y ∗ x is a bijection and (2.0.1) holds for all x, y, z ∈ X . Thus, Case (3) of the preceding lemma gives a non-degenerate solution of the Yang–Baxter equation if and only if ( X, ∗ ) is a rack. Racks are useful in defining invariants of framedlinks in the 3-space. We look for conditions under which a rack is a cycle set. Following [18], arack X is abelian if x ∗ ( y ∗ z ) = y ∗ ( x ∗ z ) VALERIY BARDAKOV AND MAHENDER SINGH for all x, y, z ∈ X . Note that this condition is equivalent to the groupInn( X ) = (cid:104) S x , x ∈ X | S x ( y ) := x ∗ y, x, y ∈ X (cid:105) of inner automorphisms of X being abelian. Proposition 2.2. If ( X, ∗ ) is a rack such that Inn( X ) is abelian, then it is a cycle set.Proof. It follows from the rack axiom that x ∗ ( y ∗ z ) = ( x ∗ y ) ∗ ( x ∗ z )and y ∗ ( x ∗ z ) = ( y ∗ x ) ∗ ( y ∗ z )for all x, y, z ∈ X . Since Inn( X ) is abelian, we get( x ∗ y ) ∗ ( x ∗ z ) = ( y ∗ x ) ∗ ( y ∗ z ) , which is desired. (cid:3) Cohomology and extensions of linear cycle sets. A morphism between linear cycle sets X and Y is a map ϕ : X → Y satisfying ϕ ( x + x (cid:48) ) = ϕ ( x ) + ϕ ( x (cid:48) ) and ϕ ( x · x (cid:48) ) = ϕ ( x ) · ϕ ( x (cid:48) ) for all x, x (cid:48) ∈ X . The kernel of ϕ is defined by Ker( ϕ ) = ϕ − (0). The notion of image , of a short exactsequence of linear cycle sets, and of linear cycle subsets are defined in the usual manner.Two linear cycle set extensions A i (cid:26) E π (cid:16) X and A i (cid:48) (cid:26) E (cid:48) π (cid:48) (cid:16) X are called equivalent if thereexists a linear cycle set isomorphism ϕ : E → E (cid:48) such that the diagram A i (cid:47) (cid:47) Id (cid:15) (cid:15) E ϕ (cid:15) (cid:15) π (cid:47) (cid:47) X Id (cid:15) (cid:15) A i (cid:48) (cid:47) (cid:47) E (cid:48) π (cid:48) (cid:47) (cid:47) X commute. A cohomology theory for linear cycle sets is developed in the recent works of Lebedand Vendramin [19, 20]. Following [19], a 2-cocycle for a linear cycle set ( X, · , +) with coefficientsin the (additively written) abelian group A consists of two maps f, g : X × X → A satisfying theconditions g ( x, y ) = g ( y, x ) , (2.1.1) g ( x, y ) + g ( x + y, z ) = g ( y, z ) + g ( x, y + z ) , (2.1.2) f ( x + y, z ) = f ( x · y, x · z ) + f ( x, z ) , (2.1.3) f ( x, y + z ) − f ( x, y ) − f ( x, z ) = g ( x · y, x · z ) − g ( y, z )(2.1.4)for all x, y, z ∈ X . Note that, if ( f, g ) is a 2-cocycle of a linear cycle set ( X, · , +) with coefficientsin an abelian group A , then conditions (2.1.1)-(2.1.2)-(2.1.3)(2.1.4) imply that f (0 , x ) = f ( x,
0) = 0 ,g (0 , x ) = g ( x,
0) = g (0 , x ∈ X . A pair of maps f, g : X × X → A is called a 2-coboundary if there exists a map λ : X → A such that f ( x, y ) = λ ( x · y ) − λ ( y ) , (2.1.5) g ( x, y ) = λ ( x + y ) − λ ( x ) − λ ( y )(2.1.6) ELLS TYPE EXACT SEQUENCE FOR SOLUTIONS OF THE YANG–BAXTER EQUATION 5 for all x, y ∈ X .A 2-cocycle ( f, g ) is called normalised if g (0 ,
0) = 0, whereas a 2-coboundary ( f, g ) is callednormalised if the map λ : X → A satisfy λ (0) = 0. We denote the group of normalised 2-cocycles by Z ( X ; A ), and the group of normalised 2-coboundaries by B ( X ; A ). The quotientZ ( X ; A ) / B ( X ; A ) is the normalised cohomology group H ( X ; A ) of X with coefficients in A .We shall also need the group of normalised 1-cocycles defined as(2.1.7) Z ( X ; A ) = (cid:8) λ : X → A | λ ( x + y ) = λ ( x ) + λ ( y ) and λ ( x · y ) = λ ( y ) for all x, y ∈ X (cid:9) . The following is an analogue of a similar classical result for groups [19, Lemma 5.2].
Lemma 2.3.
Let ( X, · , +) be a linear cycle set, A an abelian group and f, g : X × X → A twomaps. Then the set X × A with the operations ( x, a ) + ( y, b ) = (cid:0) x + y, a + b + g ( x, y ) (cid:1) , ( x, a ) · ( y, b ) = (cid:0) x · y, b + f ( x, y ) (cid:1) for a, b ∈ A , x, y ∈ X , is a linear cycle set if and only if ( f, g ) is a -cocycle. The linear cycle set of Lemma 2.3 is denoted by X ⊕ f,g A . A reformulation of Lemma 2.3 forbraces is as follows [19, Lemma 5.4]. Lemma 2.4.
Let ( X, ◦ , +) be a brace, A be an abelian group, and f, g : X × X → A be two maps.Then the set X × A with the operations ( x, a ) + ( y, b ) = (cid:0) x + y, a + b + g ( x, y ) (cid:1) , ( x, a ) ◦ ( y, b ) = (cid:0) x ◦ y, a + b + f ( x, y ) (cid:1) for a, b ∈ A , x, y ∈ X , is a brace if and only if for the corresponding linear cycle set ( X, · , +) , thepair ( f , g ) is a -cocycle, where f ( x, y ) = − f ( x, x · y ) + g ( x, y )(2.1.8) for all x, y ∈ X . A linear cycle subset X (cid:48) of a linear cycle set X is called central if x · x (cid:48) = x (cid:48) and x (cid:48) · x = x for all x ∈ X and x (cid:48) ∈ X (cid:48) . A central extension of a linear cycle set ( X, · , +) by an abelian group A is thedatum of a short exact sequence of linear cycle sets0 → A i → E π → X → , (2.1.9)where A is endowed with the trivial cycle set structure, and its image i ( A ) is central in E . Noticethat an extension(2.1.10) 0 → A → E → X → X, A ) denote the set of equivalence classes of central extensionsof X by A . The linear cycle set X ⊕ f,g A from Lemma 2.3 is a central extension of X by A in theobvious way. More precisely, we have a central extension of linear cycle sets0 → A i → X ⊕ f,g A π → X → , where i ( a ) = (0 , a ) and π ( x, a ) = x for all a ∈ A and x ∈ X . Trivially, the underlying extension0 → A i → ( X ⊕ f,g A, +) π → ( X, +) → VALERIY BARDAKOV AND MAHENDER SINGH of abelian groups is also central. As in case of groups, all central linear cycle set extensions of X by A arise in this manner [19, Lemma 5.6]. Lemma 2.5.
Let → A i → E π → X → be a central linear cycle set extension and s : X → E bea set-theoretic section of π . (1) The maps f, g : X × X → E defined by f : ( x, y ) (cid:55)→ s ( x ) · s ( y ) − s ( x · y ) ,g : ( x, y ) (cid:55)→ s ( x ) + s ( y ) − s ( x + y ) both take values in i ( A ) and ( f, g ) is a 2-cocycle. (2) The cocycle above is normalised if and only if s (0) = 0 . (3) Extensions E and X ⊕ f,g A are equivalent. (4) A cocycle ( f (cid:48) , g (cid:48) ) obtained from another section s (cid:48) of π is cohomologous to ( f, g ) . If bothcocycles are normalised, then they are cohomologous in the normalised sense. Lemma 2.5 yields a bijective correspondence(2.1.11) Ext(
X, A ) ←→ H ( X ; A ) . Thus, central extensions of linear cycle sets (and hence of braces) are completely determined bytheir second normalised cohomology groups.2.2.
Homomorphism from linear cycle set cohomology to group cohomology.
Let G bea group, A a G -module and H ( G ; A ) the second group cohomology of G with coefficients in A . Itis well-known that H ( G ; A ) classifies group extensions of G by A inducing the given action of G on A [3]. Let H sym ( G ; A ) be the subgroup of H ( G ; A ) consisting of cohomology classes of grouptheoretical symmetric 2-cocycles (that is, maps satisfying conditions (2.1.1) and (2.1.2)). Recallthat a group theoretical 2-cocycle g : G × G → A satisfying (2.1.1) is called symmetric . An easycheck shows that if both G and A are abelian groups, then H sym ( G ; A ) classifies extensions of theform 0 → A → E → X → , where E is an abelian group. There is a natural group homomorphism from linear cycle setcohomology to symmetric group cohomology. Proposition 2.6.
Let ( X, · , +) be a linear cycle set and A an abelian group viewed as a trivial ( X, +) -module. Then there is a group homomorphism Λ : H N ( X ; A ) → H sym (( X, +); A ) given by Λ[( f, g )] = [ g ] .Proof. Given a normalised 2-cocycle ( f, g ) ∈ Z ( X ; A ) for the linear cycle set ( X, · , +) with coef-ficients in the abelian group A , conditions (2.1.1) and (2.1.2) imply that g is a group theoreticalsymmetric 2-cocycle of the abelian group ( X, +) with coefficients in the abelian group A . Further,if ( f, g ) ∈ B ( X ; A ), then condition (2.1.6) imply that g is a group theoretical 2-coboundary. Thus,there is a well-defined map Λ : H N ( X ; A ) → H sym (( X, +); A ) given by Λ[( f, g )] = [ g ]. That Λ is agroup homomorphism follows from the fact that multiplication of 2-cocycles is point-wise for bothlinear cycle sets and groups. (cid:3) Given two abelian groups G and A , let Bilin( G × G, A ) denote the group of bilinear maps from G × G to A . For trivial cycle sets, the map Λ is surjective and its kernel can be determined precisely. ELLS TYPE EXACT SEQUENCE FOR SOLUTIONS OF THE YANG–BAXTER EQUATION 7
Proposition 2.7.
Let ( X, · , +) be a trivial linear cycle set and A an abelian group viewed as atrivial ( X, +) -module. Then H ( X ; A ) ∼ = Bilin( X × X, A ) × H sym (( X, +); A ) . Proof.
We begin by noting that if f, g : X × X → A is a 2-cocycle of the trivial linear cycle set( X, · , +), then conditions (2.1.1)-(2.1.2) imply that g is a group theoretical symmetric 2-cocyclewhereas conditions (2.1.3)-(2.1.4) imply that f is a bilinear map, that is, f ∈ Bilin( X × X, A ).Further, by conditions (2.1.5)-(2.1.6), ( f, g ) is a 2-coboundary if there exists a map λ : X → A suchthat f ( x, y ) = 0and g ( x, y ) = λ ( x + y ) − λ ( x ) − λ ( y )for all x, y ∈ X . Thus, it follows thatH ( X ; A ) ∼ = Bilin( X × X, A ) × H sym (( X, +); A ) , and the map Λ is simply projection onto the second factor. (cid:3) Given two abelian groups X and A with A viewed as a trivial G -module, it follows from Proposi-tion 2.7 that the group Bilin( X × X, A ) × H sym ( X ; A ) classifies meta-trivial linear cycle sets, thatis, extensions of a trivial linear cycle set by a trivial linear cycle set. Question 2.8.
What can we say about the homomorphism Λ for non-trivial linear cycle sets?3.
Action of automorphisms on cohomology of linear cycle sets
Let ( X, · , +) be a linear cycle set and A an abelian group. Let Aut( X ) denote the group ofall cycle set automorphisms of X and Aut ( A ) the usual automorphism group of A . For ( φ, θ ) ∈ Aut( X ) × Aut ( A ) and ( f, g ) ∈ Z ( X ; A ), we define ( φ,θ ) ( f, g ) = (cid:0) ( φ,θ ) f, ( φ,θ ) g (cid:1) , where ( φ,θ ) f ( x, y ) := θ (cid:0) f (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:1) and ( φ,θ ) g ( x, y ) := θ (cid:0) g (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:1) for all x, y ∈ X . Proposition 3.1.
The group
Aut( X ) × Aut ( A ) acts by automorphisms on the group H ( X ; A ) as ( φ,θ ) [ f, g ] = [ ( φ,θ ) ( f, g )] for ( φ, θ ) ∈ Aut( X ) × Aut ( A ) and ( f, g ) ∈ Z ( X ; A ) .Proof. For ( φ, θ ) ∈ Aut( X ) × Aut ( A ) and ( f, g ) ∈ Z ( X ; A ), we first show that ( φ,θ ) ( f, g ) is anormalised 2-cocycle of the cycle set X . For x, y ∈ X , we have ( φ,θ ) f ( x + y, z ) = θ (cid:0) f (cid:0) φ − ( x + y ) , φ − ( z ) (cid:1)(cid:1) = θ (cid:0) f (cid:0) φ − ( x ) + φ − ( y ) , φ − ( z ) (cid:1)(cid:1) = θ (cid:0) f (cid:0) φ − ( x ) · φ − ( y ) , φ − ( x ) · φ − ( z ) (cid:1)(cid:1) + θ (cid:0) f (cid:0) φ − ( x ) , φ − ( z ) (cid:1)(cid:1) due to (2.1.3)= θ (cid:0) f (cid:0) φ − ( x · y ) , φ − ( x · z ) (cid:1)(cid:1) + θ (cid:0) f (cid:0) φ − ( x ) , φ − ( z ) (cid:1)(cid:1) = ( φ,θ ) f ( x · y, x · z ) + ( φ,θ ) f ( x, z ) , VALERIY BARDAKOV AND MAHENDER SINGH( φ,θ ) f ( x, y + z ) − ( φ,θ ) f ( x, y ) − ( φ,θ ) f ( x, z )= θ (cid:0) f (cid:0) φ − ( x ) , φ − ( y ) + φ − ( z ) (cid:1) − f (cid:0) φ − ( x ) , φ − ( y ) (cid:1) − f (cid:0) φ − ( x ) , φ − ( z ) (cid:1)(cid:1) = θ (cid:0) g (cid:0) φ − ( x ) · φ − ( y ) , φ − ( x ) · φ − ( z ) (cid:1) − g (cid:0) φ − ( y ) , φ − ( z ) (cid:1)(cid:1) due to (2.1.4)= θ (cid:0) g (cid:0) φ − ( x · y ) , φ − ( x · z ) (cid:1)(cid:1) − θ (cid:0) g (cid:0) φ − ( y ) , φ − ( z ) (cid:1)(cid:1) = ( φ,θ ) g ( x · y, x · z ) − ( φ,θ ) g ( y, z ) , ( φ,θ ) g ( x, y ) + ( φ,θ ) g ( x + y, z ) = θ (cid:0) g (cid:0) φ − ( x ) , φ − ( y ) (cid:1) + g (cid:0) φ − ( x ) + φ − ( y ) , φ − ( z ) (cid:1)(cid:1) = θ (cid:0) g (cid:0) φ − ( y ) , φ − ( z ) (cid:1) + g (cid:0) φ − ( x ) , φ − ( y ) + φ − ( z ) (cid:1)(cid:1) due to (2.1.2)= θ (cid:0) g (cid:0) φ − ( y ) , φ − ( z ) (cid:1)(cid:1) + θ (cid:0) g (cid:0) φ − ( x ) , φ − ( y + z ) (cid:1)(cid:1) = ( φ,θ ) g ( y, z ) + ( φ,θ ) g ( x, y + z ) . Further, ( φ,θ ) g ( x, y ) = ( φ,θ ) g ( y, x ) and ( φ,θ ) g (0 ,
0) = 0. Hence, ( φ,θ ) ( f, g ) ∈ Z ( X ; A ). Now, given( φ i , θ i ) ∈ Aut( X ) × Aut ( A ) for i = 1 ,
2, we see that ( φ ,θ )( φ ,θ ) f ( x, y ) = ( φ φ ,θ θ ) f ( x, y )= θ θ (cid:0) f (cid:0) φ − φ − ( x ) , φ − φ − ( y ) (cid:1)(cid:1) = θ (cid:0) ( φ ,θ ) f (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:1) = ( φ ,θ ) (cid:0) ( φ ,θ ) f (cid:1) ( x, y )for all x, y ∈ X . Similarly, one can show that ( φ ,θ )( φ ,θ ) g ( x, y ) = ( φ ,θ ) (cid:0) ( φ ,θ ) g (cid:1) ( x, y ), andhence Aut( X ) × Aut ( A ) acts on Z ( X ; A ). It now remains to be shown that the action preservesB ( X ; A ). Let ( f, g ) ∈ B ( X ; A ). Then there exists λ : X → A such that conditions (2.1.5) and(2.1.6) holds. For x, y ∈ X , we have ( φ,θ ) f ( x, y ) = θ (cid:0) f (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:1) = θ (cid:0) λ (cid:0) φ − ( x ) · φ − ( y ) (cid:1) − λ (cid:0) φ − ( y ) (cid:1)(cid:1) = θ (cid:0) λ (cid:0) φ − ( x · y ) (cid:1)(cid:1) − θ (cid:0) λ (cid:0) φ − ( y ) (cid:1)(cid:1) = λ (cid:48) ( x · y ) − λ (cid:48) ( y )and ( φ,θ ) g ( x, y ) = θ (cid:0) λ (cid:0) φ − ( x ) + φ − ( y ) (cid:1) − λ (cid:0) φ − ( x ) (cid:1) − λ (cid:0) φ − ( y ) (cid:1)(cid:1) = λ (cid:48) ( x + y ) − λ (cid:48) ( x ) − λ (cid:48) ( y ) , where λ (cid:48) = θλφ − : X → A . Hence, ( φ,θ ) ( f, g ) ∈ B ( X ; A ) and we are done. (cid:3) Applying the orbit-stabiliser theorem to the action of Aut( X ) × Aut ( A ) on H ( X ; A ) yields Corollary 3.2. If X is a finite linear cycle set, A a finite abelian group and ( f, g ) ∈ Z ( X ; A ) ,then | H ( X ; A ) | ≥ | Aut( X ) × Aut ( A ) || (Aut( X ) × Aut ( A )) [ f,g ] | , where (Aut( X ) × Aut ( A )) [ f,g ] is the stabiliser subgroup of Aut( X ) × Aut ( A ) at [ f, g ] . ELLS TYPE EXACT SEQUENCE FOR SOLUTIONS OF THE YANG–BAXTER EQUATION 9
In general, we have Aut( X ) ≤ Aut (( X, +)) for any linear cycle set ( X, · , +) where the equalityholds for trivial linear cycle sets. Corollary 3.2 and Proposition 2.7 then yields the following. Corollary 3.3. If G and A are finite abelian groups with A viewed as a trivial G -module and ( f, g ) ∈ Z ( G ; A ) , then | H sym ( G ; A ) | ≥ | Aut ( G ) × Aut ( A ) || Bilin( G × G, A ) | | ( Aut ( G ) × Aut ( A )) [ f,g ] | , where ( Aut ( G ) × Aut ( A )) [ f,g ] is the stabiliser subgroup of Aut ( G ) × Aut ( A ) at [ f, g ] . An exact sequence relating automorphisms and cohomology
Let ( X, · , +) be a linear cycle set, A an abelian group and(4.0.1) E : 0 → A i → E π → X →
0a central extension of X by A . In view of Lemma 2.5, there exists a normalised 2-cocycle ( f, g )such that E ∼ = X ⊕ f,g A. Further, the extension E determines a unique cohomology class [ f, g ] ∈ H ( X ; A ).Fix a central extension (4.0.1) of a linear cycle set ( X, · , +) by an abelian group A and itscorresponding cohomology class [ f, g ] ∈ H ( X ; A ) as in Lemma 2.5. For each ( φ, θ ) ∈ Aut( X ) × Aut ( A ), we have ( φ,θ ) [ f, g ] ∈ H ( X ; A ). Since the group H ( X ; A ) acts freely and transitively onitself by left multiplication, there exists a unique element Θ [ f,g ] ( φ, θ ) ∈ H ( X ; A ) such thatΘ [ f,g ] ( φ, θ ) ( φ,θ ) [ f, g ] = [ f, g ] . This gives a map(4.0.2) Θ [ f,g ] : Aut( X ) × Aut ( A ) → H ( X ; A ) . We denote Θ [ f,g ] by Θ for convenience of notation. Our aim is to relate certain group of automor-phisms of E to groups Aut( X ), Aut ( A ), H ( X ; A ) and Z ( X ; A ). For this purpose, we define Aut A ( E ) = (cid:8) ψ ∈ Aut( E ) | ψ ( x, a ) = ( φ ( x ) , λ ( x )+ θ ( a )) for some ( φ, θ ) ∈ Aut( X ) × Aut( A ) and map λ : X → A (cid:9) . Proposition 4.1.
Aut A ( E ) is a subgroup of Aut( E ) .Proof. If ψ ∈ Aut A ( E ), then ψ ( x, a ) = ( φ ( x ) , λ ( x ) + θ ( a )) for some ( φ, θ ) ∈ Aut( X ) × Aut ( A ) anda map λ : X → A . Define ¯ ψ : E → E by setting¯ ψ ( x, a ) = (cid:0) φ − ( x ) , θ − (cid:0) − λ ( φ − ( x )) (cid:1) + θ − ( a ) (cid:1) for x ∈ X and a ∈ A . Bijectivity of φ and θ imply that ¯ ψ is bijective. Since ψ is a cycle setmorphism, for all x, y ∈ X and a, b ∈ A , we have (cid:0) φ ( x + y ) , λ ( x + y ) + θ (cid:0) a + b + g ( x, y ) (cid:1)(cid:1) = ψ (cid:0) x + y, a + b + g ( x, y ) (cid:1) = ψ (cid:0) ( x, a ) + ( y, b ) (cid:1) = ψ ( x, a ) + ψ ( y, b )= (cid:0) φ ( x ) , λ ( x ) + θ ( a ) (cid:1) + (cid:0) φ ( y ) , λ ( y ) + θ ( b ) (cid:1) = (cid:0) φ ( x ) + φ ( y ) , λ ( x ) + λ ( y ) + θ ( a ) + θ ( b ) + g (cid:0) φ ( x ) , φ ( y ) (cid:1)(cid:1) and (cid:0) φ ( x · y ) , λ ( x · y ) + θ (cid:0) b + f ( x, y ) (cid:1)(cid:1) = ψ (cid:0) x · y, b + f ( x, y ) (cid:1) = ψ (cid:0) ( x, a ) · ( y, b ) (cid:1) = ψ ( x, a ) · ψ ( y, b )= (cid:0) φ ( x ) , λ ( x ) + θ ( a ) (cid:1) · (cid:0) φ ( y ) , λ ( y ) + θ ( b ) (cid:1) = (cid:0) φ ( x ) · φ ( y ) , λ ( y ) + θ ( b ) + f (cid:0) φ ( x ) , φ ( y ) (cid:1)(cid:1) . This gives(4.0.3) λ ( x + y ) + θ (cid:0) g ( x, y ) (cid:1) = λ ( x ) + λ ( y ) + g (cid:0) φ ( x ) , φ ( y ) (cid:1) and(4.0.4) λ ( x · y ) + θ (cid:0) f ( x, y ) (cid:1) = λ ( y ) + f (cid:0) φ ( x ) , φ ( y ) (cid:1) for all x, y ∈ X . Replacing x, y by φ − ( x ) , φ − ( y ) in (4.0.3) and (4.0.4) gives(4.0.5) λ (cid:0) φ − ( x ) + φ − ( y ) (cid:1) + θ (cid:0) g (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:1) = λ (cid:0) φ − ( x ) (cid:1) + λ (cid:0) φ − ( y ) (cid:1) + g ( x, y )and(4.0.6) λ (cid:0) φ − ( x ) · φ − ( y ) (cid:1) + θ (cid:0) f (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:1) = λ (cid:0) φ − ( y ) (cid:1) + f ( x, y )for all x, y ∈ X . Now, we compute¯ ψ (cid:0) ( x, a ) + ( y, b ) (cid:1) = ¯ ψ (cid:0) x + y, a + b + g ( x, y ) (cid:1) = (cid:16) φ − ( x + y ) , θ − (cid:0) − λ ( φ − ( x + y )) (cid:1) + θ − (cid:0) a + b + g ( x, y ) (cid:1)(cid:17) = (cid:16) φ − ( x ) + φ − ( y ) , θ − (cid:0) − λ ( φ − ( x )) (cid:1) + θ − ( a ) + θ − (cid:0) − λ ( φ − ( y )) (cid:1) + θ − ( b ) + g (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:17) by (4.0.5)= (cid:16) φ − ( x ) , θ − (cid:0) − λ ( φ − ( x )) (cid:1) + θ − ( a ) (cid:17) + (cid:16) φ − ( y ) , θ − (cid:0) − λ ( φ − ( y )) (cid:1) + θ − ( b ) (cid:17) = ¯ ψ ( x, a ) + ¯ ψ ( y, b ) (cid:1) and ¯ ψ (cid:0) ( x, a ) · ( y, b ) (cid:1) = ¯ ψ (cid:0) x · y, b + f ( x, y ) (cid:1) = (cid:0) φ − ( x · y ) , θ − (cid:0) − λ ( φ − ( x · y )) (cid:1) + θ − (cid:0) b + f ( x, y ) (cid:1)(cid:1) = (cid:16) φ − ( x ) · φ − ( y ) , θ − (cid:0) − λ ( φ − ( y )) (cid:1) + θ − ( b ) + f (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:17) by (4.0.6)= (cid:16) φ − ( x ) , θ − (cid:0) − λ ( φ − ( x )) (cid:1) + θ − ( a ) (cid:17) · (cid:16) φ − ( y ) , θ − (cid:0) − λ ( φ − ( y )) (cid:1) + θ − ( b ) (cid:17) = ¯ ψ ( x, a ) · ¯ ψ ( y, b ) (cid:1) ELLS TYPE EXACT SEQUENCE FOR SOLUTIONS OF THE YANG–BAXTER EQUATION 11 for all x, y ∈ X and a, b ∈ A . A direct check shows that ¯ ψ ∈ Aut A ( E ) and is the inverse of ψ .Further, if ψ i ( x, a ) = ( φ i ( x ) , λ i ( x ) + θ i ( a )) for i = 1 ,
2, then ψ ψ ( x, a ) = ψ (cid:0) φ ( x ) , λ ( x ) + θ ( a ) (cid:1) = (cid:0) φ φ ( x ) , λ (cid:0) φ ( x ) (cid:1) + θ (cid:0) λ ( x ) (cid:1) + θ θ ( a ) (cid:1) = (cid:0) φ φ ( x ) , λ ( x ) + θ θ ( a ) (cid:1) , (4.0.7)where λ : X → A is given by(4.0.8) λ ( x ) = λ (cid:0) φ ( x ) (cid:1) + θ (cid:0) λ ( x ) (cid:1) for x ∈ X and a ∈ A . It follows easily that Aut A ( E ) is closed under composition, and hence is asubgroup of Aut( E ). (cid:3) In view of (4.0.7), the map Ψ : Aut A ( E ) → Aut( X ) × Aut ( A )given by Ψ( ψ ) = ( φ, θ ) is a group homomorphism. Proposition 4.2.
Im(Ψ) = Θ − { } .Proof. First note thatΘ − { } = (cid:8) ( φ, θ ) ∈ Aut( X ) × Aut ( A ) | ( φ,θ ) [ f, g ] = [ f, g ] (cid:9) , the stabiliser subgroup of Aut( X ) × Aut ( A ) at [ f, g ]. Suppose that ( φ, θ ) ∈ Θ − { } . Then, bydefinition of cohomologous 2-cocycles, there exists a map λ : X → A such that ( φ,θ ) f ( x, y ) − f ( x, y ) = λ ( x · y ) − λ ( y ) , ( φ,θ ) g ( x, y ) − g ( x, y ) = λ ( x + y ) − λ ( x ) − λ ( y )for all x, y ∈ X . The preceding equations can be written as θ (cid:0) f ( x, y ) (cid:1) − f (cid:0) φ ( x ) , φ ( y ) (cid:1) = λ (cid:0) φ ( x ) · φ ( y ) (cid:1) − λ (cid:0) φ ( y ) (cid:1) , (4.0.9) θ (cid:0) g ( x, y ) (cid:1) − g (cid:0) φ ( x ) , φ ( y ) (cid:1) = λ (cid:0) φ ( x ) + φ ( y ) (cid:1) − λ (cid:0) φ ( x ) (cid:1) − λ (cid:0) φ ( y ) (cid:1) (4.0.10)for all x, y ∈ X . We define ψ : E → E by setting ψ ( x, a ) = (cid:0) φ ( x ) , − λ (cid:0) φ ( x ) (cid:1) + θ ( a ) (cid:1) for x ∈ X and a ∈ A . Bijectivity of φ and θ implies that ψ is a bijection. We now check that ψ isa morphism of linear cycle sets. For x, y ∈ X and a, b ∈ A , we have ψ (cid:0) ( x, a ) + ( y, b ) (cid:1) = ψ (cid:0) x + y, a + b + g ( x, y ) (cid:1) = (cid:0) φ ( x + y ) , − λ (cid:0) φ ( x + y ) (cid:1) + θ (cid:0) a + b + g ( x, y ) (cid:1)(cid:1) = (cid:0) φ ( x ) + φ ( y ) , − λ (cid:0) φ ( x ) (cid:1) − λ (cid:0) φ ( y ) (cid:1) + θ ( a ) + θ ( b ) + g (cid:0) φ ( x ) , φ ( y ) (cid:1)(cid:1) by (4.0.10)= (cid:0) φ ( x ) , − λ (cid:0) φ ( x ) (cid:1) + θ ( a ) (cid:1) + (cid:0) φ ( y ) , − λ (cid:0) φ ( y ) (cid:1) + θ ( b ) (cid:1) = ψ (cid:0) ( x, a ) (cid:1) + ψ (cid:0) ( y, b ) (cid:1) . Similarly, (4.0.9) gives ψ (cid:0) ( x, a ) · ( y, b ) (cid:1) = ψ (cid:0) ( x, a ) (cid:1) · ψ (cid:0) ( y, b ) (cid:1) , and hence ψ ∈ Aut A ( E ). SinceΨ( ψ ) = ( φ, θ ), we get Θ − { } ⊆ Im(Ψ). Conversely, if ( φ, θ ) ∈ Im(Ψ), then there exists a map λ : X → A such that ψ : E → E given by ψ ( x, a ) := ( φ ( x ) , λ ( x ) + θ ( a )) lies in Aut A ( E ). The map ψ being a morphism of linear cycle sets gives the conditions (4.0.9) and (4.0.10). As seen above,this implies that ( φ,θ ) [ f, g ] = [ f, g ], which completes the proof. (cid:3) Proposition 4.3. Z ( X ; A ) ∼ = Ker(Ψ) .Proof. Observe that ψ ∈ Ker(Ψ) if and only ψ ( x, a ) = ( x, λ ( x ) + a ) for all x ∈ X and a ∈ A . Now, ψ is a morphism of linear cycle sets if and only if conditions (4.0.9) and (4.0.10) hold with φ = id X and θ = id A . These conditions take the form λ ( x · y ) = λ (cid:0) y ) , (4.0.11) λ ( x + y ) = λ (cid:0) x ) + λ ( y )(4.0.12)for x, y ∈ X , and hence λ ∈ Z ( X ; A ). Conversely, given λ ∈ Z ( X ; A ), we see that ψ : E → E defined as ψ ( x, a ) = ( x, λ ( x ) + a ) is an element of Ker(Ψ). In view of (4.0.8), it follows that themap ι : Z ( X ; A ) → Ker(Ψ)given by ι ( λ ) = ψ is an isomorphism of groups. (cid:3) Combining (4.0.2), Proposition 4.2 and Proposition 4.3 gives the following exact sequence
Theorem 4.4.
Let X be a linear cycle set, A an abelian group and E = X ⊕ f,g A the centralextension of X by A corresponding to the 2-cocycle ( f, g ) ∈ Z ( X ; A ) . Then there exists an exactsequence of groups (4.0.13) 1 −→ Z ( X ; A ) ι −→ Aut A ( E ) Ψ −→ Aut( X ) × Aut ( A ) Θ −→ H ( X ; A ) , where exactness at Aut( X ) × Aut ( A ) means that Im(Ψ) = Θ − { } . Corollary 4.5.
Let X be a linear cycle set and A an abelian group such that H ( X ; A ) is trivial.Then every automorphism in Aut( X ) × Aut ( A ) extends to an automorphism in Aut A ( E ) . Restricting the action of Aut( X ) × Aut ( A ) on H ( X ; A ) to that of its subgroups Aut( X ) and Aut ( A ) gives the following result. Corollary 4.6.
Every automorphism in
Aut( X ) [ α ] and Aut ( A ) [ α ] can be extended to an automor-phism in Aut A ( E ) . Properties of map
ΘLet ( X, · , +) be a linear cycle set and A an abelian group. Since the group Aut( X ) × Aut ( A )acts on the group H ( X ; A ), we have their semi-direct product H ( X ; A ) (cid:111) (cid:0) Aut( X ) × Aut ( A ) (cid:1) .Further, the group H ( X ; A ) acts on itself by left multiplication. Proposition 5.1. H ( X ; A ) (cid:111) (cid:0) Aut( X ) × Aut ( A ) (cid:1) acts on H ( X ; A ) by setting [ α ]( φ,θ ) [ β ] = [ α ] ( ( φ,θ ) [ β ]) for ( φ, θ ) ∈ Aut( X ) × Aut ( A ) and [ α ] , [ β ] ∈ H ( X ; A ) . ELLS TYPE EXACT SEQUENCE FOR SOLUTIONS OF THE YANG–BAXTER EQUATION 13
Proof.
For ( φ , θ ) , ( φ , θ ) ∈ Aut( X ) × Aut ( A ) and [ α ] , [ α ] , [ β ] ∈ H ( X ; A ), we compute (cid:0) [ α ]( φ ,θ ) (cid:1)(cid:0) [ α ]( φ ,θ ) (cid:1) [ β ] = (cid:0) [ α ] ( φ ,θ [ α ] (cid:1)(cid:0) ( φ ,θ )( φ ,θ ) (cid:1) [ β ]= (cid:0) [ α ] ( φ ,θ [ α ] (cid:1)(cid:0)(cid:0) ( φ ,θ )( φ ,θ ) (cid:1) [ β ] (cid:1) = [ α ] (cid:0) ( φ ,θ [ α ] (cid:0) ( φ ,θ ) (cid:0) ( φ ,θ ) [ β ] (cid:1)(cid:1)(cid:1) = [ α ] (cid:0) ( φ ,θ ) [ α ] (cid:0) ( φ ,θ ) (cid:0) ( φ ,θ ) [ β ] (cid:1)(cid:1)(cid:1) , since ( φ ,θ [ α ] ∈ H ( X ; A ) , which acts on itself by left translation= [ α ] (cid:0) ( φ ,θ ) (cid:0) [ α ] (cid:0) ( φ ,θ ) [ β ] (cid:1)(cid:1)(cid:1) , since Aut( X ) × Aut ( A ) acts by automorphisms on H ( X ; A )= (cid:0) [ α ]( φ ,θ ) (cid:1)(cid:0)(cid:0) [ α ]( φ ,θ ) (cid:1) [ β ] (cid:1) , since H ( X ; A ) acts on itself by left translation . Hence, H ( X ; A ) (cid:111) (cid:0) Aut( X ) × Aut ( A ) (cid:1) acts on H ( X ; A ). (cid:3) Let G be a group and A an abelian group equipped with an action of G . Then Z ( G ; A ) = (cid:8) f : G → A | f ( xy ) = f ( x ) x f ( y ) for all x, y ∈ G (cid:9) is called the group of and B ( G ; A ) = (cid:8) f : G → A | there exists a ∈ A such that f ( x ) = ( x a ) a − for all x ∈ G (cid:9) the group of [3, Chapter 4]. Further, a complement of a subgroup H in a group G is another subgroup K of G such that G = HK and H ∩ K = 1. The following result relating1-cocycles and complements is well-known [30, 11.1.2]. Lemma 5.2.
Let H be an abelian group and G a group acting on H by automorphisms. Thenthe map f (cid:55)→ { f ( g ) g | g ∈ G } gives a bijection from the set Z ( G ; H ) of 1-cocycles to the set { K | G = HK and H ∩ K = 1 } of complements of H in G . Theorem 5.3.
Let ( X, · , +) be a linear cycle set, A an abelian group and Θ [ α ] : Aut( X ) × Aut ( A ) −→ H ( X ; A ) the map corresponding to a cohomology class [ α ] ∈ H ( X ; A ) . Then thefollowing hold: (1) Θ [ α ] is a group theoretical 1-cocycle. (2) Any two such maps corresponding to distinct linear cycle set cohomology classes are coho-mologous as group theoretical 1-cocycles.Proof.
Suppose that Θ = Θ [ α ] for [ α ] ∈ H ( X ; A ) and g ∈ H ( X ; A ) (cid:111) (cid:0) Aut( X ) × Aut ( A ) (cid:1) . Thenfor elements [ α ] , g [ α ] ∈ H ( X ; A ), there exists a unique [ β ] ∈ H ( X ; A ) such that [ β ] [ α ] = g [ α ].This shows that [ β ] − g ∈ S [ α ] , the stabiliser subgroup of H ( X ; A ) (cid:111) (cid:0) Aut( X ) × Aut ( A ) (cid:1) at [ α ],and hence H ( X ; A ) (cid:111) (cid:0) Aut( X ) × Aut ( A ) (cid:1) = H ( X ; A ) S [ α ] . Further, since H ( X ; A ) acts freely on itself, it follows that S [ α ] is the complement of H ( X ; A )in H ( X ; A ) (cid:111) (cid:0) Aut( X ) × Aut ( A ) (cid:1) . By Lemma 5.2, let f : Aut( X ) × Aut ( A ) → H ( X ; A ) bethe unique 1-cocycle corresponding to the complement S [ α ] of H ( X ; A ) in H ( X ; A ) (cid:111) (cid:0) Aut( X ) × Aut ( A ) (cid:1) . Then S [ α ] = (cid:8) f ( φ, θ )( φ, θ ) | ( φ, θ ) ∈ Aut( X ) × Aut ( A ) (cid:9) , that is, [ α ] = f ( φ,θ )( φ,θ ) [ α ] = f ( φ,θ ) (cid:0) ( φ,θ ) [ α ] (cid:1) . Now, by definition of Θ as in (4.0.2), we obtain f ( φ, θ ) = Θ( φ, θ ), and hence Θ is a 1-cocycle.Let Θ = Θ [ α ] and Θ (cid:48) = Θ (cid:48) [ α (cid:48) ] for [ α ] , [ α (cid:48) ] ∈ H ( X ; A ). Then for any ( φ, θ ) ∈ Aut( X ) × Aut ( A ),we have Θ( φ,θ ) (cid:0) ( φ,θ ) [ α ] (cid:1) = [ α ] and Θ (cid:48) ( φ,θ ) (cid:0) ( φ,θ ) [ α (cid:48) ] (cid:1) = [ α (cid:48) ] . Since H ( X ; A ) acts transitively on itself by left multiplication, there exists a unique [ β ] ∈ H ( X ; A )such that [ β ] [ α (cid:48) ] = [ α ]. This gives [ β ] − (cid:0) Θ( φ,θ ) (cid:0) ( φ,θ ) [ β ] (cid:0) ( φ,θ ) [ α (cid:48) ] (cid:1)(cid:1)(cid:1) = Θ (cid:48) ( φ,θ ) (cid:0) ( φ,θ ) [ α (cid:48) ] (cid:1) . Since [ β ] − Θ( φ, θ ) ( φ,θ ) [ β ], Θ (cid:48) ( φ, θ ) and ( φ,θ ) [ α (cid:48) ] all lie in H ( X ; A ), which acts freely on itself, wemust have [ β ] − Θ( φ, θ ) ( φ,θ ) [ β ] = Θ (cid:48) ( φ, θ ) . Thus, Θ and Θ (cid:48) differ by a 1-coboundary, which completes the proof. (cid:3) Comparison with Wells exact sequence for groups
In [31], Wells derived an exact sequence relating 1-cocycles, automorphisms and second cohomol-ogy of groups corresponding to a given extension of groups. The sequence has found applicationsin some long standing problems on automorphisms of finite groups. We refer the reader to [23,Chapter 2] for a detailed account of the same, and recall the construction of this exact sequencefor central extension of groups. Consider a central extension(6.0.1) E (cid:48) : 0 → N → G → H → N is a trivial H -module, and hence the group Z ( H ; N )of 1-cocycles is simply the group of all homomorphisms from H to N . Let H ( H ; N ) be the secondgroup cohomology of H with coefficients in N , and g : H × H → N be a group theoretical normalised2-cocycle corresponding to the extension (6.0.1). It follows from classical extension theory of groupsthat G ∼ = H × g N, where H × g N has underlying set H × N and group operation( x, a ) + ( y, b ) = ( x + y, a + b + g ( x, y ))for x, y ∈ H and a, b ∈ N .Let Aut N ( G ) be the group of automorphisms of G keeping N invariant as a set. In view of theidentification G ∼ = H × g N , we have Aut N ( G ) = (cid:8) ψ ∈ Aut( G ) | ψ ( x, a ) = ( φ ( x ) , λ ( x ) + θ ( a )) for some ( φ, θ ) ∈ Aut( H ) × Aut( N ) and map λ : H → N (cid:9) . There is an isomorphism of groups : Z ( H, N ) → Aut N ( G )given by ( λ ) = ψ , where ψ ( x, a ) = ( x, λ ( x ) + a ) [23, Proposition 2.45]. Also, there is a naturalhomomorphism Φ : Aut N ( G ) → Aut ( H ) × Aut ( N )given by Φ( ψ ) = ( φ, θ ) . ELLS TYPE EXACT SEQUENCE FOR SOLUTIONS OF THE YANG–BAXTER EQUATION 15
As in Section 3, there is an action of
Aut ( H ) × Aut ( N ) on H ( H ; N ). In fact, given any ( φ, θ ) ∈ Aut ( H ) × Aut ( N ) and [ h ] ∈ H ( H ; N ), setting ( φ,θ ) [ h ] = [ ( φ,θ ) h ] , where ( φ,θ ) h ( x, y ) = θ (cid:0) h (cid:0) φ − ( x ) , φ − ( y ) (cid:1)(cid:1) for x, y ∈ X , defines this action. Further, the actionrestricts to an action on the subgroup H sym ( H ; N ) of H ( H ; N ) consisting of symmetric cohomologyclasses. Notice that the group H ( H ; N ) acts freely and transitively on itself by left multiplication.Now, for each ( φ, θ ) ∈ Aut ( H ) × Aut ( N ), we have cohomology classes ( φ,θ ) [ g ] , [ g ] ∈ H ( H ; N ).Thus, there exists a unique element Ω( φ, θ ) ∈ H ( H ; N ) such thatΩ( φ, θ ) ( φ,θ ) [ g ] = [ g ] . This gives a map(6.0.2) Ω :
Aut ( H ) × Aut ( N ) → H ( H ; N ) , which depends on the equivalence class of the extension E (cid:48) or equivalently on its correspondingcohomology class. Further, Ω is a 1-cocycle with respect to the action of Aut ( H ) × Aut ( N )on H ( H ; N ) [23, Corollary 2.41]. With the preceding set-up, Wells derived the following exactsequence of groups(6.0.3) 1 −→ Z ( H, N ) −→ Aut N ( G ) Φ −→ Aut ( H ) × Aut ( N ) Ω −→ H ( H ; N ) . Let L and A denote the categories of linear cycle sets and abelian groups, respectively. Thenthere is a forgetful functor F : L → A that maps a linear cycle set to its underlying abelian group. The preceding discussion shows thatthe functor F induces a map from the exact sequence (4.0.13) to (6.0.3). Theorem 6.1.
Let ( X, · , +) be a linear cycle set, A an abelian group viewed as a trivial ( X, +) -module and E = X ⊕ f,g A the central extension corresponding to the 2-cocycle ( f, g ) ∈ Z ( X ; A ) .Then the following diagram of groups commutes (cid:47) (cid:47) Z ( X ; A ) (cid:127) (cid:95) inclusion (cid:15) (cid:15) ι (cid:47) (cid:47) Aut A ( E ) Ψ (cid:47) (cid:47) (cid:127) (cid:95) inclusion (cid:15) (cid:15) Aut( X ) × Aut ( A ) Θ (cid:47) (cid:47) (cid:127) (cid:95) inclusion (cid:15) (cid:15) H ( X ; A ) Λ (cid:15) (cid:15) (cid:47) (cid:47) Z (( X, +); A ) (cid:47) (cid:47) Aut A (( E, +)) Φ (cid:47) (cid:47) Aut (( X, +)) × Aut ( A ) Ω (cid:47) (cid:47) H sym (( X, +); A ) . Extensions of bi-groupoids and dynamical cocycles
During the last decade many new examples of bi-groupoids, namely, bi-racks, bi-quandles, braces,skew braces, linear cycle sets, etc, have been introduced in connection to virtual knot theory andsolutions of the Yang–Baxter equation. Let X and S be two non-empty sets, α, α (cid:48) : X × X → Map( S × S, S ) and β, β (cid:48) : S × S → Map( X × X, X ) maps. Then X × S with the binary operations(7.0.1) ( x, s ) · ( y, t ) = (cid:0) β s,t ( x, y ) , α x,y ( s, t ) (cid:1) , (7.0.2) ( x, s ) ∗ ( y, t ) = (cid:0) β (cid:48) s,t ( x, y ) , α (cid:48) x,y ( s, t ) (cid:1) forms a bi-groupoid. Naturally, to obtain a bi-groupoid of special type it is essential to havefunctions α, α (cid:48) , β, β (cid:48) with nice properties. Using defining axioms of a cycle set, we can deduce a generalisation of [29, Lemma 2.1], which itself is a linear cycle set analogue of a similar result forquandles [1, Lemma 2.1]. Proposition 7.1.
Let X and S be two sets, α : X × X → Map( S × S, S ) and β : S × S → Map( X × X, X ) two maps. Then the set X × S with the binary operation (7.0.3) ( x, s ) · ( y, t ) = (cid:0) β s,t ( x, y ) , α x,y ( s, t ) (cid:1) , forms a cycle set if and only if the following conditions hold: (1) the map ( y, t ) (cid:55)→ (cid:0) β s,t ( x, y ) , α x,y ( s, t ) (cid:1) is a bijection for all ( x, s ) ∈ X × S , (2) β α x,y ( s,t ) ,α x,z ( s,q ) (cid:16) β s,t ( x, y ) , β s,q ( x, z ) (cid:17) = β α y,x ( t,s ) ,α y,z ( t,q ) (cid:16) β t,s ( y, x ) , β t,q ( y, z ) (cid:17) and α β s,t ( x,y ) ,β s,q ( x,z ) (cid:16) α x,y ( s, t ) , α x,z ( s, q ) (cid:17) = α β t,s ( y,x ) ,β t,q ( y,z ) (cid:16) α y,x ( t, s ) , α y,z ( t, q ) (cid:17) for all x, y, z ∈ X and s, t, q ∈ S . If one of the sets is a cycle set, then we obtain
Corollary 7.2.
Let ( X, · ) be a cycle set, S a set and α : X × X → Map( S × S, S ) a map. Thenthe set X × S with the binary operation (7.0.4) ( x, s ) · ( y, t ) = (cid:0) x · y, α x,y ( s, t ) (cid:1) , forms a cycle set if and only if the following conditions hold: (1) the map ( y, t ) (cid:55)→ (cid:0) x · y, α x,y ( s, t ) (cid:1) is a bijection for each ( x, s ) ∈ X × S , (2) α x · y,x · z (cid:16) α x,y ( s, t ) , α x,z ( s, q ) (cid:17) = α y · x,y · z (cid:16) α y,x ( t, s ) , α y,z ( t, q ) (cid:17) for all x, y, z ∈ X and s, t, q ∈ S . A map α satisfying condition (2) of the preceding corollary is referred as a dynamical cocycle of X with values in S and the cycle set structure on X × S is called a dynamical extension of X by α . One can prove a similar result for linear cyclic sets. Proposition 7.3.
Let X and S be two sets, α, α (cid:48) : X × X → Map( S × S, S ) and β, β (cid:48) : S × S → Map( X × X, X ) maps. Then the set X × S with the binary operations (7.0.5) ( x, s ) · ( y, t ) = (cid:0) β s,t ( x, y ) , α x,y ( s, t ) (cid:1) , (7.0.6) ( x, s ) + ( y, t ) = (cid:0) β (cid:48) s,t ( x, y ) , α (cid:48) x,y ( s, t ) (cid:1) forms a linear cycle set if and only if the following conditions hold: (1) the map ( y, t ) (cid:55)→ (cid:0) β s,t ( x, y ) , α x,y ( s, t ) (cid:1) is a bijection for each ( x, s ) ∈ X × S , (2) β α x,y ( s,t ) ,α x,z ( s,q ) (cid:16) β s,t ( x, y ) , β s,q ( x, z ) (cid:17) = β α y,x ( t,s ) ,α y,z ( t,q ) (cid:16) β t,s ( y, x ) , β t,q ( y, z ) (cid:17) and α β s,t ( x,y ) ,β s,q ( x,z ) (cid:16) α x,y ( s, t ) , α x,z ( s, q ) (cid:17) = α β t,s ( y,x ) ,β t,q ( y,z ) (cid:16) α y,x ( t, s ) , α y,z ( t, q ) (cid:17) , (3) β s,α (cid:48) y,z ( t,q ) (cid:16) x, β (cid:48) t,q ( y, z ) (cid:17) = β (cid:48) α x,y ( s,t ) ,α x,z ( s,q ) (cid:16) β s,t ( x, y ) , β s,q ( x, z ) (cid:17) and α x,β (cid:48) t,q ( y,z ) (cid:16) s, α (cid:48) y,z ( t, q ) (cid:17) = α (cid:48) β s,t ( x,y ) ,β s,q ( x,z ) (cid:16) α x,y ( s, t ) , α x,z ( s, q ) (cid:17) , (4) β α (cid:48) x,y ( s,t ) ,q (cid:16) β (cid:48) s,t ( x, y ) , z (cid:17) = β α x,y ( s,t ) ,α x,z ( s,q ) (cid:16) β s,t ( x, y ) , β s,q ( x, z ) (cid:17) and α β (cid:48) s,t ( x,y ) ,z (cid:16) α (cid:48) x,y ( s, t ) , q (cid:17) = α β s,t ( x,y ) ,β s,q ( x,z ) (cid:16) α x,y ( s, t ) , α x,z ( s, q ) (cid:17) ELLS TYPE EXACT SEQUENCE FOR SOLUTIONS OF THE YANG–BAXTER EQUATION 17 for all x, y, z ∈ X and s, t, q ∈ S . As before, if one of the sets is already a linear cycle set, then we have
Corollary 7.4.
Let ( X, · , +) be a linear cycle set, S a set and α, α (cid:48) : X × X → Map( S × S, S ) twomaps. Then the set X × S with the binary operations (7.0.7) ( x, s ) · ( y, t ) = (cid:0) x · y, α x,y ( s, t ) (cid:1) , (7.0.8) ( x, s ) + ( y, t ) = (cid:0) x + y, α (cid:48) x,y ( s, t ) (cid:1) forms a linear cycle set if and only if the following conditions hold: (1) the map ( y, t ) (cid:55)→ (cid:0) x · y, α x,y ( s, t ) (cid:1) is a bijection for each ( x, s ) ∈ X × S , (2) α x · y,x · z (cid:16) α x,y ( s, t ) , α x,z ( s, q ) (cid:17) = α y · x,y · z (cid:16) α y,x ( t, s ) , α y,z ( t, q ) (cid:17) , (3) α x,y + z (cid:16) s, α (cid:48) y,z ( t, q ) (cid:17) = α (cid:48) x · y,x · z (cid:16) α x,y ( s, t ) , α x,z ( s, q ) (cid:17) , (4) α x + y,z (cid:16) α (cid:48) x,y ( s, t ) , q (cid:17) = α x · y,x · z (cid:16) α x,y ( s, t ) , α x,z ( s, q ) (cid:17) ,for all x, y, z ∈ X and s, t, q ∈ S . The pair of maps α, α (cid:48) : X × X → Map( S × S, S ) satisfying conditions (2)-(4) of the precedingcorollary is called a dynamical cocycle of the linear cycle set X with values in S and the linear cycleset structure on X × S is called the dynamical extension of X by S . Taking α x,y ( s, t ) = t + f ( x, y )and α (cid:48) x,y ( s, t ) = s + t + g ( x, y )for some 2-cocycle ( f, g ) ∈ Z ( X ; A ), we see that the dynamical extension generalises the extensionobtained in Lemma 2.3. Acknowledgments.
Bardakov is supported by the Russian Science Foundation grant 19-41-02005. Singh is supported by the SwarnaJayanti Fellowship grants DST/SJF/MSA-02/2018-19and SB/SJF/2019-20/04 and Indo-Russian grant DST/INT/RUS/RSF/P-19.
References [1] N. Andruskiewitsch and M. Gra˜na,
From racks to pointed Hopf algebras , Adv. Math. 178 (2003), no. 2, 177–243.[2] D. Bachiller, F. Ced´o, E. Jespers and J. Okni´nski,
A family of irretractable square-free solutions of the Yang–Baxter equation , Forum Math. 29 (2017), no. 6, 1291–1306.[3] K. S. Brown,
Cohomology of Groups , Graduate Texts in Mathematics, vol.87 (Springer, New York, 1982), x+306pp.[4] F. Ced´o, E. Jespers and J. Okni´nski,
Retractability of set theoretic solutions of the Yang–Baxter equation , Adv.Math. 224 (6) (2010), 2472–2484.[5] F. Ced´o, E. Jespers and J. Okni´nski,
Braces and the Yang–Baxter equation , Comm. Math. Phys. 327 (1) (2014),101–116.[6] F. Ced´o, A. Smoktunowicz and L. Vendramin,
Skew left braces of nilpotent type , Proc. Lond. Math. Soc. (3) 118(2019), no. 6, 1367–1392.[7] F. Chouraqui,
Garside groups and Yang–Baxter equation , Comm. Algebra 38 (12) (2010), 4441–4460.[8] P. Dehornoy,
Set-theoretic solutions of the Yang–Baxter equation, RC-calculus, and Garside germs , Adv. Math.282 (2015), 93–127.[9] V. G. Drinfeld,
On some unsolved problems in quantum group theory , Quantum groups (Leningrad, 1990), 1–8,Lecture Notes in Math., 1510, Springer, Berlin, 1992.[10] P. Etingof and S. Gelaki,
A method of construction of finite-dimensional triangular semisimple Hopf algebras ,Math. Res. Lett. 5 (1998) 551–561. [11] P. Etingof, T. Schedler and A. Soloviev,
Set-theoretical solutions to the quantum Yang–Baxter equation , DukeMath. J. 100(2) (1999), 169–209.[12] R. Fenn, M. Jordan-Santana and L. H. Kauffman,
Biquandles and virtual links , Topology Appl. 145 (2004),157–175.[13] T. Gateva-Ivanova,
Set-theoretic solutions of the Yang–Baxter equation, Braces, and Symmetric groups , Adv.Math. 338 (2018), 649–701.[14] T. Gateva-Ivanova and M. Van den Bergh,
Semigroups of I -type , J. Algebra 206 (1998) 97–112.[15] I. Gorshkov and T. Nasybullov, Finite skew-braces with solvable additive group , J. Algebra 574 (2021), 172–183.[16] L. H. Kauffman and V. Manturov,
Virtual biquandles , Fund. Math. 188 (2005), 103–146.[17] V. Lebed,
Applications of self-distributivity to Yang–Baxter operators and their cohomology , J. Knot TheoryRamifications 27 (11) (2018), 1843012, 20 pp.[18] V. Lebed and A. Mortier,
Abelian quandles and quandles with abelian structure group , J. Pure Appl. Algebra225 (2021), no. 1, 106474, 22 pp.[19] V. Lebed and L. Vendramin,
Cohomology and extensions of braces , Pacific J. Math. 284 (2016), no. 1, 191–212.[20] V. Lebed and L. Vendramin,
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation ,Adv. Math. 304 (2017) 1219–1261.[21] J.-H. Lu, M. Yan and Y.-C. Zhu,
On the set-theoretical Yang–Baxter equation , Duke Math. J. 104 (2000), 1–18.[22] T. Nasybullov,
Connections between properties of the additive and the multiplicative groups of a two-sided skewbrace , J. Algebra 540 (2019), 156–167.[23] I. B. S. Passi, M. Singh and M. K. Yadav,
Automorphisms of finite groups , Springer Monographs in Mathematics.Springer, Singapore, 2018. xix+217 pp.[24] W. Rump,
A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation ,Adv. Math. 193 (2005), 40–55.[25] W. Rump,
Braces, radical rings, and the quantum Yang–Baxter equation , J. Algebra 307 (2007),153–170.[26] A. Smoktunowicz,
A note on set-theoretic solutions of the Yang–Baxter equation , J. Algebra 500 (2018), 3–18.[27] A. Smoktunowicz,
On Engel groups, nilpotent groups, rings, braces and the Yang–Baxter equation , Trans. Amer.Math. Soc. 370 (2018), no. 9, 6535–6564.[28] A. Soloviev,
Non-unitary set-theoretical solutions to the quantum Yang–Baxter equation , Math. Res. Lett. 7(2000), 577–596.[29] L. Vendramin,
Extensions of set-theoretic solutions of the Yang–Baxter equation and a conjecture of Gateva-Ivanova , J. Pure Appl. Algebra 220 (2016), 2064–2076.[30] D. J. S. Robinson,
A Course in the Theory of Groups , Graduate Texts in Mathematics, vol.80 (Springer, NewYork, 1982), xvii+481 pp.[31] C. Wells,
Automorphisms of group extensions , Trans. Amer. Math. Soc. 155 (1971), 189–194.
Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk 630090, Russia.Novosibirsk State Agrarian University, Dobrolyubova street, 160, Novosibirsk, 630039, Russia.
Email address : [email protected] Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER)Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India.
Email address ::