Dynamical Yang-Baxter maps and Hopf algebroids associated with s-sets
aa r X i v : . [ m a t h . QA ] J un Dynamical Yang-Baxter maps and Hopf algebroidsassociated with s-sets
Noriaki Kamiya ∗ Youichi Shibukawa † June 8, 2016
Abstract
An s-set is an algebraic generalization of the regular s-manifoldintroduced by Kowalski, one of the generalized symmetric spaces indifferential geometry. We prove that suitable s-sets give birth to dy-namical Yang-Baxter maps, set-theoretic solutions to a version of thequantum dynamical Yang-Baxter equation. As an application, Hopfalgebroids and rigid tensor categories are constructed by means of thesedynamical Yang-Baxter maps.
The quantum Yang-Baxter equation [2, 3, 35, 36] is closely related to alge-braic structures, for example, the quantum group [7, 12], the Hopf algebra[1, 21, 33], and the triple system [22]. Analogously, the quantum dynamicalYang-Baxter equation [10, 11], a generalization of this equation, producesHopf algebroids [4, 5, 6, 17, 18, 24, 25, 34]. In fact, Felder’s dynamicalR-matrix [10], a solution to the quantum dynamical Yang-Baxter equation,yields a Hopf algebroid called the elliptic quantum group [8] through theFaddeev-Reshetikhin-Takhtajan construction [9].In a similar way, suitable dynamical Yang-Baxter maps [26, 28], set-theoretic solutions to a version of the quantum dynamical Yang-Baxter ∗ Center for Mathematical Sciences, University of Aizu, Aizuwakamatsu, Fukushima9658580, Japan † Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo0600810, JapanKeywords: Dynamical Yang-Baxter maps; Hopf algebroids; S-sets; Rigid tensor cate-gories; Homogeneous pre-systems.MSC2010: Primary 16T25, 18D10, 53C35, 81R50; Secondary 20N05, 20N10, 53C30. C ∞ -manifold M with a differentiable multiplication M × M ∋ ( x, y ) x · y ∈ M such that the maps s x : M ∋ y x · y ∈ M satisfy the following:(1) s x ( x ) = x for any x ∈ M ;(2) every map s x is a diffeomorphism;(3) s x ◦ s y = s s x ( y ) ◦ s x for any x, y ∈ M ;(4) for each x ∈ M , the tangent map ( s x ) ∗ x : T x ( M ) → T x ( M ) has nofixed vector except the null vector.The aim of this paper is to construct Hopf algebroids and rigid tensorcategories by introducing a notion of the s-set (Definition 3.1), a general-ization of the regular s-manifold from the algebraic point of view. In thisconstruction, suitable s-sets give birth to the dynamical Yang-Baxter mapsthrough the homogeneous pre-systems. This paper gives another way torelate the dynamical Yang-Baxter map to differential geometry.The organization of this paper is as follows. In Section 2, we give a briefexposition of the homogeneous pre-system, the dynamical Yang-Baxter map,the Hopf algebroid, and the rigid tensor category [13, 30]. Section 3 discussesthe construction of ternary operations by means of the s-sets. In Section 4,we apply the results of Section 3 to get the dynamical Yang-Baxter maps viathe homogeneous pre-systems from suitable s-sets. The last section, Section5, is devoted to the study of the Hopf algebroids associated with the above2ynamical Yang-Baxter maps. Each Hopf algebroid can produce the rigidtensor category consisting of finite-dimensional L-operators. In this section, we summarize without proofs the relevant material on ho-mogeneous pre-systems, dynamical Yang-Baxter maps, Hopf algebroids, andrigid tensor categories [13, 30], to render this paper as self-contained as pos-sible.We first introduce quasigroups [23, 32].
Definition 2.1.
A nonempty set Q with a binary operation Q × Q ∋ ( u, v ) uv ∈ Q is called a quasigroup, iff:(1) for any u, w ∈ Q , there uniquely exists v ∈ Q such that uv = w ;(2) for any v, w ∈ Q , there uniquely exists u ∈ Q such that uv = w .That is to say, the left and the right translations on the quasigroup areboth bijective. On account of this fact, we define the map \ : Q × Q → Q by v = u \ w ⇔ uv = w ( u, v, w ∈ Q ) . (2.1)Any group is a quasigroup; on the other hand, the quasigroup is notalways associative. Example 2.2 ([30]) . Let Q := { , , , , } . We define a binary operationon this set Q by Table 1. Here 0 4 = 0. Each element of Q appears onceand only once in each row and in each column of Table 1, and this set Q ishence a quasigroup [23, Theorem I.1.3]. The binary operation on Q is notassociative, because (12)3 = 1 = 4 = 1(23). Definition 2.3.
A pair (
S, η ) of a nonempty set S and a ternary operation η : S × S × S → S is called a homogeneous pre-system [13], iff the ternaryoperation η satisfies: for any x, y, u, v, w ∈ S , η ( x, y, x ) = y ; η ( x, y, η ( u, v, w )) = η ( η ( x, y, u ) , η ( x, y, v ) , η ( x, y, w )) . (2.2)3able 1: The binary operation on Q [30].0 1 2 3 40 4 3 2 1 01 3 1 0 2 42 0 2 3 4 13 1 0 4 3 24 2 4 1 0 3This homogeneous pre-system ( S, η ) satisfying η ( x, y, z ) = η ( w, η ( x, y, w ) , z ) ( ∀ x, y, z, w ∈ S ) , (2.3)together with a suitable quasigroup, can produce a dynamical Yang-Baxtermap [13]. Let H and X be nonempty sets with a map H × X ∋ ( λ, x ) λx ∈ H . Definition 2.4.
A map σ ( λ ) : X × X → X × X ( λ ∈ H ) is a dynamicalYang-Baxter map, iff σ ( λ ) satisfies a version of the quantum dynamicalYang-Baxter equation σ ( λX (3) ) σ ( λ ) σ ( λX (3) ) = σ ( λ ) σ ( λX (3) ) σ ( λ ) ( ∀ λ ∈ H ) . (2.4)Here, the maps σ ( λX (3) ) , σ ( λ ) : X × X × X → X × X × X are definedby σ ( λX (3) )( x, y, z ) = ( σ ( λz )( x, y ) , z ); σ ( λ )( x, y, z ) = ( x, σ ( λ )( y, z )) . Remark . For a map σ ( λ ) : X × X → X × X ( λ ∈ H ), we set R ( λ )( x, y ) := σ ( λ )( y, x ) ( λ ∈ H, x, y ∈ X ). Then the following conditions are equivalent:(1) the map σ ( λ ) satisfies (2.4);(2) the map R ( λ ) satisfies R ( λ ) R ( λX (2) ) R ( λ ) = R ( λX (3) ) R ( λ ) R ( λX (1) ) (2.5)for any λ ∈ H [26, (2.1)].Throughout this paper, both (2.4) and (2.5) are called versions of the quan-tum dynamical Yang-Baxter equation (see also Remark 2.9).4et ( S, η ) be a homogeneous pre-system satisfying (2.3), and let Q bea quasigroup, isomorphic to S as sets. We denote by π : Q → S the(set-theoretic) bijection that gives this isomorphism. We define the ternaryoperation µ on S by µ ( a, b, c ) = η ( b, a, c ) ( a, b, c ∈ S ) . (2.6) Proposition 2.6.
The ternary operation µ satisfies : µ ( a, µ ( a, b, c ) , µ ( µ ( a, b, c ) , c, d )) = µ ( a, b, µ ( b, c, d )) ( ∀ a, b, c, d ∈ S ); µ ( µ ( a, b, c ) , c, d ) = µ ( µ ( a, b, µ ( b, c, d )) , µ ( b, c, d ) , d ) ( ∀ a, b, c, d ∈ S ) . For λ ∈ Q , we define the map σ ( λ ) : Q × Q → Q × Q by σ ( λ )( u, v ) = ( h ( λ, v, u ) \ (( λv ) u ) , λ \ h ( λ, v, u )) . Here, h ( λ, v, u ) = π − ( µ ( π ( λ ) , π ( λv ) , π (( λv ) u ))) ( λ, u, v ∈ Q ) and see (2.1)for h ( λ, v, u ) \ (( λv ) u ). Proposition 2.6 implies (2.4), and we have the follow-ing as a result. Proposition 2.7.
The map σ ( λ ) is a dynamical Yang-Baxter map. Furthermore, we assume that this dynamical Yang-Baxter map σ ( λ )satisfies:(1) the set S is finite (and so is the set Q );(2) for any b, c, d ∈ S , there uniquely exists a ∈ S such that µ ( a, b, c ) = d ; (2.7)(3) for any a, c, d ∈ S , there exists a unique solution b ∈ S to (2 . a, b, d ∈ S , there exists a unique solution c ∈ S to (2 . σ ( λ ) produces the Hopf algebroid A σ [30,Sections 3 and 4]. We will briefly describe it as below.Let K be an arbitrary field, and let M Q denote the K -algebra of all K -valued maps on the set Q . We define a map T a : M Q → M Q ( a ∈ Q ) by T a ( f )( λ ) = f ( λa ) ( f ∈ M Q , λ, a ∈ Q ) . (2.8)Let L ab , ( L − ) ab ( a, b ∈ Q ) be indeterminates. We define the set AQ by AQ := ( M Q ⊗ K M Q ) G { L ab | a, b, ∈ Q } G { ( L − ) ab | a, b, ∈ Q } .A σ is the quotient of the free K -algebra K h AQ i on the set AQ by two-sidedideal I σ whose generators are: 51) ξ + ξ ′ − ( ξ + ξ ′ ), cξ − ( cξ ), ξξ ′ − ( ξξ ′ ) ( c ∈ K , ξ, ξ ′ ∈ M Q ⊗ K M Q ).Here the symbol + in ξ + ξ ′ means the addition in the algebra K h AQ i ,while the symbol + in ( ξ + ξ ′ )( ∈ AQ ) is the addition in the algebra M Q ⊗ K M Q . The notations of the scalar products and products in theother generators are similar.(2) X c ∈ Q L ac ( L − ) cb − δ ab ∅ , X c ∈ Q ( L − ) ac L cb − δ ab ∅ ( a, b ∈ Q ).Here δ ab denotes Kronecker’s delta symbol.(3) ( T a ( f ) ⊗ M Q ) L ab − L ab ( f ⊗ M Q ),(1 M Q ⊗ T b ( f )) L ab − L ab (1 M Q ⊗ f ),( f ⊗ M Q )( L − ) ab − ( L − ) ab ( T b ( f ) ⊗ M Q ),(1 M Q ⊗ f )( L − ) ab − ( L − ) ab (1 M Q ⊗ T a ( f )) ( f ∈ M Q , a, b ∈ Q ).Here 1 M Q defined by 1 M Q ( λ ) = 1 ( λ ∈ Q ) is the unit of M Q (for T a ,see (2.8)).(4) P x,y ∈ Q ( σ xyac ⊗ M Q ) L yd L xb − P x,y ∈ Q (1 M Q ⊗ σ bdxy ) L cy L ax ( a, b, c, d ∈ Q ).Here σ xyac ∈ M Q is defined by σ xyac ( λ ) = ( , if σ ( λ )( x, y ) = ( a, c );0 , otherwise . (5) ∅ − M Q ⊗ M Q . Proposition 2.8.
This algebra A σ is a Hopf algebroid. For details of the Hopf algebroid including its definition, see [4, 5].This Hopf algebroid A σ produces the rigid tensor category Rep V G ( σ ) f consisting of finite-dimensional L-operators associated with the dynamicalYang-Baxter map σ [30, 31]. For the definition of Rep V G ( σ ) f , we needa tensor category V G . Here, G is the opposite group of the group of allpermutations on the set Q .An object of the category V G is a G -graded K -vector space V = ⊕ α ∈ G V α ;and its morphism f : V → W is a map f : Q → Hom K ( V, W ) satisfying f ( λ ) V α ⊂ M β ∈ G,β ( λ )= α ( λ ) W β λ ∈ Q , where Hom K ( V, W ) is the K -vector space of K -linear mapsfrom V to W . In addition, the composition f g of morphisms f and g isdefined by ( f g )( λ ) := f ( λ ) ◦ g ( λ ) ( λ ∈ Q ). V G is a tensor category. The tensor product V ⊗ W of objects V = ⊕ β ∈ G V β and W ⊕ γ ∈ G W γ is V ⊗ W = ⊕ α ∈ G ( V ⊗ W ) α , where ( V ⊗ W ) α := ⊕ β,γ ∈ G,α = γβ V β ⊗ K W γ . Here, γβ ( ∈ G ) is the multiplication of γ and β inthe group G . In addition, the tensor product f ⊗ g of morphisms f : U → V and g : W → Y is a map ( f ⊗ g )( λ ) = P α ∈ G ( P β,γ ∈ G,α = γβ ( f ⊗ g )( λ ) β,γ )( λ ∈ Q ). Here, ( f ⊗ g )( λ ) β,γ ∈ Hom K ( U β ⊗ K W γ , V ⊗ Y ) is defined by( f ⊗ g )( λ ) β,γ ( u β ⊗ w γ ) := f ( γ ( λ ))( u β ) ⊗ g ( λ )( w γ ) ( u β ∈ U β , w γ ∈ W γ ) . The unit I is the K -vector space K = ⊕ α ∈ G I α with I α = ( K , if α = 1 G (= id Q ); { } , otherwise . The left and right unit constraints with respect to this unit I are defined by l V ( λ ) = ⊕ α ∈ G l V ( λ ) α and r V ( λ ) = ⊕ α ∈ G r V ( λ ) α ( λ ∈ Q ). Here, l V ( λ ) α : ( I ⊗ V ) α = I G ⊗ K V α (= K ⊗ K V α ) ∋ a ⊗ v av ∈ V α ⊂ V ; r V ( λ ) α : ( V ⊗ I ) α = V α ⊗ K I G (= V α ⊗ K K ) ∋ v ⊗ a av ∈ V α ⊂ V. We are now in a position to construct the rigid tensor category Rep V G ( σ ) f .For a ∈ Q , we define deg( a ) ∈ G by deg( a )( λ ) := λa ( λ ∈ Q ). This definitionis unambiguous, because Q is a quasigroup. Let K Q denote the K -vectorspace with the basis Q . This K Q is an object of V G , since K Q = ⊕ α ∈ G ( K Q ) α and ( K Q ) α = ( K a ( ∼ = K ) , if ∃ a ∈ G such that α = deg( a ); { } , otherwise.This is well defined on account of the definition of the quasigroup. We canregard the map σ ( λ ) : Q × Q → Q × Q ( λ ∈ Q ) as a K -linear map on K Q ⊗ K K Q , and this σ : K Q ⊗ K Q → K Q ⊗ K Q is a morphism of thecategory V G . Remark . A version of the quantum dynamical Yang-Baxter equation(2.4) for the dynamical Yang-Baxter map σ ( λ ) is exactly the same as thebraid relation in the category V G [30, Proposition 4.5].The object of Rep V G ( σ ) f is a pair of an object V ∈ V G of finite dimen-sions and an isomorphism L V : V ⊗ K Q → K Q ⊗ V of V G satisfying( σ ⊗ id V )(id K Q ⊗ L V )( L V ⊗ id K Q ) = (id K Q ⊗ L V )( L V ⊗ id K Q )(id V ⊗ σ ) . f : ( V, L V ) → ( W, L W ) is a morphism f : V → W of V G suchthat (id K Q ⊗ f ) L V = L W ( f ⊗ id K Q ).The tensor product ( V, L V ) ⊠ ( W, L W ) is ( V ⊗ W, ( L V ⊗ id W )(id V ⊗ L W )),and the tensor product of morphisms is exactly the same as that of V G : f ⊠ g := f ⊗ g . The unit is ( I, r − K Q l K Q ).From [30, Section 5], we have the following. Proposition 2.10.
Rep V G ( σ ) f is a rigid tensor category. Let M be a non-empty set and s x : M → M a bijection for each x ∈ M .An s-set ( M, { s x } x ∈ M ) is a generalization of the regular s-manifold [15] (seeIntroduction) in the generalized symmetric spaces from the algebraic pointof view. Definition 3.1.
A pair ( M, { s x } x ∈ M ) is an s-set, iff the bijections s x satisfy s x ◦ s y = s s x ( y ) ◦ s x (3.1)for any x, y ∈ M .For a simple example, we note that any group G makes an s-set ( G, { s x } ).Here, the bijection s x ( x ∈ G ) is defined by s x ( y ) = xyx − ( y ∈ G ).In this section, we will show that every s-set ( M, { s x } ) can produceternary operations η on the set M satisfying (2 . I = ( i , i , . . . , i l ) ∈ Z l ( l ≥ w I ( X, Y ) = ( X i Y i · · · X i l − Y i l , if l is even; X i Y i · · · Y i l − X i l , if l is odd . Here, w I ( X, Y ) is an element of the quotient of the free algebra on theset { X, X − , Y, Y − } by the two-sided ideal whose generators are XX − − , X − X − , Y Y − −
1, and Y − Y − η I denote the ternary operation on an s-set ( M, { s x } ) defined by η I ( x, y, z ) = w I ( s x , s y )( z ) ( x, y, z ∈ M ) . (3.2) Theorem 3.2.
The ternary operation η := η I satisfies (2 . . roof. The proof is by induction on the length l ( w I ) := | i | + | i | + · · · + | i l | of the word w I ( X, Y ).If l ( w I ) = 0, then w I is an empty word, and η I ( x, y, z ) = w I ( s x , s y )( z ) = z as a result. An easy computation shows (2.2).If l ( w I ) = 1, then w I ( X, Y ) =
X, X − , Y, Y − . We give the proof onlyfor the case that w I ( X, Y ) = X − . Because w I ( X, Y ) = X − , η I ( x, y, z ) = s − x ( z ). By substituting s − x ( u ) into y in (3.1), s x s s − x ( u ) = s u s x , (3.3)and consequently, the right-hand-side of (2.2) is s − x s − u ( w ), which is exactlythe left-hand-side of (2.2).If l ( w I ) ≥
2, then there exists a word w ′ ( X, Y ) whose length is lessthan l ( w I ) such that w I ( X, Y ) = Xw ′ ( X, Y ), or w I ( X, Y ) = X − w ′ ( X, Y ),or w I ( X, Y ) =
Y w ′ ( X, Y ), or w I ( X, Y ) = Y − w ′ ( X, Y ). For example,if w I ( X, Y ) = X − Y XY , then we set w ′ ( X, Y ) = X − Y XY , whichsatisfies w I ( X, Y ) = X − w ′ ( X, Y ).We prove only for the case that w I ( X, Y ) = X − w ′ ( X, Y ). Since w I ( X, Y ) = X − w ′ ( X, Y ), η I ( x, y, z ) = s − x w ′ ( s x , s y )( z ), and the right-hand-side of (2.2)is s − s − x w ′ ( s x ,s y )( u ) w ′ ( s s − x w ′ ( s x ,s y )( u ) , s s − x w ′ ( s x ,s y )( v ) ) s − x w ′ ( s x , s y )( w ) . (3.4)Substituting w ′ ( s x , s y )( u ) into u in (3.3) gives s s − x w ′ ( s x ,s y )( u ) = s − x s w ′ ( s x ,s y )( u ) s x , (3.5)and, in the same manner, we can see that s s − x w ′ ( s x ,s y )( v ) = s − x s w ′ ( s x ,s y )( v ) s x .Now (3.4) becomes s − s − x w ′ ( s x ,s y )( u ) w ′ ( s − x s w ′ ( s x ,s y )( u ) s x , s − x s w ′ ( s x ,s y )( v ) s x ) s − x w ′ ( s x , s y )( w ) . (3.6) Lemma 3.3.
Let
X, Y, Z, X − , Y − , Z − be indeterminates satisfying XX − = X − X = Y Y − = Y − Y = ZZ − = Z − Z = 1 , and let w ( X, Y ) be a word of X , X − , Y , and Y − . Then w ( ZXZ − , ZY Z − ) = Zw ( X, Y ) Z − . Here, we regard w ( X, Y ) as an element of the quotient ofthe free algebra on the set { X, X − , Y, Y − , Z, Z − } by the two-sided idealwhose generators are XX − − , X − X − , Y Y − − , Y − Y − , ZZ − − ,and Z − Z − . ZX ± Z − ) i = ZX ± i Z − for any i ∈ Z .On account of this lemma, (3.6) is s − s − x w ′ ( s x ,s y )( u ) s − x w ′ ( s w ′ ( s x ,s y )( u ) , s w ′ ( s x ,s y )( v ) ) w ′ ( s x , s y )( w ) . (3.7)We define the ternary operation η ′ on M by η ′ ( x, y, z ) = w ′ ( s x , s y )( z )( x, y, z ∈ M ). Because of the fact that w ′ ( s w ′ ( s x ,s y )( u ) , s w ′ ( s x ,s y )( v ) ) w ′ ( s x , s y )( w ) = η ′ ( η ′ ( x, y, u ) , η ′ ( x, y, v ) , η ′ ( x, y, w )) , the induction hypothesis, and (3.5), (3.7) is s − s − x w ′ ( s x ,s y )( u ) s − x η ′ ( x, y, η ′ ( u, v, w ))= s − x s − w ′ ( s x ,s y )( u ) w ′ ( s x , s y ) w ′ ( s u , s v )( w ) . (3.8)We will prove the following claim later. Claim 3.4.
For any x, y, u ∈ M , s w ′ ( s x ,s y )( u ) ◦ w ′ ( s x , s y ) = w ′ ( s x , s y ) ◦ s u . (3.9)By virtue of this claim, the right-hand-side of (3.8) is exactly the sameas the left-hand-side of (2.2), which is the desired conclusion. Proof of Claim . . The proof is similar to that of the theorem; we willprove it by induction on the length l ( w ′ ) of the word w ′ ( X, Y ).The proof of the cases that l ( w ′ ) = 0 , l ( w ′ ) ≥
2, then there exists a word w ′′ ( X, Y ) whose length is less than l ( w ′ ) such that w ′ ( X, Y ) = Xw ′′ ( X, Y ), or w ′ ( X, Y ) = X − w ′′ ( X, Y ), or w ′ ( X, Y ) =
Y w ′′ ( X, Y ), or w ′ ( X, Y ) = Y − w ′′ ( X, Y ). We prove only forthe case w ′ ( X, Y ) = X − w ′′ ( X, Y ). Since w ′ ( X, Y ) = X − w ′′ ( X, Y ), theleft-hand-side of (3.9) is s s − x w ′′ ( s x ,s y )( u ) s − x w ′′ ( s x , s y ) . (3.10)By substituting w ′′ ( s x , s y )( u ) into u in (3.3), (3.10) is s − x s w ′′ ( s x ,s y )( u ) w ′′ ( s x , s y ).By the induction hypothesis, this is s − x w ′′ ( s x , s y ) s u = w ′ ( s x , s y ) s u , whichis exactly the right-hand-side of (3.9). This establishes the formula.10 Dynamical Yang-Baxter maps from s-sets
This section is devoted to the construction of the dynamical Yang-Baxtermaps (Definition 2.4) via the homogeneous pre-systems (Definition 2.3) bymeans of suitable s-sets (Definition 3.1).Let R be a ring with the unit 1( = 0), M a left R -module, and r aninvertible element of the ring R . We define s x : M → M ( x ∈ M ) by s x ( y ) = (1 − r ) x + ry ( y ∈ M ) . Proposition 4.1. ( M, { s x } ) is an s-set. In fact, the inverse of the map s x is s − x ( y ) = (1 − r − ) x + r − y, and it is a simple matter to show (3.1).Let I = ( i , i , . . . , i l ) ∈ Z l ( l ≥ I ( X ) the followingpolynomial of the variables X and X − .Φ I ( X ) = ( P lj =1 ( − j X P jm =1 i m , if l is even;1 + P l − j =1 ( − j X P jm =1 i m , if l is odd . (4.1) Proposition 4.2. η I ( x, y, z ) = (Φ I ( r ) − r d ) x + (1 − Φ I ( r )) y + r d z for any x, y, z ∈ M . Here, η I ( x, y, z ) is defined in (3 . and d := i + i + · · · + i l .Proof. The proof of the proposition is by induction on l . For the proof ofthe l = 2 case, we need Lemma 4.3.
For any integer i , s ix ( z ) = (1 − r i ) x + r i z ( ∀ x, z ∈ M ) . As a corollary of this lemma, s ix s jy ( z ) = (1 − r i ) x + ( r i − r i + j ) y + r i + j z ( i, j ∈ Z , x, y, z ∈ M ) , which immediately induces the l = 2 case. The rest of the proof is straight-forward.As a corollary, we find Corollary 4.4.
If the invertible element r ∈ R satisfies that Φ I ( r ) = 0 in R , then ( M, η ) is a homogeneous pre-system ( see Definition . satisfying (2 . . η I ( x, y, z ) = − r d x + y + r d z ( x, y, z ∈ M ) , (4.2)if Φ I ( r ) = 0. Example 4.5.
Let k ( ≥
2) be a positive integer, and let r ( ∈ C ) be a primitive k -th root of unity. We denote by Φ k the cyclotomic polynomial of level k .If k = 6, then Φ I ( X ) := Φ ( X ) = 1 − X + X satisfies (4.1) ( I = (1 , I ( r ) = 0, any C -vector space V produces a homogeneous pre-system satisfying (2 .
3) on account of Corollary 4.4.If k = 10 , , , , , , , , , , , , , , , ,
40, then anyprimitive k -th root r of unity can also give birth to a homogeneous pre-system satisfying (2 . Remark . The s-set ( V, { s x } ) for any finite-dimensional C -vector space V in the above example is a regular s-manifold of order k (see [15, DefinitionII.43]). Here, we regard V as an R -vector space of 2 dim C V dimensions. Example 4.7.
Let R := Z / Z . The invertible element r := 2 ∈ R is aroot of Φ (2 , ( X ) = 1 − X + X , and hence r with any (left) R -module M gives a homogeneous pre-system satisfying (2 .
3) according to Corollary 4.4.In fact, η (2 , ( x, y, z ) = − x + y + 3 z . In a similar fashion, the invertibleelement 3 ∈ R also provides with such a homogeneous pre-system, because3 is a root of Φ (2 , − ( X ) = 1 − X + X . Example 4.8.
Let R := Z / Z . The invertible element r := 2 ∈ R is also aroot of Φ (2 , , ( X ) = Φ (2 , ( X ) = 1 − X + X , and hence r with any (left) R -module M gives a homogeneous pre-system satisfying (2 .
3) according toCorollary 4.4. In this case, η (2 , , ( x, y, z ) = − x + y + z , because r d = r = 1in R . On account of (2.6), this η (2 , , yields the ternary operation µ in [30,(4 . Q be a quasigroup (Definition 2.1), isomorphic to the set M as sets.If the element r ( ∈ R ) satisfies Φ I ( r ) = 0, then it follows from Proposition2.7 and Corollary 4.4 that this s-set ( M, { s x } ) with the quasigroup Q givesbirth to the dynamical Yang-Baxter map σ ( λ ) (Definition 2.4). Let R be a ring with the unit 1( = 0), M a left R -module, r ( ∈ R ) an invert-ible element satisfying Φ I ( r ) = 0 (for Φ I , see (4.1)), and Q a quasigroup,isomorphic to the set M as sets (see Definition 2.1).12n this last section, we will construct Hopf algebroids by means of thedynamical Yang-Baxter maps σ ( λ ) in Section 4. In order to define the Hopfalgebroid, we restrict our attention to the case that M is finite and that | M | > | Q | > R = Z / Z in Examples4.7 and 4.8 is a field and that any (left) R -module M is an R -vector space.Hence, each M ( = { } ) in Examples 4.7 and 4.8 of finite dimensions is finiteand satisfies | M | > Proposition 5.1.
The ternary operation µ (2 . on M satisfies :(1) for any b, c, d ∈ M , there exists a unique solution a ∈ M to (2 . for any a, c, d ∈ M , there exists a unique solution b ∈ M to (2 . for any a, b, d ∈ M , there exists a unique solution c ∈ M to (2 . . The proof is clear from (4.2) and the fact that the element r is invertiblein the ring R .By virtue of Proposition 2.8, we have the following. Theorem 5.2. A σ is a Hopf algebroid. The homogeneous pre-system η (2 , , in Example 4.8 yields a Hopf alge-broid A σ in [30].Furthermore, we have the following from Proposition 2.10. Theorem 5.3.
Rep V G ( σ ) f is a rigid tensor category. In view of Proposition 5.1 (2) (see [30, Proposition 4.5]), ( K Q, σ ) ∈ Rep V G ( σ ) f , and this object is not the unit of Rep V G ( σ ) f , because | Q | > V G ( σ ) f is hence non-trivial (see [31, Remark3.9]). Acknowledgments
The second author was supported in part by KAKENHI (26400031).
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