aa r X i v : . [ m a t h . R A ] A ug Construction of Hopf algebroids
Yudai Otsuto ∗ Youichi Shibukawa † Abstract
For arbitrary algebras L , we construct Hopf algebroids A σ withbase rings L by means of σ abcd ∈ L satisfying suitable properties. The quantum group [6, 13] has produced a revival of interest in the Hopfalgebra [1, 21, 34], and much attention is now directed to its generalization[16, 17, 24, 36].After pioneering works [11, 12] by Hayashi about face algebras, alsocalled weak Hopf algebras, there are two generalizations of the Hopf algebra:one is the × L -Hopf algebra [26] and the other is the Hopf algebroid [4, 5],on which we focus in this paper.Motivated by the FRT construction of a q-analogue of the function space[7, 8], one of the author constructed Hopf algebroids A σ [15, 32, 33] bymeans of dynamical Yang-Baxter maps σ [14, 18, 19, 20, 25, 28, 29, 30, 31].This σ is a set-theoretic solution to a version of the quantum dynamicalYang-Baxter equation [9, 10], a generalization of the quantum Yang-Baxterequation [2, 3, 37, 38].Let K be an arbitrary field, H a nonempty and finite set, and let M H ( K )denote the K -algebra consisting of maps from the set H to K . In the aboveconstruction of Hopf algebroids [32], the algebra A σ we first obtained isnothing but a weak Hopf algebra, because the following two conditions areequivalent [27, Section 5] by taking account of the fact that the Hopf al-gebroid is a × L -Hopf algebra [5]: (1) A σ is a weak Hopf algebra; (2) the ∗ Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo0600810, Japan; [email protected] † Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo0600810, Japan; [email protected]: Hopf algebroids.MSC2010: Primary 16T20, 20G42, 81R50; Secondary 20N05, 20N10. M H ( K ) of the Hopf algebroid A σ is separable with an idempotentFrobenius system. Here, for a commutative ring k , we say that a k -algebra L is separable with an idempotent Frobenius system [27, Section 3], iff thereexist a k -linear map φ : L → k and an element e = P e (1) ⊗ e (2) ∈ L ⊗ k L satisfying that l = P φ ( le (1) ) e (2) = P e (1) φ ( e (2) l ) for any l ∈ L and that P e (1) e (2) = 1 L . For construction of the Hopf algebroid, we need an extraprocess [32], in which we change from the finite set H to an infinite one.This change implies that the base ring M H ( K ) is not separable with anidempotent Frobenius system.The purpose of this paper is to give a simpler way to construct Hopfalgebroids that are not weak Hopf algebras, even though the set H is finite.To achieve this purpose, we tried to generalize M H ( K ) to the K -algebra M H ( R ) consisting of maps from the set H to an arbitrary K -algebra R [22].If R is not separable with an idempotent Frobenius system, then so is thealgebra M H ( R ), and the Hopf algebroid A σ with M H ( R ) is not a weak Hopfalgebra as a result (See also Section 7).The aim of this paper is to generalize the algebra M H ( R ) to an arbitraryalgebra L in this construction; and we will consequently clarify propertiesof σ that can produce the Hopf algebroid A σ .The organization of this paper is as follows. Section 2 presents a leftbialgebroid A σ . This algebra A σ is also a right bialgebroid, which is provedin Section 3. In Section 4, we construct an anti-automorphism S on thealgebra A σ , which implies that A σ is a Hopf algebroid in Section 5. Section6 deals with a sufficient condition for the existence of the anti-automorphism S on A σ . In Section 7, we provide with examples of Hopf algebroids A σ bymeans of the sufficient condition in Section 6. A σ In this section, we introduce a left bialgebroid A σ , which is a main subjectof this paper.Let A and L be associative unital rings, and let s L : L → A and t L : L op → A be ring homomorphisms satisfying s L ( l ) t L ( l ′ ) = t L ( l ′ ) s L ( l ) ( ∀ l, l ′ ∈ L ) . (2.1)Here, L op is the opposite ring of L . These two homomorphisms give an L -bimodule structure in A , which is denoted by L A L , through the followingaction: l · a · l ′ = s L ( l ) t L ( l ′ ) a ( l, l ′ ∈ L, a ∈ A ) . (2.2)2or f ∈ L , the notations ρ l ( f ) and ρ r ( f ) respectively mean the left andthe right multiplications by f : ρ l ( f ) : L ∋ g f g ∈ L ; ρ r ( f ) : L ∋ g gf ∈ L. (2.3)These are elements of End k ( L ) and make L an L -bimodule: l · f · l ′ = ρ l ( l ) ρ r ( l ′ ) f ( l, l ′ , f ∈ L ) . Definition 2.1.
Let ( L A L , ∆ L , π L ) be a comonoid in the tensor categoryof L -bimodules. A sextuplet A L = ( A, L, s L , t L , ∆ L , π L ) is called a leftbialgebroid, iff a (1) t L ( l ) ⊗ a (2) = a (1) ⊗ a (2) s L ( l ) , (2.4)∆ L (1 A ) = 1 A ⊗ A , (2.5)∆ L ( ab ) = ∆ L ( a )∆ L ( b ) , (2.6) π L (1 A ) = 1 L , (2.7) π L ( as L ( π L ( b ))) = π L ( ab ) = π L ( at L ( π L ( b ))) (2.8)for any l ∈ L and a, b ∈ A . Here, 1 A is the unit element of the ring A and 1 L is that of the ring L . We use Sweedler’s notation in (2.4): ∆ L ( a ) = a (1) ⊗ a (2) ∈ A ⊗ L A . The right-hand-side of (2.6) is well defined because of(2.4).The left bialgebroid is also called Takeuchi’s × L -bialgebra [35].From now on, the symbols k and L respectively denote a commutativering with the unit 1 k and a k -algebra with the unit 1 L . That is to say, L is a k -module with an associative multiplication that is bilinear and has the unitelement 1 L . The letter G means a group, and, for any α ∈ G , let T α : L → L be a k -algebra automorphism satisfying T α ◦ T α − = id L ( ∀ α ∈ G ) . (2.9)We will denote by deg a map from a finite set X to the group G .Let Gen denote the set ( L ⊗ k L op ) ` { L ab : a, b ∈ X } ` { ( L − ) ab : a, b ∈ X } . Here, L ab and ( L − ) ab are indeterminates, and L op is the opposite alge-bra of L . h Gen i means the monoid consisting of all (finite) words includingthe empty word ∅ whose set of alphabets is Gen . The binary operation ofthis monoid is a concatenation of words. Let σ abcd ∈ L ( a, b, c, d ∈ X ) and wedenote by I σ the two-sided ideal of the free k -algebra k h Gen i = ⊕ w ∈h Gen i kw whose generators are the following. 31) ξ + ξ ′ − ( ξ + ξ ′ ), cξ − ( cξ ), ξξ ′ − ( ξξ ′ ) ( ∀ c ∈ k, ξ, ξ ′ ∈ L ⊗ k L op ).Here, the symbol + in ξ + ξ ′ means the addition in the algebra k h Gen i ,while the symbol + in ( ξ + ξ ′ )( ∈ Gen ) is the addition in the algebra L ⊗ k L op . The notations of the scalar products and products in theother generators are similar.(2) X c ∈ X L ac ( L − ) cb − δ ab ∅ , X c ∈ X ( L − ) ac L cb − δ ab ∅ ( ∀ a, b ∈ X ).Here, δ ab denotes Kronecker’s delta symbol; δ ab = ( k if a = b ;0 k otherwise.(3) ( T deg( a ) ( f ) ⊗ L ) L ab − L ab ( f ⊗ L ),(1 L ⊗ T deg( b ) ( f )) L ab − L ab (1 L ⊗ f ),( f ⊗ L )( L − ) ab − ( L − ) ab ( T deg( b ) ( f ) ⊗ L ),(1 L ⊗ f )( L − ) ab − ( L − ) ab (1 L ⊗ T deg( a ) ( f )) ( ∀ f ∈ L (= L op ) , a, b ∈ X ).(4) P x,y ∈ X ( σ xyac ⊗ L ) L yd L xb − P x,y ∈ X (1 L ⊗ σ bdxy ) L cy L ax ( ∀ a, b, c, d ∈ X ).(5) ∅ − L ⊗ L .Let us denote by A σ the quotient A σ = k h Gen i /I σ . (2.10) Theorem 2.2.
If the elements σ abcd ∈ L satisfy ρ l ( σ bdac ) ◦ T deg( d ) ◦ T deg( b ) = ρ r ( σ bdac ) ◦ T deg( c ) ◦ T deg( a ) (2.11) for any a, b, c, d ∈ X , then the quotient A σ (2 . is a left bialgebroid. The maps s L : L → A σ and t L : L op → A σ are defined as follows: s L : L ∋ l l ⊗ L + I σ ∈ A σ ; (2.12) t L : L op ∋ l L ⊗ l + I σ ∈ A σ . (2.13)These are k -algebra homomorphisms satisfying (2.1), and A σ is consequentlyan L -bimodule with the action (2.2).In order to define the map ∆ L , we need a k -algebra homomorphism∆ : k h Gen i → A σ ⊗ k A σ whose definition on the generators is as follows:∆( ξ ) = ( s L ⊗ k t L )( ξ ) ( ξ ∈ L ⊗ k L op );∆( L ab ) = X c ∈ X L ac + I σ ⊗ L cb + I σ ( a, b ∈ X );∆(( L − ) ab ) = X c ∈ X ( L − ) cb + I σ ⊗ ( L − ) ac + I σ ( a, b ∈ X ) . (2.14)4e write I for the right ideal of A σ ⊗ k A σ whose generators are t L ( l ) ⊗ A σ − A σ ⊗ s L ( l ) ( ∀ l ∈ L ). Proposition 2.3. I is a k -module. If a ∈ I σ is any generator (1) – (5) oftwo-sided ideal I σ , then ∆( a ) ∈ I . In addition, ∆( k h Gen i ) I ⊂ I . From this proposition, ∆( I σ ) ⊂ I , which induces a k -module homo-morphism e ∆ : A σ → ( A σ ⊗ k A σ ) /I . By taking account of the fact that( A σ ⊗ k A σ ) /I ∼ = A σ ⊗ L A σ as Z -modules, this e ∆ induces the Z -modulehomomorphism ∆ L : A σ → A σ ⊗ L A σ , which is also an L -bimodule homo-morphism. Proof of Proposition . . We give the proof only for that ∆( k h Gen i ) I ⊂ I .It is sufficient to show that ∆( v )( t L ( l ) ⊗ A σ − A σ ⊗ s L ( l )) ∈ I for any v ∈ Gen and l ∈ L , since I is a right ideal.If v = ( L − ) ab , then, on account of (2.9) and the generators (3) of theideal I σ , ∆( v )( t L ( l ) ⊗ A σ − A σ ⊗ s L ( l ))= X c ∈ X ( L − ) cb (1 L ⊗ l ) + I σ ⊗ ( L − ) ac + I σ − X c ∈ X ( L − ) cb + I σ ⊗ ( L − ) ac ( l ⊗ L ) + I σ = X c ∈ X (1 L ⊗ T deg( c ) − ( l ))( L − ) cb + I σ ⊗ ( L − ) ac + I σ − X c ∈ X ( L − ) cb + I σ ⊗ ( T deg( c ) − ( l ) ⊗ L )( L − ) ac + I σ = X c ∈ X ( t L ( T deg( c ) − ( l )) ⊗ A σ − A σ ⊗ s L ( T deg( c ) − ( l ))) ×× (( L − ) cb + I σ ⊗ ( L − ) ac + I σ ) ∈ I . The proof is easy for the other v ∈ Gen .The next task is to define the map π L : A σ → L . For this purpose, wefirst construct the k -algebra homomorphism ε : k h Gen i →
End k ( L ), whichis defined on the generators as follows: ε ( L ab ) = δ ab T deg( a ) , ε (( L − ) ab ) = δ ab T deg( a ) − ( a, b ∈ X ) , and ε on L ⊗ k L op is the (unique) k -linear map satisfying ε ( l ⊗ l ′ ) = ρ l ( l ) ρ r ( l ′ )( l, l ′ ∈ L ). For ρ l and ρ r , see (2.3). 5 roposition 2.4. ε ( I σ ) = { } .Proof. It suffices to prove that ε ( a ) = 0 for any generator a of the two-sidedideal I σ . We give the proof only for the case that a is a generator (3) or (4).Because T α is a k -algebra homomorphism satisfying (2.9), ε ( a ) = 0 forthe case that a is a generator (3).By virtue of (2.11), ε ( a ) = 0 for the case that a is a generator (4).According to this proposition, ε induces the k -algebra homomorphism ε : A σ → End k ( L ), and the definition of the map π L is that π L : A σ ∋ a ε ( a )(1 L ) ∈ L. This π L is an L -bimodule homomorphism.The triplet ( A σ , ∆ L , π L ) is a comonoid in the tensor category of L -bimodules. Moreover, the sextuplet A σ = ( A σ , L, s L , t L , ∆ L , π L ) satisfies(2.4)–(2.8) in Definition 2.1, and A σ is thus a left bialgebroid, which com-pletes the proof of Theorem 2.2. A σ In this section, we clarify the condition that makes the algebra A σ in theprevious section to be a right bialgebroid, and show that this conditionimplies (2.11); hence, A σ is a left and right bialgebroid under the condition.Let A and L ′ be associative unital rings, and let s L ′ : L ′ → A and t L ′ : L ′ op → A be ring homomorphisms satisfying s L ′ ( r ) t L ′ ( r ′ ) = t L ′ ( r ′ ) s L ′ ( r ) ( ∀ r, r ′ ∈ L ′ ) . (3.1)These two homomorphisms give an L ′ -bimodule structure in A , which isdenoted by L ′ A L ′ , through the following action: r · a · r ′ = as L ′ ( r ′ ) t L ′ ( r ) ( r, r ′ ∈ L ′ , a ∈ A ) . (3.2) Definition 3.1.
Let ( L ′ A L ′ , ∆ L ′ , π L ′ ) be a comonoid in the tensor categoryof L ′ -bimodules. A sextuplet A L ′ = ( A, L ′ , s L ′ , t L ′ , ∆ L ′ , π L ′ ) is called a rightbialgebroid, iff s L ′ ( r ) a (1) ⊗ a (2) = a (1) ⊗ t L ′ ( r ) a (2) , (3.3)∆ L ′ (1 A ) = 1 A ⊗ A , (3.4)∆ L ′ ( ab ) = ∆ L ′ ( a )∆ L ′ ( b ) , (3.5)6 L ′ (1 A ) = 1 L ′ , (3.6) π L ′ ( s L ′ ( π L ′ ( a )) b ) = π L ′ ( ab ) = π L ′ ( t L ′ ( π L ′ ( a )) b ) (3.7)for any r ∈ L ′ and a, b ∈ A . Here, a (1) ⊗ a (2) is Sweedler’s notation: ∆ L ′ ( a ) = a (1) ⊗ a (2) . The right-hand-side of (3.5) is well defined because of (3.3). Theorem 3.2.
If the elements σ abcd ∈ L satisfy T deg( a ) − ◦ T deg( c ) − ◦ ρ l ( σ bdac ) = T deg( b ) − ◦ T deg( d ) − ◦ ρ r ( σ bdac ) (3.8) for any a, b, c, d ∈ X , then the algebra A σ (2 . is a right bialgebroid for L ′ = L op ( For ρ l and ρ r , see (2 . . The maps s L (2.12) and t L (2.13) define s L op : L op → A σ and t L op : L → A σ : s L op := t L ; t L op := s L . These are k -algebra homomorphisms satisfying(3.1), and A σ is consequently an L op -bimodule with the action (3.2).In order to define the map ∆ L op , we make use of the k -algebra homo-morphism ∆ : k h Gen i → A σ ⊗ k A σ in (2.14). We write I ′ for the left ideal of A σ ⊗ k A σ whose generators are s L op ( l ) ⊗ A σ − A σ ⊗ t L op ( l ) ( ∀ l ∈ L op (= L )).This I ′ is a k -module and ∆( I σ ) ⊂ I ′ , which induces a k -module homomor-phism e ∆ ′ : A σ → ( A σ ⊗ k A σ ) /I ′ . The proof is similar to that of Proposition2.3. Because ( A σ ⊗ k A σ ) /I ′ ∼ = A σ ⊗ L op A σ as Z -modules, this e ∆ ′ impliesthe Z -module homomorphism ∆ L op : A σ → A σ ⊗ L op A σ , which is also an L op -bimodule homomorphism.The next task is to define a map π L op : A σ → L op . We construct the k -algebra anti-homomorphism ε ′ : k h Gen i →
End k ( L op ) whose definition onthe generators is as follows: ε ′ ( L ab ) = δ ab T deg( a ) − , ε ′ (( L − ) ab ) = δ ab T deg( a ) ( a, b ∈ X ) , and ε ′ on L ⊗ k L op is the (unique) k -linear map satisfying ε ′ ( l ⊗ l ′ ) = ρ l ( l ′ ) ρ r ( l )( l, l ′ ∈ L op (= L )). According to the fact that ε ′ ( I σ ) = { } , ε ′ induces the k -algebra anti-homomorphism ε ′ : A σ → End k ( L op ), and the definition ofthe map π L op is the following. π L op : A σ ∋ a ε ′ ( a )(1 L ) ∈ L op . This π L op is an L op -bimodule homomorphism.The triplet ( A σ , ∆ L op , π L op ) is a comonoid in the tensor category of L op -bimodules. In addition, the sextuplet A σ = ( A σ , L op , s L op , t L op , ∆ L op , π L op )satisfies (3.3)–(3.7) in Definition 3.1, and A σ is hence a right bialgebroid.7 roposition 3.3. This right bialgebroid A σ satisfying (3 . is also a leftbialgebroid.Proof. It suffices to prove that (3.8) implies (2.11). From (3.8), T deg( a ) − ◦ T deg( c ) − ◦ ρ l ( σ bdac )(1 L ) = T deg( b ) − ◦ T deg( d ) − ◦ ρ r ( σ bdac )(1 L ) , which is exactly the same as T deg( a ) − ◦ T deg( c ) − ( σ bdac ) = T deg( b ) − ◦ T deg( d ) − ( σ bdac ) . (3.9)By means of (3.8) and (3.9), the left-hand-side of (2.11) is T deg( c ) ◦ T deg( a ) ◦ T deg( a ) − ◦ T deg( c ) − ◦ ρ l ( σ bdac ) ◦ T deg( d ) ◦ T deg( b ) = T deg( c ) ◦ T deg( a ) ◦ T deg( b ) − ◦ T deg( d ) − ◦ ρ r ( σ bdac ) ◦ T deg( d ) ◦ T deg( b ) = T deg( c ) ◦ T deg( a ) ◦ ρ r ( T deg( b ) − ◦ T deg( d ) − ( σ bdac ))= ρ r ( T deg( c ) ◦ T deg( a ) ◦ T deg( b ) − ◦ T deg( d ) − ( σ bdac )) ◦ T deg( c ) ◦ T deg( a ) = ρ r ( σ bdac ) ◦ T deg( c ) ◦ T deg( a ) . This establishes the formula.
Remark . We note that (3.8) is equivalent to (2.11) and (3.9). σ Let σ abcd ∈ L ( a, b, c, d ∈ X ) satisfying (3.9) for any a, b, c, d ∈ X , and wewrite σ = ( σ abcd ) a,b,c,d ∈ X . This section deals with a property of σ that makesthe algebra A σ a Hopf algebroid. Definition 4.1. σ is called rigid (cf. [7, Section 4.5]), iff, for any a, b ∈ X ,there exist x ab , y ab ∈ A σ such that X c ∈ X (( L − ) cb + I σ ) x ac = X c ∈ X x cb (( L − ) ac + I σ )= X c ∈ X ( L cb + I σ ) y ac = X c ∈ X y cb ( L ac + I σ )= δ ab A σ . Proposition 4.2.
The following conditions are equivalent :81) σ is rigid ;(2) There exists a k -algebra anti-automorphism S : A σ → A σ such that S ( f ⊗ L + I σ ) = 1 L ⊗ f + I σ , S (1 L ⊗ f + I σ ) = f ⊗ L + I σ ( ∀ f ∈ L ) ,S ( L ab + I σ ) = ( L − ) ab + I σ ( ∀ a, b ∈ X ) . Proof.
The condition (2) implies (1), if we set x ab = S (( L − ) ab + I σ ) and y ab = S − ( L ab + I σ ) ( a, b ∈ X ).Next we show that the condition (1) induces (2). Because k h Gen i isfree, there uniquely exists a k -algebra homomorphism S : k h Gen i → A opσ such that S ( l ⊗ l ′ ) = l ′ ⊗ l + I σ ( l, l ′ ∈ L (= L op )); S ( L ab ) = ( L − ) ab + I σ ( a, b ∈ X ); S (( L − ) ab ) = x ab ( a, b ∈ X ) . (4.1)This S : k h Gen i → A σ is a k -algebra anti-homomorphism.We claim that the following is true. Claim 4.3. S ( I σ ) = { } . Assuming this claim for the moment, we complete the proof. This claimimmediately induces a k -algebra anti-homomorphism S : A σ → A σ definedby S ( a + I σ ) = S ( a ) ( a ∈ k h Gen i ). This S is the desired one. Proof of Claim . . It is sufficient to show S ( a ) = 0 for any generator ofthe two-sided ideal I σ . We give the proof only for the case that a is thegenerator (4).Let x ′ , y ′ , x ′′ , y ′′ ∈ X . From the generator (4) of I σ ,0 A σ = X a,b,c,d ∈ X (( L − ) x ′′ a + I σ )(( L − ) y ′′ c + I σ ) ×× ( X x,y ∈ X ( σ xyac ⊗ L ) L yd L xb + I σ − X x,y ∈ X (1 L ⊗ σ bdxy ) L cy L cx + I σ ) ×× (( L − ) bx ′ + I σ )(( L − ) dy ′ + I σ ) . (4.2)On account of the generators (2) and (3) of I σ , the right-hand-side of (4.2)9s X a,c ∈ X (( L − ) x ′′ a + I σ )(( L − ) y ′′ c + I σ )( σ x ′ y ′ ac ⊗ L + I σ ) − X b,d ∈ X (( L − ) bx ′ + I σ )(( L − ) dy ′ + I σ ) ×× (1 L ⊗ T deg( d ) T deg( b ) T deg( x ′′ ) − T deg( y ′′ ) − ( σ bdx ′′ y ′′ ) + I σ ) . It follows from (3.9) that T deg( d ) T deg( b ) T deg( x ′′ ) − T deg( y ′′ ) − ( σ bdx ′′ y ′′ ) = σ bdx ′′ y ′′ ,and (4.2) is0 A σ = X a,c ∈ X (( L − ) x ′′ a + I σ )(( L − ) y ′′ c + I σ )( σ x ′ y ′ ac ⊗ L + I σ ) − X b,d ∈ X (( L − ) bx ′ + I σ )(( L − ) dy ′ + I σ )(1 L ⊗ σ bdx ′′ y ′′ + I σ ) . Therefore, by virtue of (4.1), S ( X x,y ∈ X ( σ xyac ⊗ L ) L yd L xb − X x,y ∈ X (1 L ⊗ σ bdxy ) L cy L ax )= X x,y ∈ X S ( L xb ) S ( L yd ) S ( σ xyac ⊗ L ) − X x,y ∈ X S ( L ax ) S ( L cy ) S (1 L ⊗ σ bdxy )=0 A σ , and the proof is complete. A σ In this section, we introduce the notion of the Hopf algebroid and proceedwith the study of the left and right bialgebroid A σ .Let A L = ( A, L, s L , t L , ∆ L , π L ) be a left bialgebroid, together with ananti-automorphism S of the ring A , and let L ′ be a ring isomorphic to theopposite ring L op . We will denote by ν : L op → L ′ this isomorphism.According to (2.2), the ring A has the left L -module structure denotedby L A and the right L -module structure denoted by A L : L A : l · a = s L ( l ) a ; A L : a · l = t L ( l ) a ( l ∈ L, a ∈ A ) . Moreover, the ring A has left and right L ′ -module structures written by L ′ A and A L ′ respectively: L ′ A : r · a = as L ( ν − ( r )); A L ′ : a · r = aS ( s L ( ν − ( r ))) ( r ∈ L ′ , a ∈ A ) .
10f the maps S , s L , and t L satisfy S ◦ t L = s L , (5.1)then there uniquely exists a Z -module map S A ⊗ L A : A L ⊗ L A → A L ′ ⊗ L ′ A such that S A ⊗ L A ( a ⊗ b ) = S ( b ) ⊗ S ( a ) ( a, b ∈ A ).From (5.1), S ( a (1) ) a (2) makes sense. Here, ∆ L ( a ) = a (1) ⊗ a (2) is Sweedler’snotation. If the maps S , s L , t L , and π L satisfy (5.1) and S ( a (1) ) a (2) = t L ◦ π L ◦ S ( a ) ( ∀ a ∈ A ) , (5.2)then there uniquely exists a Z -module map S A ⊗ L ′ A : A L ′ ⊗ L ′ A → A L ⊗ L A such that S A ⊗ L ′ A ( a ⊗ b ) = S ( b ) ⊗ S ( a ) ( a, b ∈ A ).We write ∆ L ′ for S A ⊗ L A ◦ ∆ L ◦ S − . Definition 5.1.
A pair ( A L , S ) of a left bialgebroid and an anti-automorphism S of the ring A satisfying (5.1), (5.2), and(∆ L ⊗ id A ) ◦ ∆ L ′ = (id A ⊗ ∆ L ′ ) ◦ ∆ L , (∆ L ′ ⊗ id A ) ◦ ∆ L = (id A ⊗ ∆ L ) ◦ ∆ L ′ is a Hopf algebroid, iff there exists the inverse S − A ⊗ L ′ A of S A ⊗ L ′ A such that S A ⊗ L A ◦ ∆ L ◦ S − = S − A ⊗ L ′ A ◦ ∆ L ◦ S. If σ satisfies (3 . A σ is a left bialgebroid and a right bialgebroid on ac-count of Proposition 3.3; moreover, (3.9) holds (See the proof of Proposition3.3). Theorem 5.2.
The algebra A σ with the k -algebra anti-automorphism S inProposition . is a Hopf algebroid for a rigid σ satisfying (3 . . The proof of this theorem is similar to that of Theorem 3.9 in [32].
In this section, we continue the study of the rigid σ in Section 4. We willwrite ˜ σ abcd = T deg( d ) − ( σ abcd ) ∈ L for a, b, c, d ∈ X . Theorem 6.1.
Under the following conditions (1) – (5) , σ satisfying (3 . isrigid. (1) For any a, b, c, d ∈ X , there exists i ∗ (˜ σ ) abcd ∈ L such that X a,b ∈ X i ∗ (˜ σ ) wazb ˜ σ byax = X a,b ∈ X ˜ σ wazb i ∗ (˜ σ ) byax = δ wx δ yz L ; 112) For a, b ∈ X , we will write Q ab := X u ∈ X i ∗ (˜ σ ) ubua ∈ L . Then there exists Q − ab ∈ L such that P b ∈ X Q ab Q − bc = δ ac L ;(3) For a, b ∈ X , let Q ′ ab denote the element X u ∈ X i ∗ (˜ σ ) buau ∈ L . Then thereexists Q ′− ab ∈ L such that P b ∈ X Q ′− bc Q ′ ab = δ ac L ;(4) For a, b ∈ X , we will set Q ′′ ab := X u ∈ X T deg( b ) ( i ∗ (˜ σ ) buau ) ∈ L . Then thereexists Q ′′− ab ∈ L such that P b ∈ X Q ′′− ab Q ′′ bc = δ ac L ;(5) For a, b ∈ X , let us denote by Q ′′′ ab the element X u ∈ X T deg( a ) − ( i ∗ (˜ σ ) uaub ) ∈ L . Then there exists Q ′′′− ab ∈ L such that P b ∈ X Q ′′′ ab Q ′′′− bc = δ ac L . From the assumptions (1)–(5) in this theorem, x ab = X c,d ∈ X ( Q ac ⊗ Q − db ) L cd + I σ = X c,d ∈ X ( Q ′′− ac ⊗ Q ′′ db ) L cd + I σ ,y ab = X c,d ∈ X ( Q ′− db ⊗ Q ′ ac )( L − ) cd + I σ = X c,d ∈ X ( Q ′′′ bd ⊗ Q ′′′− ca )( L − ) cd + I σ for any a, b ∈ X .This theorem is proved in much the same way as Theorem 4.1 in [32]. According to the thesis [22], this last section deals with examples of σ =( σ abcd ) a,b,c,d ∈ X satisfying (3.8) and (1)–(5) in Theorem 6.1 (See also the paper[32, Section 4]). Therefore, by means of σ , we can construct the Hopfalgebroid A σ . Definition 7.1.
A non-empty set QG with a binary operation is a quasi-group, iff it satisfies:(1) there uniquely exists the element a ∈ QG such that ab = c for any b, c ∈ QG ; 12able 1: The binary operation on QG b ∈ QG such that ab = c for any a, c ∈ QG .The unique element a ∈ QG in the condition (1) will be denoted by c/b , andthe unique b ∈ QG in (2) will be denoted by a \ c .For example, any group is a quasigroup. However, the quasigroup is notalways associative. Example 7.2.
Let QG = { , , , , } denote the set of five elements,together with the binary operation in Table 1 [32, Section 4]. Here 02 = 2.Because each element of QG appears once and only once in each row andin each column of Table 1, QG with this binary operation is a quasigroup[23, Theorem I.1.3]. This is not associative, since (12)3 = 1 = 4 = 1(23).Let QG be a finite quasigroup with at least two elements, M a setisomorphic to the set QG , and µ : M × M × M → M a ternary operationon the set M . We write π : QG → M for a bijection between the sets QG and M . The requirement on the ternary operation µ is the following:(QG1) µ ( a, µ ( a, b, c ) , µ ( µ ( a, b, c ) , c, d )) = µ ( a, b, µ ( b, c, d )) for any a, b, c, d ∈ M ;(QG2) µ ( µ ( a, b, c ) , c, d ) = µ ( µ ( a, b, µ ( b, c, d )) , µ ( b, c, d ) , d ) for any a, b, c, d ∈ M ;(QG3) for any b, c, d ∈ M , there uniquely exists a ∈ M such that µ ( a, b, c ) = d ;(QG4) for any a, c, d ∈ M , there uniquely exists b ∈ M such that µ ( a, b, c ) = d ; 13QG5) for any a, b, d ∈ M , there uniquely exists c ∈ M such that µ ( a, b, c ) = d .For any finite quasigroup QG satisfying | QG | > M isomorphic to the set QG , the ternary operation µ on M defined by µ ( a, b, c ) = a − b + c ( a, b, c ∈ M ) enjoys all of the above conditions.We set H := QG and X := QG . Let G denote the opposite group ofthe symmetric group on the set H . For a ∈ QG , deg( a ) ∈ G is given by λ deg( a ) = λa ( λ ∈ H = QG ).Let R be a k -algebra. Here k is a commutative ring. We define the map σ abcd : H → R by σ abcd ( λ ) = R , if c = π − ( µ ( π ( λ ) , π ( λb ) , π (( λb ) a ))) \ (( λb ) a )and d = λ \ π − ( µ ( π ( λ ) , π ( λb ) , π (( λb ) a )));0 R , otherwise . Let L denote the k -algebra consisting of maps from the set H to R . Theproduct of this k -algebra L is defined by ( f g )( λ ) = f ( λ ) g ( λ ) ( f, g ∈ L, λ ∈ H ). It follows that L ∼ = R | H | as k -algebras. Theorem 7.3.
This σ = ( σ abcd ) a,b,c,d ∈ X is rigid. The proof of this theorem is straightforward; in fact, we can show that σ satisfies (3.8) and (1)–(5) in Theorem 6.1. Corollary 7.4. A σ is a Hopf algebroid. If R is not separable with an idempotent Frobenius system (For thedefinition, see Introduction), then so is L ( ∼ = R | H | ). As a result, A σ is not aweak Hopf algebra on account of [27, Theorem 5.1]. Acknowledgments
The work was supported in part by JSPS KAKENHI Grant Number JP17K05187.
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