On Non-trivially Associated Tensor Categories EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 4, 2018, 1027-1045 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On Non-trivially Associated Tensor Categories B. Al-harbi1, W. M. Fakieh2, M. M. Al-Shomrani2,∗ 1 Department of Mathematics, Al-Baha University, Al-Baha, Saudi Arabia 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia Abstract. The purpose of this article is to provide mathematical formulas for some operations on the objects of a non-trivially associated tensor category constructed from a factorization of a group into a subgroup and a set of left coset representatives. A detailed example is provided. 2010 Mathematics Subject Classifications: 16W50, 13A02, 16D25 Key Words and Phrases: Non-trivially associated tensor categories, algebras in tensor cate- gories, coalgebras in tensor categories, dual of algebras and coalgebras 1. Introduction In [4], Beggs form a set M of left coset representatives for the left action of a sub- group G of a group X on the group X. Moreover, he defined an operation on M which has a left identity and satisfies the right division property. This binary operation is not associative. However, associativity can be obtained by a ”cocycle” τ : M ×M −→ G. By using this cocycle, one can construct a non-trivial associator for a category C whose objects are the M -graded right representations of G. Every object in this category has a dual. Consequently, it is possible to define an evaluation and a coevaluation maps to make the category into a rigid tensor category. If we assume that the binary operation on M satisfies the left division property, then the grading and group action can be combined into the action of an algebra A on the objects in the category. It turns out that A itself is in C, and that the multiplication is associative. It is well known that for every factorization X = GM of a group into two subgroups G and M , a Hopf algebra H = KMBJ K(G) can be constructed, where K is a field, KM is the group Hopf algebra of M and K(G) is the Hopf algebra functions on G. In the symbol KMBJ K(G), the B part means that KM acts on K(G), and the J part means that K(G) coacts on KM , [3]. Moreover, if A is an algebra (resp. a coalgebra) in a rigid ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3222 Email addresses: b.s.alharbi@hotmail.com (B. Al-harbi), wfakieh@kau.edu.sa (W. Fakieh), malshomrani@hotmail.com (M. Al-Shomrani) http://www.ejpam.com 1027 c© 2018 EJPAM All rights reserved. B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1028 tensor category, then its dual A∗ is a coalgebra (resp. an algebra) in the same category. In [1], Al-shomrani reproved this result by using specific definitions in terms of diagrams that had been used in [2], [5], [7]and [8] . In this article we obtain mathematical formulas for some operations on the objects of a non-trivially associated tensor category constructed from a factorization of a group into a subgroup and a set of left coset representatives. We consider the same non-trivially associated tensor category C as defined in [4]. Throughout this article, we use the same formulas and ideas from [4] which is itself based on [3], [5] and [6], but is mostly self-contained in terms of notation and definitions. In addition, we assume that all groups mentioned, unless otherwise stated, are finite, and that all vector spaces are finite dimensional over a field k, which will be denoted by 1 as an object in the category. Moreover we are going to restrict ourselves to the finite case of algebras, coalgebras and Hopf algebras although many results are still true in the infinite case (see [10]). 2. Preliminaries In this section, we include some definitions and results that will be used later in this article. Definition 2.1. [11] A K-algebra is a triple (A,µA, ηA) consisting of a vector space A over a field K and K-linear maps µA : A ⊗ A −→ A and ηA : K −→ A such that the following diagrams commute: −−−−−−−→ −−−−−−−−→ y y K ⊗A A⊗A A A ηA ⊗ IA IA ∼= µA ←−−−−−−−− ←−−−−−−−−− A⊗K A IA ⊗ ηA IA y∼= −−−−−−→ −−−−−−−→ y y A⊗A⊗A A⊗A A⊗A A µA ⊗ IA µA IA ⊗ µA µA Figure 1: Unit and the associative property on A. Here the map IA : A −→ A is the identity map and the maps IA⊗µA : A⊗A⊗A −→ A⊗A and µA ⊗ IA : A⊗A⊗A −→ A⊗A are defined by a⊗ b⊗ c 7−→ a⊗ µA(b⊗ c) and a⊗ b⊗ c 7−→ µA(a⊗ b)⊗ c, respectively, for all a, b, c ∈ A. The maps IA⊗ ηA, ηA⊗ IA are defined by a⊗k 7−→ a⊗ηA(k), k⊗a 7−→ ηA(k)⊗a for all k ∈ K, a ∈ A, respectively. These commuted diagrams can be represented in terms of equations as follows for all k ∈ K and a, b, c ∈ A: µA(IA ⊗ µA)(a⊗ b⊗ c) = µA(µA ⊗ IA)(a⊗ b⊗ c) (1) B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1029 and µA(IA ⊗ ηA)(a⊗ k) = ka = µA(ηA ⊗ IA)(k ⊗ a) (2) The map µA is the multiplication map and ηA is the unit map. The associative property follows from (1) and the unit property follows from (2). We say that the K-algebra A is commutative if µAτ = µA, where τ is the twist map which is defined by τ(a⊗ b) = b⊗ a for a, b ∈ A. Definition 2.2. [11] A K-coalgebra is a triple (C,∆C , εC) consisting of a vector space C over a field K and K-linear maps ∆C : C −→ C ⊗ C and εC : C −→ K such that the following diagrams commute: ←−−−−−−− ←−−−−−−−− x x k ⊗ C C ⊗ C C C εC ⊗ IC IC 1⊗− ∆C −−−−−−−−→ −−−−−−−−−→ C ⊗ k C IC ⊗ εC IC x−⊗ 1 ←−−−−− ←−−−−−−− x x C ⊗ C ⊗ C C ⊗ C C ⊗ C C ∆C ⊗ IC ∆C IC ⊗∆C ∆C Figure 2: Counit and the coassociative property on C. Here the map IC : C −→ C is the the identity map on C. Also, the maps IC ⊗∆C : C⊗C −→ C⊗C⊗C and ∆C⊗IC : C⊗C −→ C⊗C⊗C are defined by a⊗b 7−→ a⊗∆C(b) and a ⊗ b 7−→ ∆C(a) ⊗ b, for all a, b ∈ C, respectively. In addition, the maps − ⊗ 1 and 1⊗− are defined by c 7−→ c⊗ 1 and c 7−→ 1⊗ c, respectively. These commuted diagrams can be represented in terms of equations as follows for all c ∈ C: (IC ⊗∆C)∆C(c) = (∆C ⊗ IC)∆C(c) (3) and (εC ⊗ IC)∆C(c) = 1⊗ c, (IC ⊗ εC)∆C(c) = c⊗ 1 . (4) The maps ∆C and εC are called the comultiplication and counit maps on the coalgebra C, respectively. The coassociative property is presented by equation (3) and the counit property is presented by equation (4). A K-coalgebra C is cocommutative if τ(∆C(c)) = ∆C(c), for all c ∈ C. We use the notation of Sweedler [9] to write ∆C(c) = ∑ (c) c(1) ⊗ c(2). Since (1 ⊗ c) = c = (c ⊗ 1), equation (4) implies that ∑ (c) εC(c(1))c(2) = c = ∑ (c) εC(c(2))c(1). Moreover, we have (IC⊗∆C)∆C(c) = (IC⊗∆C) (∑ (c) c(1)⊗c(2) ) = ∑ (c) c(1)⊗∆C(c(2)) = ∑ (c),(c(2)) c(1)⊗c(2)(1)⊗c(2)(2) B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1030 and (∆C⊗IC)∆C(c) = (∆C⊗IC) (∑ (c) c(1)⊗c(2) ) = ∑ (c) ∆C(c(1))⊗c(2) = ∑ (c),(c(1)) c(1)(1)⊗c(1)(2)⊗c(2). But, (IC ⊗ ∆C)∆C = (∆C ⊗ IC)∆C by (3). So, the expressions in both of the above equations are equal. The common value in both is denoted by∑ (c) c(1) ⊗ c(2) ⊗ c(3). In general we write ∆n−1(c) = ∑ (c) c(1) ⊗ . . . . ⊗ c(n) . ∆n−1(c) is the element obtained by applying the coassociativity (n− 1) times. Definition 2.3. [11] A K-vector space H over a field K is a bialgebra if (H,µH , ηH) is an algebra, (H,∆H , εH) is a coalgebra and either of the following equivalent conditions holds: 1) ∆H and εH are algebra maps. 2) µH and ηH are coalgebra maps. Corollary 2.4. [11] Let K be a filed and let Vi, 1 ≤ i ≤ n, be a finite set of vector spaces over K. Then V ∗1 ⊗ V ∗2 ⊗ ...⊗ V ∗n ⊆ (V1 ⊗ V2 ⊗ ...⊗ Vn)∗. Definition 2.5. [4] For a group X and a subgroup G, we call M ⊂ X a set of left coset representatives if for every x ∈ X there is a unique s ∈ M such that x ∈ Gs. The decomposition x = us is called the unique factorization of x where u ∈ G and s ∈M . In what follows, M ⊂ X is assumed to be a set of left coset representatives for the subgroup G ⊂ X. In addition, the identity in X will be denoted by e. Definition 2.6. [4] For s, t ∈ M we define τ(s, t) ∈ G and s · t ∈ M by the unique factorization st = τ(s, t)(s · t) in X. The functions . : M ×G→ G and / : M ×G→ M are also defined by the unique factorization su = (s . u)(s / u) for s, s / u ∈ M and u, s . u ∈ G. It was shown in [4] that the binary operation (M, ·) has a unique left identity em ∈M and also has the right division property (i.e. there is a unique solution p ∈ M to the equation p · s = t for all s, t ∈M). If e ∈M then em = e is also a right identity [4]. The next proposition will be used at many places in this article: Proposition 2.7. [4] For t, s, p ∈M and u, v ∈ G, the following identities between (M, ·) and τ are satisfied: s . (t . u) = τ(s, t) ( (s · t ) . u)τ ( s / (t . u), t / u )−1 and (s · t) / u = ( s / (t . u) ) · (t / u) , s . uv = (s . u) ( (s / u) . v ) and s / uv = (s/) / v , τ(p, s)τ(p · s, t) = ( p . τ(s, t) ) τ ( p / τ(s, t), s · t ) and ( p / τ(s, t) ) · (s · t) = (p · s) · t . B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1031 In what follows, unless otherwise stated, we assume that e ∈ M for the sake of sim- plicity. In [4], it was proved that for all t ∈M and v ∈ G, the following identities hold: e / v = e , e . v = v , t . e = e , t / e = t . Let X = GM be a factorization of a finite group as defined before, the category C is defined as the following [4]: Take a category C of finite dimensional vector spaces over a field K, whose objects are right representations of the group G and have M -gradings. The action for the representation is written as /̄ : V ×G→ V . In addition it is supposed that the action and the grading satisfy the compatibility condition, i.e. 〈ξ/̄u〉 = 〈ξ〉 / u where ξ ∈ Vs corresponds to 〈ξ〉 = s. The morphisms in the category C is defined to be linear maps that preserve both of grading and action, i.e. for a morphism ϑ : V → W we have 〈ϑ(ξ)〉 = 〈ξ〉 and ϑ(ξ)/̄u = ϑ(ξ/̄u) for all ξ ∈ V and u ∈ G. C can be made into a tensor category by taking V ⊗W to be the usual vector space tensor product, with actions and gradings given by 〈ξ ⊗ η〉 = 〈ξ〉 · 〈η〉 and (ξ ⊗ η)/̄u = ξ/̄(〈η〉 B u)⊗ η/̄u . There is an associator ΦUVW : (U ⊗ V )⊗W → U ⊗ (V ⊗W ) given by Φ((ξ ⊗ η)⊗ ζ) = ξ/̄τ(〈η〉, 〈ζ〉)⊗ (η ⊗ ζ) . Now, for the rigidity of C, suppose that (M, ·) has right inverses, i.e. for every s ∈M there is an sR ∈M so that s · sR = e and consider V = ⊕ s∈M Vs, where ξ ∈ Vs corresponds to 〈ξ〉 = s. Now take the dual vector space V ∗, and set V ∗ sL = {α ∈ V ∗ : α|Vt = 0 ∀t 6= s}. Then V ∗ = ⊕ s∈M V ∗ sL , and we define 〈α〉 = sL when α ∈ V ∗ sL , where sL is the left inverse of s in M . The evaluation map ev : V ∗ ⊗ V → K is defined by ev(α, ξ) = α(ξ). Considering the action /̄u, if we apply evaluation to α/̄(〈ξ〉 B u) ⊗ ξ/̄u we should get α(ξ)/̄u = α(ξ). So we define ( α/̄(〈ξ〉 B u) ) (ξ/̄u) = α(ξ), or if we put η = ξ/̄u we get( α/̄ ( (〈η〉 C u−1) B u )) (η) = α(η/̄u−1) = ( α/̄(〈η〉 B u−1)−1 ) (η) . If this is rearranged to give α/v, we get the following formula: (α/̄v)(η) = α ( η/̄τ(〈η〉L, 〈η〉)−1(〈η〉L B v−1)τ(〈η〉L C v−1, (〈η〉L C v−1)R) ) . (5) For the coevaluation map to be defined, a basis {ξ} of each Vs is taken and a corresponding dual basis {ξ̂} of each V ∗ sL , i.e. η̂(ξ) = δξ,η. Then these bases are put together for all s ∈M to get the following definition, which is a morphism in C [4]: coev(1) = ∑ ξ∈basis ξ/̄τ(〈ξ〉L, 〈ξ〉)−1 ⊗ ξ̂ . The algebra A in the tensor category C is constructed such that the group action and the grading in the definition of C can be combined. We consider a single object A in C, a vector space spanned by a basis δs⊗u for s ∈M and u ∈ G. For any object V in C define a map /̄ : V ⊗A→ V by ξ/̄(δs ⊗ u) = δs,〈ξ〉ξ/̄u . This map is a morphism in C only if 〈ξ〉 · 〈δs ⊗ u〉 = 〈ξ/̄u〉, i.e. s · 〈δs ⊗ u〉 = s/u, where 〈ξ〉 = s. If we put a = 〈δs ⊗ u〉, the action of v ∈ G is given by (δs ⊗ u)/̄v = δsC(aBv) ⊗ (a B v)−1uv . B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1032 In the remaining of this article, when an algebra A in C is mentioned, it is meant to refer to this construction. Proposition 2.8. [4] The formula of the multiplication µA for A in C is given by (δs ⊗ u)(δt ⊗ v) = δt,sCuδsCτ(a,b) ⊗ τ(a, b)−1uv, where a = 〈δs ⊗ u〉 and b = 〈δt ⊗ v〉. Proposition 2.9. [4] Multiplication µA : A⊗ A −→ A is a morphism and associative in C. Also there are an identity I for the multiplication and an algebra map εA : A −→ K in the category given by IA = ∑ t δt ⊗ e, εA(δs ⊗ u) = δs,e. The identity IA has the trivial action on the objects of C. Also the action of h ∈ A on the object K is just multiplication by εA(h), and εA(I) = 1, the identity element in K. Proposition 2.10. [1] Define a basis s⊗ δu of A∗ with evaluation map given by ev ( (s⊗ δu)⊗ (δt ⊗ v) ) = δs,t δu,v , for s, t ∈ M and u, v ∈ G. Then the M -grade and the G-action on A∗ are defined as follows: 〈s⊗ δu〉 = 〈δs ⊗ u〉L, and for any w ∈ G (s⊗ δu)/̄(〈s⊗ δu〉R B w) = s/(〈s⊗ δu〉R B w)⊗ δ (〈s⊗δu〉RBw)−1uw . Proposition 2.11. [1] If A is an algebra in a rigid tensor category, then its dual A∗ is a coalgebra in the category using the following definitions: @@ �� @@ �� �� @@ � � @ @ @ @ �� A∗A∗ A∗ =�� @@ ∗ A∗ A∗ A∗ , A∗ @@ �� � �� ηA=� �� εA∗ A∗ Figure 3: Comultiplication and counit on A∗. B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1033 Proposition 2.12. [1] If C is a coalgebra in a rigid tensor category, then its dual C∗ is an algebra in the category using the following definitions: �� @@ �� @@ @@ �� @ @ � � � �@@ C∗C∗ C∗ =@@ �� ∗ C∗C∗ C∗ , �� @@ C∗ � �� εC = C∗ ���� ηC∗ Figure 4: Multiplication and unit on C∗. B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1034 3. Results In this section, we consider an algebra A and a coalgebra C in the rigid tensor category C as defined before as well as their duals in the same category. We provide mathematical formulas for some operations on the dual objects of C. Precisely, formulas for the multipli- cation µC∗ on C∗, the counit εA∗ on A∗ and the unit ηC∗ on C∗ are obtained. Moreover, the unit property and the counit property for ηC∗ and εA∗ , respectively, are checked. Proposition 3.1. Let C be a coalgebra in the category C. Then the multiplication µC∗ on C∗ for any elements α ′ = (t1⊗ δv1) and α = (t2⊗ δv2) in C∗ for v1, v2 ∈ G and t1, t2 ∈M, can be given by µC∗(α⊗ α ′ ) = δt1Cv1,t2 ( t1 C τ(a1, a2) ⊗ δτ(a1,a2)−1v1v2 ) , with τ(a2, a L) = e where a1 = 〈δt1 ⊗ v1〉, a2 = 〈δt2 ⊗ v2〉 and a = a1 · a2. Proof. From Proposition 2.12, we know that �� @@ �� @@ @@ �� @ @ � � � �@@ C∗C∗ C∗ =@@ �� ∗ C∗C∗ C∗ For α, α ′ ∈ C∗, we follow the above figure from top to bottom and calculate the following: Put coev(1) = β ⊗ γ for some β ∈ C and γ ∈ C∗with 4C (β) = β1 ⊗ β2, α ⊗ α′ = γ, ev (α⊗ β2) = 1 and ev ( α ′ ⊗ β1 ) = 1 that imply 〈β〉·〈γ〉 = e, 〈α〉·〈β2〉 = e, 〈 α ′ 〉 ·〈β1〉 = e, 〈β〉 = 〈β1〉 · 〈β2〉 and 〈α〉.〈α′〉 = 〈γ〉. We start with (α⊗ α′)⊗ coev(1) = (α⊗ α′)⊗ (β ⊗ γ). (6) Applying the associator Φ on the right hand side of (6) and then the comultiplication on β give α/̄τ(〈α′〉, 〈β〉 · 〈γ〉)⊗ ( α′ ⊗ (β ⊗ γ) ) = α⊗ ( α′ ⊗ (β ⊗ γ) ) = α⊗ ( α ′ ⊗ ( (β1 ⊗ β2)⊗ γ )) , (7) since α/̄τ((〈α′〉, 〈β〉 · 〈γ〉) = α/̄τ(〈α′〉, e) = α/̄e = α and 4C (β) = β1 ⊗ β2. Now, Applying the associator Φ and then the associator inverse Φ−1 on the right hand side of (7) give α⊗ ( α ′ ⊗ ( β1/̄τ(〈β2〉, 〈γ〉 ) ⊗ (β2 ⊗ γ) ) , B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1035 α⊗ (( α ′ /̄τ(〈β′〉, 〈β2〉.〈γ〉)−1 ⊗ β′ ) ⊗ (β2 ⊗ γ) ) , (8) where β ′ = β1/̄τ(〈β2〉, 〈γ〉). Next, we apply the evaluation map on ( ( α ′ /̄τ(〈β′〉, 〈β2〉.〈γ〉)−1 ⊗ β′ ) of (8) to get α ′ /̄τ(〈β′〉, 〈β2〉 · 〈γ〉)−1(β ′ ) (9) = α ′ (β ′ /̄τ(〈β′〉l, 〈β′〉)−1(〈β′〉l.τ(〈β′〉, 〈β2〉·〈γ〉))τ(〈β′〉L C τ(〈β′〉, 〈β2〉·〈γ〉), (〈β ′〉L C τ(〈β′〉, 〈β2〉·〈γ〉))R)). To make this equation simpler we need to do following calculations: We first show that 〈β′〉 = (〈β2〉 · 〈γ〉)L as follows: 〈β′〉 · (〈β2〉 · 〈γ〉) = ( 〈β1〉 C τ(〈β2〉, 〈γ〉) ) · (〈β2〉 · 〈γ〉) = (〈β1〉 · 〈β2〉) · 〈γ〉 = 〈β〉 · 〈γ〉 = e. But we know 〈α〉 · 〈α′〉 = 〈γ〉 and 〈α〉 · 〈β2〉 = e, that imply 〈β2〉 · 〈γ〉 = 〈α′〉 . Hence, 〈β′〉 = (〈β2〉.〈γ〉)L = 〈α′〉L. Substituting in (9) gives α ′ /̄τ(〈α′〉L, 〈α′〉)−1(β ′ ) = (10) α ′ (β ′ /̄τ(〈β′〉l, 〈β′〉)−1(〈β′〉l.τ(〈α′〉l, 〈α′〉))τ(〈β′〉l C τ(〈α′〉l, 〈α′〉), (〈β′〉l C τ(〈α′〉l, 〈α′〉))R)). Next, we need to do the following calculations:( 〈α′〉 LL /τ(〈α′〉 L , 〈α′〉) ) · (〈α′〉 L · 〈α′〉) = (〈α′〉 LL · 〈α′〉 L ) · 〈α′〉, which implies that 〈α′〉 LL C τ(〈α′〉 L , 〈α′〉) = 〈α′〉. Thus, we can consider the following 〈α′〉 LL 〈α′〉 L 〈α′〉 = 〈α′〉 LL τ(〈α′〉 L , 〈α′〉) = ( 〈α′〉)LL B τ(〈α′〉 L , 〈α′〉) ) 〈α′〉, which implies that 〈α′〉 LL 〈α′〉 L = τ(〈α′〉 LL , 〈α′〉 L ) = 〈α′〉LL B τ(〈′α〉L, 〈α′〉). Now, substituting in equation (10) gives α ′ /̄τ(〈α′〉L, 〈α′〉)−1(β ′ ) = α ′ (β ′ /̄τ(〈α′〉, 〈α′〉R)). After applying the evaluation map and since τ(〈α′〉, 〈α′〉R) = e, (8) becomes α⊗ ( (α ′ (β ′ /̄τ(〈α′〉, 〈α′〉R))⊗ (β2 ⊗ γ) ) = α⊗ ( α ′ (β ′ )⊗ (β2 ⊗ γ) ) . (11) B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1036 We now apply the associator inverse Φ−1 on (11) to get α⊗ ( α′(β ′ )/̄τ(〈β2〉, 〈γ〉)−1 ⊗ β2 ) ⊗ γ. Applying the associator inverse again gives( α/̄τ(〈β′′〉, 〈γ〉)−1 ⊗ β′′ ) ⊗ γ, (12) where β′′ = α ′ (β ′ )/̄τ(〈β2〉, 〈γ〉)−1⊗ β2 = α′ ( β1 C τ(〈β2〉, 〈γ〉) ) /̄τ(〈β2〉, 〈γ〉)−1⊗ β2 = α′(β1)⊗ β2, which implies that 〈β′′〉 = (〈α′〉 · 〈β1〉) · 〈β2〉 = e · 〈β2〉 = 〈β2〉. Now, we apply the evaluation map on (12) to get(( α/̄τ(〈β2〉, 〈γ〉)−1 ) (β′′) ) (γ) = α ( β′′/̄τ(〈β2〉L, 〈β2〉)−1 ( 〈β2〉L B τ(〈β2〉, 〈γ〉) ) τ ( 〈β2〉L C τ(〈β2〉, 〈γ〉), (〈β2〉L C τ(〈β2〉, 〈γ〉))R )) (γ) (13) = α ( β′′/̄(〈β2〉L B τ(〈β2〉, 〈γ〉) ) (γ). Considering the equality of the diagram, we should have µC∗(α⊗ α′) = α ( β′′/̄ ( 〈β2〉L B τ(〈β2〉, 〈γ〉) )) (γ), (14) where β′′ = α′(β1)⊗ β2. But, from the definition of the coevaluation map, we know that coev(1) = ∑ ξ∈ basis of V ξ /̄ τ( 〈ξ〉L, 〈ξ〉 )−1 ⊗ ξ̂, . So we can put β = ξ /̄ τ( 〈ξ〉L, 〈ξ〉 )−1 and γ = ξ̂, that imply that 〈β〉 = 〈ξ〉/ τ( 〈ξ〉L, 〈ξ〉 )−1 and 〈γ〉 = 〈ξ̂〉 = 〈ξ〉L. Thus, if we apply the coproduct on β, we get ∆C(β) = ∆C(ξ/ τ( 〈ξ〉L, 〈ξ〉 )−1) = ξ1/̄τ(〈ξ1〉L, 〈ξ1〉)−1 ⊗ ξ2/̄τ(ξL2 , ξ2)−1. Consequently, we can write β1 = ξ1/ τ( 〈ξ1〉L, 〈ξ1〉 )−1 and β2 = ξ2/̄τ(〈ξ2〉L, 〈ξ2〉)−1, with 〈β1〉 = 〈ξ1〉 C τ(〈ξ1〉L, 〈ξ1〉)−1 and 〈β2〉 = 〈ξ2〉 C τ(〈ξ2〉L, 〈ξ2〉)−1. Now, let ξ = δt⊗ v , γ = t⊗ δv , ξ1 = δt1 ⊗ v1 and ξ2 = δt2 ⊗ v2, with a = 〈ξ〉 = 〈δt⊗ v〉, aL = 〈γ〉 = 〈t⊗ δv〉, a1 = 〈ξ1〉 = 〈δt1 ⊗ v1〉 and a2 = 〈ξ2〉 = 〈δt2 ⊗ v2〉. B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1037 As τ(〈ξ1〉L, 〈ξ1〉)−1 = e−1 = e and τ(〈ξ2〉L, 〈ξ2〉)−1 = e−1 = e, it follows that β1 = ξ1/̄e = ξ1 and β2 = ξ2/̄e = ξ2, which means that β′′ = α′(δt1 ⊗ v1)⊗ (δt2 ⊗ v2). If we put α′ = t1⊗δv1 in the right hand side of the above equation, then it can be rewritten as β′′ = ev ( (t1 ⊗ δv1)⊗ (δt1 ⊗ v1) ) ⊗ (δt2 ⊗ v2) = δt1,t1δv1,v1(δt2 ⊗ v2) = (δt2 ⊗ v2). Also, if we put q = 〈β2〉L B τ(〈β2〉, 〈γ〉) = aL2 B τ(a2, a L), then β′′/̄q = (δt2 ⊗ v2)/̄q = (δt2C(a2Bq) ⊗ (a2 B q) −1v2q). Now we substitute these simplified parts in equation (14) to get µC∗(α⊗ α ′ ) = α ( (δt2C(a2Bq) ⊗ (a2 B q) −1v2q) ) (γ). If we put α = t2 ⊗ δv2 , the above equation becomes µC∗(α⊗ α ′ ) = ev ( (t2 ⊗ δv2)⊗ (δt2C(a2Bq) ⊗ (a2 B q) −1v2q) ) (γ) = δt2,t2C(a2Bq)δv2,(a2Bq)−1v2q)(γ) (15) which implies that t2 = t2 C (a2 B q) = t2 C ( a2 B ( aL2 B τ(a2, a L) )) , v2 = (a2 B q) −1v2q = ( a2 B (aL2 B τ(a2, a L) )−1 v2 ( aL2 B τ(a2, a L) ) . To have these equations satisfied we should have τ(a2, a L) = e. Hence, aL2 B τ(a2, a L) = e and a2 B (aL2 B τ(a2, a L)) = e. On the other hand, we know that δt ⊗ v = (δt1 ⊗ v1)⊗ (δt2 ⊗ v2) = δt2,t1Cv1δt1Cτ(a1,a2) ⊗ τ(a1, a2)−1v1v2 Thus, v = τ(a1, a2)−1v1v2 and t = t1 C τ(a1, a2). Therefore, µC∗(α⊗ α ′ ) = δt1Cv1,t2 ( t1 C τ(a1, a2) ⊗ δτ(a1,a2)−1v1v2 ) . To confirm our calculation we show t C v = t · a knowing that t1 C v1 = t1 · a1, t2 C v2 = t2 · a2, t1 C v1 = t2, and a = a1 · a2. We start with the right hand side as follows: t · a = t1 C τ(a1, a2) · (a1 · a2) = (t1 · a1) · a2 = (t1 C v1) · a2 = t2 · a2 = t2 C v2. On the other hand, t C v = t1 C τ(a1, a2) C τ(a1, a2)−1 C v1v2 = t1 C v1v2 = t1 C v1 C v2 = t2 C v2. � B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1038 Proposition 3.2. Let A be an algebra in the category C. Then the counit εA∗ on A∗ for any element α = (s⊗ δu) ∈ A∗ is given by εA∗(s⊗ δu) = δu,e, for u ∈ G and s ∈M . Proof. From Proposition (2.11), we know that A∗ @@ �� � �� ηA=� �� εA∗ A∗ Figure 5: Definition of counit on A∗. We follow figure 5 from top to bottom and start with the following for α ∈ A∗ and k ∈ K : α = α⊗ k. (16) Knowing that ηA : K −→ A, by definition (2.1), we apply the map (IA∗ ⊗ ηA) on equation (16) to get (IA∗ ⊗ ηA)(α⊗ k) = IA∗(α)⊗ ηA(k) = α⊗ β, (17) where β = (δs ⊗ e) ∈ A . Now, we put α = (s ⊗ δu) and apply the evaluation map on the right hand side of equation (17) to have ev(α⊗ β) = ev((s⊗ δu)⊗ (δs ⊗ e)) = δu,eδs,s = δu,e. Finally, considering the left hand side of the equality in figure 5 gives εA∗(s⊗ δu) = δu,e. � Proposition 3.3. Let C be a coalgebra in the category C . Then the unit ηC∗ on C∗ can be given by ηC∗(1K) = ∑ v∈G e⊗ δv, where 1K is the unity of K. B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1039 Proof. From Proposition (2.12), we know that �� @@ C∗ � �� εC = C∗ ���� ηC∗ Figure 6: Definition of unit on C∗. We follow figure 6 from top to bottom and start by considering the following: coev(1) = β ⊗ γ, (18) for β ∈ C and γ ∈ C∗, which implies 〈β〉 · 〈γ〉 = e. But, from the definition of the coevaluation map, we know coev(1) = ∑ ξ∈ basis of V ξ /̄ τ( 〈ξ〉L, 〈ξ〉 )−1 ⊗ ξ̂. We let β = ξ /̄ τ( 〈ξ〉L, 〈ξ〉 )−1, γ = ξ̂ and w = τ(〈ξ〉L, 〈ξ〉)−1. If ξ = δt ⊗ v, γ = t⊗ δv, then a = 〈ξ〉 = 〈δt ⊗ v〉, and aL = 〈γ〉 = 〈t⊗ δv〉 . Hence, β = (δt ⊗ v)/̄w = δtC(aBw) ⊗ (a B w)−1vw. Now, applying the map (εC ⊗ IC∗) on equation (18) gives εC(β)⊗ IC∗(γ) = ∑ v∈G δtC(aBw),e ⊗ (t⊗ δv) = ∑ v∈G δtC(aBw),e (t⊗ δv). (19) To get a nonzero solution we should have t C (a B w) = e⇒ t C (a B w) C (a B w)−1 = e C (a B w)−1 ⇒ t = e which leads to a = 〈δt ⊗ v〉 = 〈δe ⊗ v〉 = e. Thus, equation (19) can be rewritten as εC(β)⊗ IC∗(γ) = ∑ v∈G e⊗ δv. Finally, considering the left hand side of the equality in figure 6 gives ηC∗(1k) = ∑ v∈G e⊗ δv. � B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1040 In the next propositions we will check the unit property and the counit property for ηC∗ and εA∗ respectively. Proposition 3.4. Let A be an algebra in the category C. Then the counit property for the counit on A∗ is satisfied, i.e. (εA∗ ⊗ IA∗)∆A∗(t⊗ δv) = (IA∗ ⊗ εA∗)∆A∗(t⊗ δv) for any element γ = (t⊗ δv) ∈ A∗ with v ∈ G, t ∈M. �� @@ ∗ � �� εA∗ = = �� @@ ∗ � �� εA∗ A∗ A∗ A∗ A∗ A∗ A∗ Figure 7: Counit property on A∗. Proof. As A is an algebra in the category C, it has a unit map ηA : K −→ A satisfying µA(IA ⊗ ηA)(β ⊗ k) = kβ = µA(ηA ⊗ IA)(k ⊗ β). We consider the dual map η∗A : A∗ −→ K∗ = K and let εA∗ : A∗ −→ K denote the restriction of η∗A to A∗. Now, for γ = (t⊗ δv) ∈ A∗, k ∈ K, we have εA∗(γ)(k) = γ ( ηA(k) ) = γ ( ηA(1K)k ) = γ(1A)(k). (20) Hence, εA∗(γ) = γ(1A). Next, let µ∗A : A∗ −→ (A⊗A)∗ be the transpose of the multiplication map µA defined as µ∗A(γ)(β1 ⊗ β2) = γ(µA(β1 ⊗ β2)) = γ(β1.β2). (21) It is known that µ∗A(A∗) ⊆ A∗⊗A∗ [11]. Let ∆A∗ denote the restriction of µ∗A toA∗. Then ∆A∗ : A∗ −→ A∗ ⊗A∗ is a K-linear map defined as ∆A∗(γ) = µ∗A(γ), for γ ∈ A∗. (22) B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1041 Thus, for γ = (t⊗ δv) ∈ A∗, β = (δs ⊗ u) ∈ A and k ∈ K, we have (εA∗ ⊗ IA∗)∆A∗(γ)(k ⊗ β) = (η∗A ⊗ I∗A)µ∗A(γ)(k ⊗ β) = µ∗A(γ)(ηA ⊗ IA)(k ⊗ β) = γ ( µA(ηA(k)⊗ β) ) = γ ( µA ( (δt ⊗ e)⊗ (δs ⊗ u) )) = γ ( δs,tCeδtCτ(a,b) ⊗ τ(a, b)−1eu ) = γ ( δs,tδtCτ(e,b) ⊗ τ(e, b)−1eu ) = γ(δs ⊗ u) = γ(β) = γ ( µA(β ⊗ ηA(k)) ) = µ∗A(γ)(IA ⊗ ηA)(β ⊗ k) = (I∗A ⊗ η∗A)µ∗A(γ)(β ⊗ k) = (IA∗ ⊗ εA∗)∆A∗(γ)(β ⊗ k), where a = 〈δt ⊗ e〉 = e and b = 〈δs ⊗ u〉. We have used the following calculations: τ(a, b) = τ(e, b) = e, τ(a, b)−1 = τ(e, b)−1 = e−1 = e, t C τ(a, b) = t C e = t , τ(a, b)−1eu = u and t = s. Therefore, εA∗ satisfies the counit property as claimed. � Proposition 3.5. let C be a coalgebra in category C. Then the unit property on C∗ is satisfied, i.e. µC∗(IC∗ ⊗ ηC∗) ( γ ⊗ k ) = µC∗(ηC∗ ⊗ IC∗) ( k ⊗ γ ) for any element γ = (t⊗ δv) ∈ C∗ with v ∈ G, t ∈M, and k ∈ K. C∗ @@ �� ���� ηC∗ ∗ = = C∗ @@ �� ���� ηC∗ ∗ C∗ C∗ C∗ C∗ Figure 8: Unit property on C∗. B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1042 Proof. As C is a coalgebra in the category C, it has a counit map εC : C −→ K satisfying the counit property, i.e∑ (β) εC(β(1))β(2) = β = ∑ (β) εC(β(2))β(1). The transpose of the counit map of C is ε∗C : K∗ −→ C∗ which is defined by ε∗C(γ)(β) = γ ( εC(β) ) , for γ ∈ K∗, β ∈ C. If we identify K with K∗, we get ε∗C : K −→ C∗ defined as ε∗C(k)(β) = k ( εC(β) ) = kεC(β), (23) for k ∈ K, β ∈ A. Now, we use the same techniques as in the proof of the previous proposition to have ηC∗ = ε∗C and µC∗ = ∆∗C . We define the following maps: IC∗ ⊗ ηC∗ : C∗ ⊗K −→ C∗ ⊗C∗ by γ ⊗ k 7−→ γ ⊗ ηC∗(k), and ηC∗ ⊗ IC∗ : K ⊗ C∗ −→ C∗ ⊗ C∗ by k ⊗ γ 7−→ ηC∗(k)⊗ γ. Next, it is known that the transpose of ∆C is a K-linear map ∆∗C : (C ⊗C)∗ −→ C∗, defined by ∆∗C(ψ)(β) = ψ(∆C(β)), (24) for ψ ∈ (C ⊗ C)∗, β ∈ C. Also, by Corollary 2.4, we have C∗ ⊗ C∗ ⊆ (C ⊗ C)∗. Hence, ∆∗C leads to a K-linear map µC∗ : C∗ ⊗ C∗ −→ C∗, defined by µC∗(γ1 ⊗ γ2)(β) = ∆∗C(γ1 ⊗ γ2)(β) = (γ1 ⊗ γ2)(∆C(β)) = ∑ (β) γ1(β1)⊗ γ2(β2). (25) Thus, if we put γ = (t⊗δv), β = (δs⊗u), β1 = (δs1⊗u1) and β2 = (δs2⊗u2) with β = B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1043 β1 ⊗ β2 and 〈β〉 = 〈β1〉 · 〈β2〉, for k ∈ K, β, β1, β2 ∈ C and γ ∈ C∗, we get µC∗(IC∗ ⊗ ηC∗)(γ ⊗ k)(β) = ∆∗C ( γ ⊗ ε∗C(k) ) (β) = ( γ ⊗ ε∗C(k) )( ∆C(β) ) = ( γ ⊗ ε∗C(k) ) (β1 ⊗ β2) = ∑ (β) γ(β1)⊗ ε∗C(k)(β2) = ∑ (β) γ(β1)⊗ k ( εC(β2) ) = ∑ (β) γ(β1)k ( εC(β2) ) = k ∑ (β) γ(β1)εC(β2) = k ∑ (β) γ(δs1 ⊗ u1)δs2,e = k ∑ (β) δs2,eγ(δs1 ⊗ u1) = k ∑ (β) εC(β2)γ(β1) = k ∑ (β) γ ( εC(β2)β1 ) = kγ (∑ (β) εC(β2)β1 ) = kγ(β) = kγ (∑ (β) β2εC(β1) ) = k ∑ (β) γ ( β2εC(β1) ) = k ∑ (β) γ(δs2 ⊗ u2)δs1,e = k ∑ (β) δs1,eγ(δs2 ⊗ u2) = k ∑ (β) εC(β1)γ(β2) = ∑ (β) k ( εC(β1) ) γ(β2) = ∑ (β) ε∗C(k)(β1)γ(β2) = ∑ (β) k ( εC(β1) ) ⊗ γ(β2) = ∑ (β) ε∗C(k)(β1)⊗ γ(β2) = ∑ (β) (ε∗C(k)⊗ γ)(β1 ⊗ β2) = ( ε∗C(k)⊗ γ ) ∆C(β) = ∆C∗ ( ε∗C(k)⊗ γ ) (β) = ∆C∗ ( ε∗C ⊗ IC∗ ) (k ⊗ γ)(β) = µC∗ ( ηC∗ ⊗ IC∗ ) (k ⊗ γ)(β). In the above calculations we have used equations (23), (24) and (25), the facts that k ( εC(β2) ) ∈ K and C ⊗ K ∼= C, Proposition 2.9 and Definition 2.2. Therefore, ηC∗ satisfies the unit property as required. � Example 1. Let X be the dihedral group D6 = 〈 x, y : x6 = y2 = 1, xy = yx5 〉 and let G be the non-normal subgroup {1, x3, y, x3y}. If we choose M = {1, x, x5} to be the set of left coset representatives, then the ·, τ , the action B and the coaction C , are given by the following tables: · 1 x x5 1 1 x x5 x x x5 1 x5 x5 1 x τ 1 x x5 1 1 1 1 x 1 x3 1 x5 1 1 x3 B. Al-harbi, W. M. Fakieh, M. M. Al-Shomrani / Eur. J. Pure Appl. Math, 11 (4) (2018), 1027-1045 1044 s B u 1 x3 y x3y 1 1 x3 y x3y x 1 x3 y x3y x5 1 x3 y x3y s C u 1 x3 y x3y 1 1 1 1 1 x x x x5 x5 x5 x5 x5 x x We take our field to be the binary field F = {0, 1}. We check multiplication µC∗ in Proposition 3.1. For two elements α′ = (t1 ⊗ δv1) and α = (t2 ⊗ δv2) in C∗ with v1, v2 ∈ G, t1, t2 ∈ M , if we put t1 = x, t2 = x5 in M, v1 = y, v2 = x3 in G, then α = ( x5 ⊗ δx3 ) , α′ = ( x ⊗ δy ) , a2 = 〈δt2 ⊗ v2〉, a1 = 〈δt1 ⊗ v1〉, and a = a1 · a2. We start by calculating the following: t2 · a2 = t2 C v2, x5 · a2 = x5 C x3 ⇒ x5 · a2 = x5 ⇒ a2 = 1, and t1 · a1 = t1 C v1, x · a1 = x C y ⇒ x · a1 = x5 ⇒ a1 = x. Also, a = a1 · a2 ⇒ a = x · 1 = x ⇒ aL = x5. The following calculations are needed as well: t1 C τ(a1, a2) = x C τ(x, 1) = x C 1 = x, τ(a1, a2)−1 v1 v2 = τ(x, 1)−1 v1 v2 = 1 y x3 = x3y, and t1 C v1 = x C y = x5, and t2 = x5. Now, we substitute in the formula of µC∗ as follows: µC∗(α⊗ α ′ ) = δt1Cv1,t2 ( t1 C τ(a1, a2) ⊗ δτ(a1,a2)−1v1v2 ) , REFERENCES 1045 µC∗ ( (x5 ⊗ δx3)⊗ (x⊗ δy) ) = δx5,x5 ( x⊗ δx3y ) = x⊗ δx3y ∈ C∗. Next, we check the counit εA∗ in Proposition 3.2, for any element α = (s ⊗ δu) ∈ A∗ with s ∈M and u ∈ G as follows: Choose s = x. If u = e = 1, then εA∗(x⊗ δ1) = δ1,1 = 1 ∈ F. If u 6= e, for example u = y, then εA∗(x⊗ δy) = δy,1 = 0 ∈ F. Finally, we check the unit ηC∗ in Proposition 3.3, for 1 ∈ F. If we let t = 1 ∈ M, v = y ∈ G, then ηC∗(1) = 1⊗ δy. References [1] M M Al-Shomrani. 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