Hom-L-R-Smash Biproduct and the Category of Hom–Yetter–Drinfel'd–Long Bimodules

2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Journal of Algebra and Its Applications Vol. 17, No. 7 (2018) 1850133 (19 pages) c World Scientiﬁc Publishing Company DOI: 10.1142/S0219498818501335

Hom-L-R-smash biproduct and the category of Hom–Yetter–Drinfel’d–Long bimodules

Daowei Lu Department of Mathematics, Jining University Qufu, Shandong 273155, P. R. China [email protected]

Xiaohui Zhang∗ School of Mathematical Sciences, Qufu Normal University Qufu, Shandong 273165, P. R. China [email protected]

Received 28 September 2016 Accepted 20 June 2017 Published 18 July 2017

Communicated by A. Leroy

Let (H, αH ) be a Hom-bialgebra. In this paper, we ﬁrstly introduce the notion of Hom- L-R smash coproduct (CH,αC ⊗ αH ), where (C, αC ) is a Hom-coalgebra. Then for a Hom-algebra and Hom-coalgebra (D, αD), we introduce the notion of Hom-L-R- admissible pair (H, D). We prove that (DH, αD ⊗ αH ) becomes a Hom-bialgebra under Hom-L-R smash product and Hom-L-R smash coproduct. Next, we will introduce a prebraided monoidal category HLR(H) of Hom–Yetter–Drinfel’d–Long bimodules and show that Hom-L-R-admissible pair (H, D) actually corresponds to a bialgebra in the category HLR(H), when αH and αD are involutions. Finally, we prove that when H J. Algebra Appl. Downloaded from www.worldscientific.com is ﬁnite dimensional Hom-Hopf algebra, HLR(H) is isomorphic to the Yetter–Drinfel’d H⊗H∗ ∗ −1∗ by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. category H⊗H∗ YD as braid monoidal categories where (H ⊗ H ,αH ⊗ αH )isthe tensor product Hom–Hopf algebra.

Keywords: Hom–Hopf algebra; Braided monoidal category; Hom-L-R smash biproduct; Hom–Yetter–Drinfel’d–Long bimodule.

Mathematics Subject Classiﬁcation: 16S40, 16T15

0. Introduction Panaite and Van Oystaeyen in [12] introduced the notion of L-R-admissible pair (H, D), where H is a bialgebra and D is an algebra and coalgebra, and D ⊗ H becomes a bialgebra under L-R-smash product and L-R-smash coproduct, denoted by DH and called L-R-smash biproduct. It was also proved that D is a bialgebra in the prebraided monoidal category LR(H) of Yetter–Drinfel’d–Long bimodules

∗Corresponding author.

1850133-1 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

and satisfies certain condition. When H is finite dimensional, Lu and Wang in [7] H⊗H∗ pointed out that LR(H)andH⊗H∗ YD are isomorphic as monoidal categories. In particular, when H is a finite dimensional Hopf algebra, this isomorphism is braided. From a physical viewpoint, Hom-type structures are important because they are related to vertex operator algebras and string theory. Hom–Lie algebras were introduced in [5] to describe the structures on some q-deformations of the Witt and the Virasoro algebras, both of which are important in vertex operator algebras and string theory. Also, in this paper Hom-type algebras have been introduced in the form of Hom–Lie algebras, where the Jacobi identity was twisted along a linear endomorphism. Meanwhile, Hom-associative algebras have been suggested in [10] to give rise to a Hom–Lie algebra using the commutator bracket. Other Hom-type structures such as Hom-coalgebras, Hom-bialgebras, Hom–Hopf algebras as well as their properties have been considered in [4, 11, 13]. Yetter–Drinfel’d module for Hom-bialgebras was introduced in [8], and it was shown that the category of Hom–Yetter–Drinfel’d modules was a prebraided monoidal category. Motivated by these constructions, it is natural to ask whether there is analogue of such constructions as in [12] and whether the main result in [7] still holds in the Hom-setting? This paper is organized as follows: In Sec. 1, we will recall basic notions and constructions of Hom–Hopf algebra. In Sec. 2, we will firstly construct Hom-L-R-smash coproduct (CH,αC ⊗ αH ), where C is an H-bicomodule Hom-coalgebra. Then we will define Hom-L-R- admissible pair (H, D) and prove that (DH, αD ⊗ αH ) becomes a Hom-bialgebra under Hom-L-R-smash product and Hom-L-R-smash coproduct. Then we will construct the category HLR(H) of Hom–Yetter–Drinfel’d–Long bimodules over a Hom-bialgebra (H, αH ), and show that when αH is an involution, Hom-L-R- admissible pair actually corresponds to a bialgebra satisfying certain condition in HLR J. Algebra Appl. Downloaded from www.worldscientific.com the category (H). In Sec. 3, we prove that when H is finite dimensional Hom–Hopf algebra,

by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. H⊗H∗ HLR(H) is isomorphic to the Yetter–Drinfel’d category H⊗H∗ YD as braid monoidal categories. Throughout this paper, all the vector spaces, tensor product and homomor- phisms are over a ﬁxed ﬁeld k. For a coalgebra C,wewillusetheHeyneman– Sweedler’s notation ∆(c)=c1 ⊗ c2, for any c ∈ C (summation omitted).

1. Preliminaries In this section, we will recall the basic deﬁnitions on Hom–Hopf algebras, Hom- modules and Hom-comodules and the construction of Hom-L-R-smash product. A unital Hom-associative algebra is a triple (A, µ, α), where α : A → A and µ : A ⊗ A → A are linear maps, with notation µ(a ⊗ b)=ab such that for any a, b, c ∈ A,

α(1A)=1A, 1Aa = α(a)=a1A,α(a)(bc)=(ab)α(c). (1.1)

1850133-2 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

A linear map f :(A, µA,αA) → (B,µB ,αB) is called a morphism of Hom-associative algebra if αB ◦ f = f ◦ αA, f(1A)=1B and f ◦ µA = µB ◦ (f ⊗ f). Note that as shown in [1], from Eq. (1.1), we can obtain

α(ab)=1A(ab)=α(1A)(ab)=(1Aa)α(b)=α(a)α(b), which implies α is an algebra map. A counital Hom-coassociative coalgebra is a triple (C, ∆,ε,α)whereα : C → C, ε : C → k,and∆:C → C ⊗ C are linear maps such that ε ◦ α = ε, (ε ⊗ id) ◦ ∆=α =(id⊗ ε) ◦ ∆, (∆ ⊗ α) ◦ ∆=(α ⊗ ∆) ◦ ∆. Similarly, we also have α is a coalgebra map. A linear map f :(C, ∆C ,αC) → (D, ∆D,αD) is called a morphism of Hom- coassociative coalgebra if αD ◦ f = f ◦ αC , εD ◦ f = εC and ∆D ◦ f =(f ⊗ f) ◦ ∆C. In what follows, we will always assume all Hom-algebras are unital and Hom- coalgebras are counital. Further, for any Hom-type (co)algebras and Hom-type (co)modules, the Hom structure map α is assumed to be a bijective map. A Hom-bialgebra is a quadruple (H, µ, ∆,α), where (H, µ, α)isaHom- associative algebra and (H, ∆,α) is a Hom-coassociative coalgebra such that ∆ and ε are morphisms of Hom-associative algebra. A Hom–Hopf algebra (H, µ, ∆,α) is a Hom-bialgebra H with a linear map S : H → H(called antipode) such that

S ◦ α = α ◦ S, S(h1)h2 = h1S(h2)=ε(h)1, for any h ∈ H.ForS, we have the following properties:

S(h)1 ⊗ S(h)2 = S(h2) ⊗ S(h1), S(gh)=S(h)S(g),ε◦ S = ε. For any Hopf algebra H and any Hopf algebra endomorphism α of H,thereexists J. Algebra Appl. Downloaded from www.worldscientific.com a Hom–Hopf algebra Hα =(H, α ◦ µ, 1H , ∆ ◦ α, ε, S, α).

by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. Let (A, αA) be a Hom-associative algebra, M a linear space and αM : M → M a linear map. A left A-module structure on (M,αM ) consists of a linear map A ⊗ M → M, a ⊗ m → am, such that

1A m= αM (m),

αM (am)=αA(a) αM (m),

αA(a) (bm)=(ab) αM (m), for any a, b ∈ A and m ∈ M. Similarly we can deﬁne the right (A, α)-modules. Let (M,µ)and(N,ν)betwo left (A, α)-modules, then a linear map f : M → N is a called left A-module map if f(am)=af(m) for any a ∈ A, m ∈ M and f ◦ µ = ν ◦ f.Moreover,if(M,µ)is both a left A-module (via action )andrightA-module (via action ) and satisﬁes

(am) αA(b)=αA(a) (mb),

1850133-3 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

for all a, b ∈ A, m ∈ M,wecallM an A-bimodule. Naturally, A is a bimodule over itself. Let (C, αC ) be a Hom-coassociative coalgebra, M a linear space and αM : M → M a linear map. A right C-comodule structure on (M,αM ) consists of a linear map ρ : M → M ⊗ C such that

(id ⊗ εC) ◦ ρ = αM ,

(αM ⊗ αC ) ◦ ρ = ρ ◦ αM ,

(ρ ⊗ αC ) ◦ ρ =(αM ⊗ ∆) ◦ ρ.

Let (M,µ)and(N,ν)betworight(C, γ)-comodules, then a linear map g : M → N is a called right C-comodule map if g ◦ µ = ν ◦ g and ρN ◦ g =(g ⊗ id) ◦ ρM . Let (H, µH , ∆H ,αH ) be a Hom-bialgebra. A Hom-algebra (A, µA,αA) is called a left H-module Hom-algebra if (A, αA)isaleftH-module, with the action H ⊗ A → A, h ⊗ a → ha, such that

2 αH (h) (ab)=(h1 a)(h2 b),

h1A = ε(h)1A,

for any h ∈ H and a, b ∈ A. Similarly, we have right H-module Hom-algebra. Moreover, A is called H-bimodule Hom-algebra if A as an H-bimodule is both left and right H-module Hom-algebra. Recall from [9], let (H, αH ) be a Hom-bialgebra, (A, αA)anH-bimodule Hom- algebra, with the left and right actions H ⊗ A → A, h ⊗ a → ha and A ⊗ H → A, a ⊗ h → ah, respectively. Then we have Hom-L-R smash product (AH, αA ⊗ αH ) (denote a ⊗ h := ah), with the multiplication:

−1 −2 −2 −1 −1 (ah)(a h )=(αA (a) αH (h2))(αH (h2) αA (a ))αH (h2h1), J. Algebra Appl. Downloaded from www.worldscientific.com for all a, a ∈ A, h, h ∈ H. by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. Let (H, αH ) be a Hom-bialgebra, (C, αC ) a Hom-coalgebra. Recall from [6], C is called a right H-comodule Hom-coalgebra if C arightH-comodule, with the comodule structure map ρ : C → C ⊗ H, c → c(0) ⊗ c(1), such that the following conditions hold:

εC(c(0))c(1) = εC (c)1H , 2 c(0)1 ⊗ c(0)2 ⊗ αH (c(1))=c1(0) ⊗ c2(0) ⊗ c1(1)c2(1),

for all c ∈ C. Similarly, one could deﬁne the left H-comodule Hom-coalgebra.

2. Hom-L-R-Smash Biproduct

Deﬁnition 2.1. Let (C, αC ) be a Hom-coalgebra, (M,αM )bebothright C-comodule via ρ : M → M ⊗ C, m → m[0] ⊗ m[1] and left C-comodule via

1850133-4 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

λ : M → C ⊗ M, m → m(−1) ⊗ m(0).Wesay(M,αM )isaC-bicomodule if

αC (m(−1)) ⊗ m(0)[0] ⊗ m(0)[1] = m[0](−1) ⊗ m[0](0) ⊗ αC(m[1]),

for all m ∈ M.

Remark 2.2. (1) Obviously any Hom-coalgebra is a bicomodule over itself via its comultiplication. (2) Let C be a coalgebra, αC a coalgebra endomorphism of C, M a C-bicomodule with the coactions M → C ⊗ M,m → m−1 ⊗ m0 and M → M ⊗ C, m → m{0} ⊗ m{1}. αM : M → M is a linear map satisfying the conditions αM (m)−1 ⊗ αM (m)0 = αC (m−1) ⊗ αM (m0)andαM (m){0} ⊗ αM (m){1} = αM (m{0}) ⊗

αC (m{1}) for all m ∈ M.Then(M,αM ) becomes a CαC -bicomodule, with the

coactions M → CαC ⊗ M,m → αC (m−1) ⊗ αM (m0)andM → M ⊗ CαC ,m→ αM (m{0}) ⊗ αC (m{1}).

Deﬁnition 2.3. Let (H, αH ) be a Hom-bialgebra, (C, αC ) a Hom-coalgebra. Then C is called a H-bicomodule Hom-coalgebra if C is both left and right H-comodule Hom-coalgebra.

Remark 2.4. Let H be a bialgebra and C a H-bicomodule coalgebra. By appropri- ate bialgebra endomorphism αH of H and coalgebra endomorphism αC of C and the

same procedure of Remark 2.2, CαC becomes an HαH -bicomodule Hom-coalgebra. If we assume moreover that the maps αC and αH are bijective, and denote by CH the L-R-smash coproduct between C and H,thenαC ⊗ αH is a coalgebra endo-

morphism of CH and the Hom-coalgebras (CH)αC ⊗αH and CαC HαH coincide.

∗ −1∗ Example 2.5. Let (H, αH ) be a Hom-bialgebra. From [6], we know that (H ,αH ) is a Hom-coalgebra with the coproduct J. Algebra Appl. Downloaded from www.worldscientific.com −2 f1(h)f2(g)=f(αH (gh)), by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. ∗ ∗ −1∗ for any f ∈ H ,andg,h ∈ H. For all p, q ∈ Z,(H ,αH ) becomes a H-bicomodule Hom-coalgebra with coactions given by

−2 p f(0)(h)f(−1) = f(αH (h1))αH (h2), (2.1) −2 q f[0](h)f[1] = f(αH (h2))αH (h1). (2.2)

Proposition 2.6. Let (H, αH ) be a Hom-bialgebra, (C, αC ) a H-bicomodule Hom- coalgebra. Then we have a Hom-coalgebra (CH,αC ⊗ αH ), equal to C ⊗ H as a vector space, with the comultiplicaiton

−1 −2 −1 −1 −1 −2 ∆(ch)=αC (c1[0])αH (c2(−1))αH (h1) ⊗ αC (c2(0))αH (h2)αH (c1[1]), (2.3)

for all c ∈ C, h ∈ H. We call Hom-coalgebra (CH,αC ⊗ αH ) Hom-L-R-smash coproduct.

1850133-5 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

Proof. For all c ∈ C, h ∈ H,ﬁrstly∆◦ (αC ⊗ αH )=(αC ⊗ αH ⊗ αC ⊗ αH )∆. Then

(∆ ⊗ αC ⊗ αH )∆(ch) −1 −2 −1 −1 =∆(αC (c1[0])αH (c2(−1))αH (h1)) ⊗ c2(0)h2αH (c1[1]), −2 −3 −3 −2 = αC (c1[0]1[0])αH (c1[0]2(−1))[αH (c2(−1)1)αH (h11)] −2 −3 −2 −3 ⊗ αC (c1[0]2(0))[αH (c2(−1)2)αH (h12)]αH (c1[0]1[1]) −1 ⊗ c2(0)h2αH (c1[1]) −2 −3 −3 −2 = αC (c11[0][0])αH (c12[0](−1))[αH (c2(−1)1)αH (h11)] −2 −3 −2 −3 ⊗ αC (c12[0](0))[αH (c2(−1)2)αH (h12)]αH (c11[0][1]) −3 ⊗ c2(0)h2αH (c11[1]c12[1]) −2 −3 −2 −1 = αC (c11[0][0])[αH (c12(−1))αH (c2(−1))]αH (h11) −2 −2 −2 −4 ⊗ αC (c12(0)[0])αH (c2(0)(−1))[αH (h12)αH (c11[1]1)] −1 −1 −4 −3 ⊗ αC (c2(0)(0))[αH (h2)αH (c11[1]2]αH (c12(0)[1]) −3 = c1[0][αH (c21(−1)c22(−1))]h1 −2 −3 −2 −3 ⊗ αC (c21(0)[0])αH (c22(0)(−1))[αH (h21)αH (c1[1]1)] −2 −2 −3 −3 ⊗ αC (c22(0)(0))[αH (h22)αH (c1[1]2]αH (c21(0)[1]) −1 −2 −3 −2 −3 = c1[0]αH (c2(−1))h1 ⊗ αC (c2(0)1[0])αH (c2(0)2(−1))[αH (h21)αH (c1[1]1)] −2 −2 −3 −3 ⊗ αC (c2(0)2(0))[αH (h22)αH (c1[1]2]αH (c2(0)1[1])

C ⊗ H ⊗ J. Algebra Appl. Downloaded from www.worldscientific.com =(α α ∆)∆(ch).

by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. And (ε ⊗ id)∆(ch) −1 −2 −1 −1 −1 −2 = εC (αC (c1[0]))εH (αH (c2(−1))αH (h1))αC (c2(0))αH (h2)αH (c1[1]) −1 −1 −2 = εC (c1[0])εH (c2(−1))εH (h1))αC (c2(0))αH (h2)αH (c1[1])

= αC (c)αH (h).

Similarly (id ⊗ ε)∆(ch)=αC (c)αH (h). Finally ε ◦ (αC ⊗ αH )=ε. The proof is completed.

Example 2.7. By the construction in Example 2.4, we have a Hom-L-R-smash ∗ −1∗ coproduct (H H, αH ⊗ αH ) with the comultiplication ∗ −2 −1 ∗ −1 −2 ∆(fh)=αH (f1[0])αH (f2(−1))αH (h1) ⊗ αH (f2(0))αH (h2)αH (f1[1]), for all h ∈ H, f ∈ H∗.

1850133-6 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

Deﬁnition 2.8. Let (H, αH ) be a Hom-bialgebra, (D, αD)anH-bimodule Hom- algebra and an H-bicomodule Hom-coalgebra. For all h ∈ H, a, b ∈ D, consider the following list of conditions:

εD(1) = 1,εD(ab)=εD(a)εD(b), (2.4)

εD(ha)=εD(ah)=εH (h)εD(a), (2.5)

ρ(1D)=1D ⊗ 1H ,λ(1D)=1H ⊗ 1D, (2.6) ρ(ab)=ρ(a)ρ(b),λ(ab)=λ(a)λ(b), (2.7)

∆D(1D)=1D ⊗ 1D, (2.8)

∆D(ha)=h1 a1 ⊗ h2 a2, ∆D(ha)=a1 h1 ⊗ a2 h2, (2.9) −2 −2 ∆(ab)=a1αD (a2(−1) b1[0]) ⊗ αD (a2(0) b1[1])b2, (2.10) 2 2 (h1 a)(−1)αH (h2) ⊗ (h1 a)(0) = αH (h1)αH (a(−1)) ⊗ αH (h2) · a(0), (2.11)

(ha)[0] ⊗ (ha)[1] = αH (h) a[0] ⊗ αH (a[1]), (2.12) 2 2 (ah2)[0] ⊗ αH (h1)(ah2)[1] = a[0] αH (h1) ⊗ αH (a[1])αH (h2), (2.13)

(ah)(−1) ⊗ (ah)(0) = αH (a(−1)) ⊗ a(0) αH (h), (2.14) −2 −2 −2 −2 αD (a[0]) αH (b(−1)) ⊗ αH (a[1]) αD (b(0))=a ⊗ b. (2.15)

If all these conditions hold, we will call (H, D) a Hom-L-R-admissible pair.

Proposition 2.9. Let (H, αH ) be a Hom-bialgebra, (D, αD) a Hom-algebra and Hom-coalgebra, and (H, D) a Hom-L-R-admissible pair. Then we have a Hom- bialgebra (DH, αD ⊗ αH ) under Hom-L-R-smash product and Hom-L-R-smash

J. Algebra Appl. Downloaded from www.worldscientific.com coproduct. We will call DH Hom-L-R-smash biproduct. by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. Proof. For all a, b ∈ A, h, g ∈ H, on one hand,

∆((ah)(bg)) −1 −2 −2 −1 −1 =∆((αD (a) αH (g2))(αH (h1) αD (b))αH (h2g1)) −1 −1 −2 −2 −1 = αD [((αD (a) αH (g2))(αH (h1) αD (b)))1[0]] −2 −1 −2 −2 −1 −2 αH [((αD (a) αH (g2))(αH (h1) αD (b)))2(−1)]αH (h21g11) −1 −1 −2 −2 −1 ⊗ αD [((αD (a) αH (g2))(αH (h1) αD (b)))2(0)] −2 −2 −1 −2 −2 −1 αH (h22g12)αH [((αD (a) αH (g2))(αH (h1) αD (b)))1[1]] −1 −1 −2 −3 −1 −2 −2 = αD [(αD (a) αH (g2))1[0][αH ((αD (a) αH (g2))2(−1)h11) αD (b1[0])][0]] −3 −2 −1 −2 −2 −2 −1 {αH [(αD ((αD (a) αH (g2))2(0)) αD (b1[1])(αH (h12) αB (b2)))(−1)]

1850133-7 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

−2 −1 −1 −2 −1 −2 × αH (h21)}αH (g11) ⊗ αD [(αD ((αD (a) αH (g2))2(0)) −2 −2 −1 −2 −2 αD (b1[1])(αH (h12) αD (b2)))(0)]αH (h22g12)αH −1 −2 −3 −1 −2 −2 × [(αD (a) αH (g2))1[1][αH ((αD (a) αH (g2))2(−1)h11) αD (b1[0])][1]] −1 −1 −2 −3 −1 −2 = αD [(αD (a1) αH (g21))[0][α ((αD (a2) αH (g22))(−1)h11) −2 −3 −2 −1 −2 αD (b1[0])][0]]{αH [(αD ((αD (a2) αH (g22))(0)) −2 −2 −1 −2 −1 αD (b1[1]))(−1)(αH (h12) αD (b2))(−1)]αH (h21)}αH (g11) −1 −2 −1 −2 −2 −2 ⊗ αD [(αD ((αD (a2) αH (g22))(0)) αD (b1[1]))(0)(αH (h12) −1 −2 −2 −1 −2 −3 −1 αD (b2))(0)]αH (h22g12)αH [(αD (a1) αH (g21))[1][αH ((αD (a2) −2 −2 αH (g22))(−1)h11) αD (b1[0])][1]] −1 −1 −2 −2 −2 = αD [(αD (a1) αH (g21))[0](αH (a2(−1)h11) αD (b1[0][0]))] −3 −1 −2 −1 −2 −1 {αH [αH (a2(0)(−1))(αH (h12) αD (b2))(−1)]αH (h21)}αH (g11) −1 −2 −2 −2 −1 ⊗ αD [(αD (a2(0)(0)) αH (g22b1[1]))(αH (h12) αD (b2))(0)] −2 −2 −1 −2 −1 αH (h22g12)αH [(αD (a1) αH (g21))[1]αH (b1[0][1])] −1 −1 −2 −2 −2 = αD [(αD (a1) αH (g21))[0](αH (a2(−1)h11) αD (b1[0][0]))] −3 −3 −1 (αH (a2(0)(−1))αH (h12b2(−1)))αH (g11) −1 −2 −2 −1 −1 ⊗ αD [(αD (a2(0)(0)) αH (g22b1[1]))(αH (h21) α (b2(0)))] −1 −3 −1 −2 −3 αH (h22)[(αH (g12(αD (a1) αH (g21))[1]))αH (b1[0][1])] −1 −1 −1 −2 −2 = αD [(αD (a1[0]) αH (g12))(αH (a2(−1)h11) αD (b1[0][0]))] −3 −3 −1 (α (a2(0)(−1))α (h12b2(−1)))α (g11)

J. Algebra Appl. Downloaded from www.worldscientific.com H H H ⊗ −1 −2 −2 −1 −1 by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. αD [(αD (a2(0)(0)) αH (g22b1[1]))(αH (h21) αD (b2(0)))] −1 −3 −3 αH (h22)[(αH (a1[1]g21)αD (b1[0][1])] −2 −2 −3 −3 =[(αD (a1[0]) αH (g12))(αH (a2(−1)h11) αD (b1[0][0]))] −3 −3 −1 (αH (a2(0)(−1))αH (h12b2(−1)))αH (g11) −3 −3 −2 −2 ⊗ [(αD (a2(0)(0)) αH (g22b1[1]))(αH (h21) αD (b2(0)))] −1 −3 −3 αH (h22)[(αH (a1[1]g21)αH (b1[0][1])]. On the other hand, ∆(ah)∆(bg) −1 −2 −1 −1 −2 −1 =[αD (a1[0])αH (a2(−1))αH (h1)][αD (b1[0])αH (b2(−1))αH (g1)] −1 −1 −2 −1 −1 −2 ⊗ [αD (a2(0))αH (h2)αH (a1[1])][αD (b2(0))αH (g2)αH (b1[1])]

1850133-8 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

−2 −4 −3 −4 −3 −2 =(αD (a1[0]) (αH (b2(−1)2)αH (g12)))((αH (a2(−1)1)αH (h11)) αD (b1[0])) −3 −2 −3 −2 (αH (a2(−1)2)αH (h12))(αH (b2(−1)1)αH (g11)) −2 −3 −4 −3 −4 −2 ⊗ (αD (a2(0)) (αH (g22)αH (b1[1]2)))((αH (h21)αH (a1[1]1)) αD (b2(0))) −2 −3 −2 −3 (αH (h22)αH (a1[1]2))(αH (g21)αH (b1[1]1)) −3 −4 −3 −3 −3 =(αD (a1[0][0]) (αH (b2(0)(−1))αH (g12)))((αH (a2(−1))αH (h11)) −3 −3 −2 −2 −2 αD (b1[0][0]))(αH (a2(0)(−1))αH (h12))(αH (b2(−1))αH (g11)) −3 −3 −3 −3 −4 ⊗ (αD (a2(0)(0)) (αH (g22)αH (b1[1])))((αH (h21)αH (a1[0][1])) −3 −2 −2 −2 −3 αD (b2(0)(0)))(αH (h22)αH (a1[1]))(αH (g21)αH (b1[0][1])) −4 −4 −2 −3 −3 =(αD (a1[0][0]) αH (b2(0)(−1)) αH (g12)))((αH (a2(−1))αH (h11)) −3 −3 −2 −2 −2 αD (b1[0][0]))(αH (a2(0)(−1))αH (h12))(αH (b2(−1))αH (g11)) −3 −3 −3 −2 −4 ⊗ (αD (a2(0)(0)) (αH (g22)αH (b1[1])))(αH (h21) (αH (a1[0][1]) −4 −2 −2 −2 −3 αD (b2(0)(0)))(αH (h22)αH (a1[1]))(αH (g21)αH (b1[0][1])) −2 −2 −3 −3 −3 =(αD (a1[0]) αH (g12)))((αH (a2(−1))αH (h11)) αD (b1[0][0])) −3 −2 −2 −2 (αD (a2(0)(−1))αH (h12))(αH (b2(−1))αH (g11)) −3 −3 −3 −2 −2 ⊗ (αD (a2(0)(0)) (αH (g22)αH (b1[1])))(αH (h21) αD (b2(0))) −2 −2 −2 −3 (αH (h22)αH (a1[1]))(αH (g21)αH (b1[0][1])). That is, ∆ is a Hom-algebra map. It is straightforward to verify that ε is also a Hom-algebra map. Thus (DH, αD ⊗ αH ) is a Hom-bialgebra. The proof is completed.

Example 2.10. Let D =span{1,x} over a ﬁxed ﬁeld k with chark =2.Then

J. Algebra Appl. Downloaded from www.worldscientific.com deﬁne β as a k-linear automorphism of D by

by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. β(1D)=1D,β(x)=−x. Deﬁne the multiplication on D by 2 1D1D =1D, 1Dx = x1D = −x, x =0, then (D, β) is a Hom-algebra. For D, deﬁne the Hom-coalgebra structure by

∆(1D)=1D ⊗ 1D, ∆(x)=(−x) ⊗ 1+1⊗ (−x),

ε(1D)=1,ε(x)=0. Let H =span{1,g|g2 =1} be the group algebra. Obviously, (H, id) is a Hom– Hopf algebra. Deﬁne the left action : H ⊗ D → D of H on D by

1H 1D =1D, 1H x= −x, g 1D =1D,gx= x. It is easy to see that D is a left H-module Hom-algebra.

1850133-9 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

Deﬁne the right action : D ⊗ H → D of H on D by

1D 1H =1D,x1H = −x, 1D g=1D,xg= x.

Then D is a right H-module Hom-algebra. Moreover, D is an H-bimodule. Deﬁne the left coaction λ : D → H ⊗ D of H on D by

λ(1D)=1H ⊗ 1D,λ(x)=g ⊗ (−x),

then D is a left H-comodule Hom-coalgebra. Deﬁne the right coaction ρ : D → D ⊗ H of H on D by

ρ(1D)=1D ⊗ 1H ,ρ(x)=(−x) ⊗ g,

then D is a right H-comodule Hom-coalgebra. Moreover D is an H-bicomodule. It is straightforward to see that (H, D) is a Hom-L-R-admissible pair. Hence, we have Hom-L-R-smash biproduct (DH, β⊗id) with the following Hom-bialgebra structure:

• Multiplication:

1D1H 1Dg x1H xg

1D1H 1D1H 1Dg −x1H −xg

1Dg 1Dg 1D1H xg x1H

x1H −x1H xg 00

xg −xg x1H 00

• Comultiplication:

J. Algebra Appl. Downloaded from www.worldscientific.com ∆(1D1H )=1D1H ⊗ 1D1H , ∆(1Dg)=1Dg ⊗ 1Dg, by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. ∆(x1H )=(−x)1H ⊗ 1Dg +1Dg ⊗ (−x) ⊗ 1H ,

∆(xg)=(−x)g ⊗ 1D1H +1D1H ⊗ (−x) ⊗ g.

Next, we will describe the Hom-L-R-admissible pair from the categorical point. Let (H, αH ) be a Hom-bialgebra. We will introduce a prebraided monoidal category associated to H, denoted by HLR(H). The objects of HLR(H) are vector spaces M endowed with H-bimodule and H-bicomodule structures (denoted by h ⊗ m → hm,m⊗ h → mh,m→ m(−1) ⊗ m(0),m→ m[0] ⊗ m[1], for all h ∈ H, m ∈ M), such that M is a left-left Yetter–Drinfel’d module, a left-right Long module, a right-right Yetter–Drinfel’d module and a right-left Long module, i.e.

2 2 (h1 m)(−1)αH (h2) ⊗ (h1 m)(0) = αH (h1)αH (m(−1)) ⊗ αH (h2) m(0), (2.16)

(hm)[0] ⊗ (hm)[1] = αH (h) m[0] ⊗ αH (m[1]), (2.17)

1850133-10 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

2 2 (mh2)[0] ⊗ αH (h1)(mh2)[1] = m[0] αH (h1) ⊗ αH (m[1])αH (h2), (2.18)

(mh)(−1) ⊗ (mh)(0) = αH (m(−1)) ⊗ m(0) αH (h), (2.19) for all h ∈ H, m ∈ M. The morphisms in HLR(H)areH-bilinear and H-bicolinear maps. One can verify that HLR(H) is a monoidal category with the following structures: for all M,N ∈HLR(H), and m ∈ M,n ∈ N,h ∈ H,

h · (m ⊗ n)=h1 m⊗ h2 n, −2 (m ⊗ n)(−1) ⊗ (m ⊗ n)(0) = αH (m(−1)n(−1)) ⊗ m(0) ⊗ n(0),

(m ⊗ n) · h = mh1 ⊗ nh2, −2 (m ⊗ n)[0] ⊗ (m ⊗ n)[1] = m[0] ⊗ n[0] ⊗ αH (m[1]n[1]).

Proposition 2.11. The category HLR(H) is prebraided monoidal category. For all M,N ∈HLR(H), and m ∈ M,n ∈ N, the braiding is given by ⊗ → ⊗ ⊗ → −2 ⊗ −2 cM,N : M N N M, m n αN (m(−1) n[0]) αM (m(0) n[1]). Moreover, if H has a bijective antipode S, HLR(H) becomes braided with the inverse of c given by −1 ⊗ → ⊗ cM,N : N M M N, −2 −1 −2 −1 n ⊗ m → αM (m(0) S (n[1])) ⊗ αN (S (m(−1)) n[0]).

Proof. Firstly for all M,N ∈HLR(H)andm ∈ M,n ∈ N, h ∈ H,

cM,N (h(m ⊗ n))

= cM,N (h1 m⊗ h2 n) J. Algebra Appl. Downloaded from www.worldscientific.com −2 −2 1 2 ⊗ 1 2 by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. = αN ((h m)(−1) (h n)[0]) αM ((h m)(0) (h n)[1]) −2 −2 = αN ((h1 m)(−1) (αH (h2) n[0]) ⊗ αM ((h1 m)(0) αH (n[1])) −2 −1 2 −2 = αN (αH ((h1 m)(−1)αH (h2)) αN (n[0])) ⊗ αM ((h1 m)(0) αH (n[1])) −2 −2 = αN (aH (h1)m(−1) αN (n[0])) ⊗ αM ((αH (h2) m(0)) αH (n[1])) −2 −2 = h1 αN (m(−1) n[0]) ⊗ h2 αM (m(0) n[1]),

thus cM,N is a left H-module map. Similarly, one can verify that cM,N is a right H-module map.

cM,N (m ⊗ n)(−1) ⊗ cM,N (m ⊗ n)(0) −4 −2 = αH ((m(−1) n[0])(−1)αH (m(0)(−1))) ⊗ αN ((m(−1) n[0])(0)) −2 ⊗ αM (m(0)(0) αH (n[1]))

1850133-11 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

−5 2 −3 = αH ((m(−1)1 αN (n[0]))(−1)αH (m(−1)2)) ⊗ αN ((m(−1)1 αN (n[0]))(0)) −2 ⊗ αM (αM (m(0)) αH (n[1])) −3 −2 −2 = αH (m(−1)1n[0](−1)) ⊗ αN (m(−1)2 n[0](0)) ⊗ αM (αM (m(0)) αH (n[1])) −2 −2 −2 = αH (m(−1)n(−1)) ⊗ αN (m(0)(−1) n(0)[0]) ⊗ αM (m(0)(0) n(0)[1]) −2 ⊗ ⊗ = αH (m(−1)n(−1)) cM,N (m(0) n(0))

=(m ⊗ n)(−1) ⊗ cM,N ((m ⊗ n)(0)),

thus cM,N is a left H-comodule map. Similarly, one can verify that cM,N is a right H-comodule map. It is straightforward to check that c satisﬁes the hexagonal equations. Hence, it is a prebraiding in HLR(H). Furthermore, if H has bijective antipode S, −1 ◦ ⊗ −1 −2 ⊗ −2 cM,N cM,N (m n)=cM,N (αN (m(−1) n[0]) αM (m(0) n[1])) −2 ⊗ −2 −1 ⊗ =(αM αN )cM,N (m(−1) n[0] m(0) n[1]) −4 −4 −1 =(αM ⊗ αN )((m(0) n[1])(0) S (m(−1) n[0])[1] −1 ⊗ S (m(0) n[1])(−1) (m(−1) n[0])[0]) −4 −4 −1 =(αM ⊗ αN )((m(0)(0) αH (n[1])) S (αH (n[0][1])) −1 ⊗ S (αH (m(0)(−1))) (αH (m(−1)) n[0][0]) −4 −4 2 −1 =(αM ⊗ αN )(αM (m(0)) n[1]2S (n[1]1)) −1 2 ⊗ (S (m(−1)2)m(−1)1 αN (n[0])) = m ⊗ n. J. Algebra Appl. Downloaded from www.worldscientific.com −1 ◦ ◦ −1 Thus cM,N cM,N =idM⊗N . Similarly, we have cM,N cM,N =idN⊗M .Thenc is by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. invertible; thus it is a braiding. The proof is completed.

H H Remark 2.12. We denote by H YD and YDH the categories of left-left and right- H right Hom–Yetter–Drinfel’d modules, respectively. Assume that V ∈ H YD and W ∈ H YDH ,thenV ⊗ W ∈HLR(H) with the structure given in Proposition 2.12. In H H particular, for W = k and respectively V = k, we obtain that H YD and YDH are subcategories of HLR(H), and one can see that they are actually braided subcategories, that is, the braiding of HLR(H) restricts to the usual braidings of H H H YD and YDH . We can state now the categorical interpretation of Hom-L-R-admissible pair.

2 Proposition 2.13. Let (H, αH ) be a Hom-bialgebra with αH =id, and (D, αD) a 2 vector space with αD =id.Then(H, D) is a Hom-L-R-admissible pair if and only if D is a bialgebra in HLR(H) satisfying the compatible condition (1.15).

1850133-12 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

Proof. The proof is straightforward and left to the reader.

3. Categorical Equivalence

Throughout this section, we assume that the Hom-bialgebra (H, αH ) is ﬁnite dimensional.

Lemma 3.1. Let (H, αH ) be a Hom-bialgebra. Then we have a functor H⊗H∗ F : HLR(H) → H⊗H∗ YD given for any object M ∈HLR(H) and any morphism ϑ by F (M)=M and F (ϑ)=ϑ, where H ⊗ H∗ is a Hom-bialgebra with tensor product and tensor coproduct.

Proof. For all M ∈HLR(H), ﬁrst of all, deﬁne the left action of H ⊗ H∗ on M by −1 (h ⊗ f) · m = f,m[1]hαM (m[0]), (2.20) for all h ∈ H, f ∈ H∗ and m ∈ M.ThenM is a left H ⊗ H∗-module. Indeed for all h, h ∈ H, f, f ∈ H∗ and m ∈ M, (h ⊗ f)(h ⊗ f ) · αM (m) = ff ,m[1]hh m[0] −1 −1 −1 = f,αH (m[1]1)f ,αH (m[1]2)αH (h) (h αM (m[0])) −1 −2 = f,αH (m[0][1])f ,m[1]αH (h) (h αM (m[0][0])) −1 −1 −1 −1 = f,αH ((h · αM (m[0]))[1])f ,m[1]αH (h) αM ((h αM (m[0]))[0]) ⊗ −1∗ · −1 J. Algebra Appl. Downloaded from www.worldscientific.com = f ,m[1] (αH (h) αH (f)) (h αM (m[0])) −1∗ by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. =(αH (h) ⊗ αH (f)) · ((h ⊗ f ) · m). And

(1 ⊗ ε) · m = ε, m[1]m[0] = αM (m), as claimed. Next for all m ∈ M, deﬁne the left coaction of H ⊗ H∗ on M by −2∗ i −1 ρ(m)=m−1 ⊗ m0 = m(−1) ⊗ αH (h ) ⊗ αM (m(0)) hi, (2.21) i where {hi} and {h } are dual bases of H. Then on one hand, −2∗ i (∆H⊗H∗ ⊗ αH )ρ(m)= m(−1)1 ⊗ αH (h1) ⊗ m(−1)2

−2∗ i ⊗ αH (h2) ⊗ m(0) αH (hi). Evaluating the right side of the equation on id ⊗ g ⊗ id ⊗ h ⊗ id, we obtain −3 m(−1)1 ⊗ m(−1)2 ⊗ m(0) αH (gh).

1850133-13 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

On the other hand −1∗ −3∗ i −1 (αH ⊗ αH ⊗ ρ)ρ(m)= αH (m(−1)) ⊗ αH (h ) ⊗ (αM (m(0)) hi)(−1)

−2∗ j −1 −1 ⊗ αH (h ) ⊗ αM ((αM (m(0)) hi)(0)) hj −3∗ i −2∗ j = αH (m(−1)) ⊗ αH (h ) ⊗ m(0)(−1) ⊗ αH (h )

−1 −1 ⊗ αM (αM (m(0)(0)) αH (hi)) hj −3∗ i −2∗ j = m(−1)1 ⊗ αH (h ) ⊗ m(−1)2 ⊗ αH (h )

−1 ⊗ (αM (m(0)) hi) hj. Evaluating the right side of the equation on id ⊗ g ⊗ id ⊗ h ⊗ id, we obtain −3 m(−1)1 ⊗ m(−1)2 ⊗ m(0) αH (gh). Since g,h ∈ H were arbitrary, we have −1∗ (∆H⊗H∗ ⊗ αM )ρ =(αH ⊗ αH ⊗ ρ)ρ. And since ∗ (ε ⊗ ε ⊗ id)ρ(m)=ε(m(−1))m(0) = αM (m). M is a left H ⊗ H∗-comodule. Finally 2 −2∗ [(h1 ⊗ f1) · m]−1(αH (h2) ⊗ αH (f2)) ⊗ [(h1 ⊗ f1) · m]0 −1 2 −2∗ −1 =(h1 αM (m[0]))−1f1,m[1](αH (h2) ⊗ αH (f2)) ⊗ (h1 αM (m[0]))0 −1 2 −2∗ i −2∗ = f1,m[1](h1 αM (m[0]))(−1)αH (h2) ⊗ αH (h )αH (f2)

−1 −1 ⊗ αM ((h1 αM (m[0]))(0)) hi J. Algebra Appl. Downloaded from www.worldscientific.com 2 ⊗ −2∗ i −2∗ ⊗ −2 by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. = f1,m[1] αH (h1)m[0](−1) αH (h )αH (f2) (h2 αM (m[0](0))) hi. Evaluating the right side of the equation on id ⊗ g ⊗ id, we obtain −2 −6 2 −2 −4 f,αH (m[1])αH (g2)αH (h1)m[0](−1) ⊗ (h2 αM (m[0](0))) αH (g1). And 2 −2∗ −1∗ −1∗ (αH (h1) ⊗ αH (f1))(αH ⊗ αH )m−1 ⊗ (αH (h2) ⊗ αH (f2)) · m0 2 −2∗ −3∗ i = αH (h1)αH (m(−1)) ⊗ αH (f1)αH (h )

−1 −1 −1 −1 ⊗f2,αH ((αM (m(0)) hi)[1])αH (h2) αM ((αM (m(0)) hi)[0]). Evaluating the right side of the equation on id ⊗ g ⊗ id, we obtain 2 −1 −1 αH (h1)αH (m(−1)) ⊗ αH (h2) αM ((αM (m(0)) hi)[0]) −4 i −5 −1 −1 f1,αH (g1)h ,αH (g2)f2,αH ((αM (m(0)) hi)[1])

1850133-14 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

2 −2 −4 = αH (h1)αH (m(−1)) ⊗ αH (h2) αM ((m(0) αH (g2))[0]) −4 −2 −4 f,αH (αH (g1)(m(0) αH (g2))[1]) 2 −2 −5 = αH (h1)αH (m(−1)) ⊗ αH (h2) (αM (m(0)[0]) αH (g1)) −3 −6 f,αH (m(0)[1])αH (g2) 2 −2 −5 = αH (h1)m[0](−1) ⊗ αH (h2) (αM (m[0](0)) αH (g1)) −2 −6 f,αH (m[1])αH (g2). Therefore M is a left-left Yetter–Drinfeld module over H ⊗H∗. It is straightforward H⊗H∗ to verify that any morphism in HLR(H) is also a morphism in H⊗H∗ YD. The proof is completed.

Lemma 3.2. Let (H, α) be a Hom-bialgebra. Then we have a functor H⊗H∗ H⊗H∗ G : H⊗H∗ YD → HLR(H) given for any object M ∈ H⊗H∗ YD and any morphism θ by G(M)=M and G(θ)=θ.

H⊗H∗ ∗ Proof. For any M ∈ H⊗H∗ YD,denotetheleftH ⊗ H -coaction on M by

m → m−1 ⊗ m0, for all m ∈ M. Deﬁne the H-bimodule and H-bicomodule structures as follows: hm=(h ⊗ ε) · m, (2.22) ∗ ρL(m)=m(−1) ⊗ m(0) =(id⊗ ε )(m−1) ⊗ m0, (2.23) 2 mh= (ε ⊗ id)m−1,αH (h)m0, (2.24) i J. Algebra Appl. Downloaded from www.worldscientific.com ρR(m)=m[0] ⊗ m[1] = (1 ⊗ h ) · m ⊗ hi. (2.25)

by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. for all h ∈ H. Obviously, M is a left H-module. And ∗ (∆ ⊗ αM )ρL(m) = ∆((id ⊗ ε )(m−1)) ⊗ α(m0) ∗ ∗ =(id⊗ ε )(m−11) ⊗ (id ⊗ ε )(m−12) ⊗ α(m0) ∗ ∗ =(αH ⊗ ε )(m−1) ⊗ (id ⊗ ε )(m0−1) ⊗ m00 −1∗ =(α ⊗ αH ⊗ ρL)ρL(m). The counit is straightforward. Thus, M is a left H-comodule. For all h, h ∈ M, −1∗ 2 αM (m) hh = (ε ⊗ αH )m−1,αH (hh )αM (m0) = (ε ⊗ id)m−1,α(hh )αM (m0) 3 3 = (ε ⊗ id)m−11,αH (h)(ε ⊗ id)m−12,αH (h )αM (m0)

1850133-15 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

2 3 = (ε ⊗ id)m−1,αH (h)(ε ⊗ id)m0−1,αH (h )m00 2 = (ε ⊗ id)m−1,αH (h)m0 αH (h ) =(mh) αH (h ). The unit is obvious. Thus M is a right H-module. Since −1∗ i (αM ⊗ ∆)ρR(m)= (1 ⊗ αH (h )) · αM (m) ⊗ hi1 ⊗ hi2 −1∗ i j 2 2 = (1 ⊗ αH (h h )) · αM (m) ⊗ αH (hi) ⊗ αH (hj) ∗ i j = (1 ⊗ αH (h )h ) · αM (m) ⊗ hi ⊗ αH (hj)

=(ρR ⊗ αH )ρR(m). Thus M is a right H-comodule. Moreover (hm) αH (h )=((h ⊗ ε) · m) αH (h ) 3 = (ε ⊗ id)((h ⊗ ε) · m)−1,αH (h )((h ⊗ ε) · m)0 ∗ −1 = (ε ⊗ id)[(id ⊗ αH )(((αH (h1) ⊗ ε) · m)−1(αH (h2) ⊗ ε))], 3 −1 × αH (h )((αH (h1) ⊗ ε) · m)0 ∗ −1∗ = (ε ⊗ id)[(id ⊗ αH )((αH (h1) ⊗ ε)(αH ⊗ αH )m−1)], 3 × αH (h )((h2 ⊗ ε) · m0) 2 = (ε ⊗ id)m−1,αH (h )(αH (h) ⊗ ε) · m0 = αH (h) (mh). Thus, M is an H-bimodule. And

J. Algebra Appl. Downloaded from www.worldscientific.com (ρL ⊗ αH )ρR(m)

by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. ∗ i i = (id ⊗ ε )((1 ⊗ h ) · m)−1 ⊗ ((1 ⊗ h ) · m)0 ⊗ αH (hi) −1 ∗ ∗ i i = (αH ⊗ ε )[((1 ⊗ αH (h1)) · m)−1(1 ⊗ h2)]

∗ i ⊗ ((1 ⊗ αH (h1)) · m)0 ⊗ αH (hi) −1 ∗ −1∗ i −1∗ = (αH ⊗ ε )((1 ⊗ αH (h1))(αH ⊗ αH )m−1)

i ⊗ (1 ⊗ h2) · m0 ⊗ αH (hi) ∗ −1∗ i = (αH ⊗ ε )(m−1) ⊗ (1 ⊗ αH (h )) · m0 ⊗ αH (hi) ∗ i = (αH ⊗ ε )(m−1) ⊗ (1 ⊗ h ) · m0 ⊗ hi

=(αH ⊗ ρR)ρL(m). Thus, M is an H-bicomodule.

1850133-16 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

We now prove (2.16). For all h ∈ H, m ∈ M, 2 (h1 m)(−1)αH (h2) ⊗ (h1 m)(0) 2 =((h1 ⊗ ε) · m)(−1)αH (h2) ⊗ ((h1 ⊗ ε) · m)(0) ∗ 2 =(id⊗ ε )(((h1 ⊗ ε) · m)−1(αH (h2) ⊗ ε)) ⊗ ((h1 ⊗ ε) · m)0 ∗ 2 −1∗ =(id⊗ ε )((αH (h1) ⊗ ε)(αH ⊗ αH )m−1) ⊗ (αH (h2) ⊗ ε) · m0 2 ∗ = αH (h1)(αH ⊗ ε )m−1 ⊗ (αH (h2) ⊗ ε) · m0 2 = αH (h1)αH (m(−1)) ⊗ αH (h2) m(0). We now prove (2.17):

(hm)[0] ⊗ (hm)[1] =((h ⊗ ε) · m)[0] ⊗ ((h ⊗ ε) · m)[1] i = (1 ⊗ h ) · ((h ⊗ ε) · m) ⊗ hi i = (αH (h) ⊗ h ) · αM (m) ⊗ hi i = (αH (h) ⊗ h ) · αM (m) ⊗ hi i = (αH (h) ⊗ ε) · ((1 ⊗ h ) · m) ⊗ αH (hi)

= αH (h) m[0] ⊗ αH (m[1]). We now prove (2.18): On one hand, 2 (mh2)[0] ⊗ αH (h1)(mh2)[1] 2 2 = (ε ⊗ id)m−1,αH (h2)m0[0] ⊗ αH (h1)m0[1] 2 i 2 = (ε ⊗ id)m−1,αH (h2)(1 ⊗ h ) · m0 ⊗ αH (h1)hi. ∗ J. Algebra Appl. Downloaded from www.worldscientific.com Evaluating the right side on id ⊗ f for all f ∈ H ,wehave 2 2∗ 4 by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. (ε ⊗ id)m−1,αH (h2)(1 ⊗ αH (f2)) · m0f1(αH (h1)) 2∗ 4 2∗ = (ε ⊗ id)(1 ⊗ αH (f1))m−1,αH (h)(1 ⊗ αH (f2)) · m0 ∗ −1∗ = (ε ⊗ id)(((1 ⊗ αH (f1)(αH ⊗ αH )m−1))), 5 2∗ × αH (h)(1 ⊗ αH (f2)) · m0 3∗ ∗ = (ε ⊗ id)(((1 ⊗ αH (f1)) · m)−1(1 ⊗ αH (f2))), 5 3∗ × αH (h)((1 ⊗ αH (f1)) · m)0. On the other hand, 2 m[0] αH (h1) ⊗ αH (m[1])αH (h2) i 2 = ((1 ⊗ h ) · m) αH (h1) ⊗ αH (hi)αH (h2) i 3 i 2 = (ε ⊗ id)(((1 ⊗ h ) · m)−1),αH (h1)((1 ⊗ h ) · m)0 ⊗ αH (hi)αH (h2).

1850133-17 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

D. Lu & X. Zhang

Evaluating the right side on id ⊗ f,wehave

i 3 i 3 4 (ε ⊗ id)((1 ⊗ h ) · m)−1,αH (h1)((1 ⊗ h ) · m)0f1,αH (hi)f2,αH (h2) 3∗ 3 = (ε ⊗ id)((1 ⊗ αH (f1)) · m)−1,αH (h1) 3∗ ∗ 3 × ((1 ⊗ αH (f1)) · m)0αH (f2),αH (h2) 3∗ ∗ 5 3∗ = (ε ⊗ id)(((1 ⊗ αH (f1)) · m)−1(1 ⊗ αH (f2))),αH (h)((1 ⊗ αH (f1)) · m)0.

2 2 Hence, (mh2)[0] ⊗ αH (h1)(mh2)[1] = m[0] αH (h1) ⊗ αH (m[1])αH (h2)sincef was arbitrary. We now prove (2.19):

(mh)(−1) ⊗ (mh)(0) 2 ∗ = (ε ⊗ id)m−1,αH (h)(id ⊗ ε )(m0−1) ⊗ m00 ∗ 2 ∗ = (ε ⊗ αH )m−11,αH (h)(id ⊗ ε )(m−12) ⊗ αM (m0) 3 ∗ = (ε ⊗ id)m−11,αH (h)(id ⊗ ε )(m−12) ⊗ αM (m0) ∗ 3 =(id⊗ ε )(m−11) ⊗(ε ⊗ id)m−12,αH (h)αM (m0) ∗ 3 =(αH ⊗ ε )(m−1) ⊗(ε ⊗ id)m0−1,αH (h)m00

= αH (m(−1)) ⊗ m(0) αH (h).

Therefore M ∈HLR(H). It is straightforward to verify that any morphism in H⊗H∗ H⊗H∗ YD is also a morphism in HLR(H). The proof is completed.

To summarize, we have the main result of this section.

J. Algebra Appl. Downloaded from www.worldscientific.com Theorem 3.3. Let (H, αH ) be Hom-algebra. Then we have an isomorphism of

by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. prebraided monoidal category ∼ H⊗H∗ HLR(H) = H⊗H∗ YD.

Furthermore, when H has a bijective antipode, this isomorphism is braided.

Proof. Easy to see that functor F is monoidal. For all M,N ∈HLR(H)and m ∈ M,n ∈ N,

−2 2∗ −1 −1 (αH ⊗ αH )m[−1] · αN (n) ⊗ αM (m[0]) (5.3.2) −2 ∗ i −1 −2 −1 = (αH (m(−1)) ⊗ αH (h )) · αN (n) ⊗ αM (m(0)) αH (hi) (5.3.6) −2 −2 −2 −2 = αH (m(−1)) αN (n0) ⊗ αM (m(0)) αH (n1).

Hence, F is prebraided. The proof is completed.

1850133-18 2nd Reading

July 14, 2017 15:15 WSPC/S0219-4988 171-JAA 1850133

Category of Hom–Yetter–Drinfel’d–Long bimodules

Acknowledgments The work was partially supported by the NSF of China (No. 11371088), the TianYuan Special Funds of the National Natural Science Foundation of China (No. 11626138), the NSF of Shandong Province (No. ZR2016AQ03), and the Fundamen- tal Research Funds for the Central Universities (No. KYLX15 0109).

References [1] J. N. Alonso Alvarez,´ J. M. Fern´andez Vilaboa and R. Gonz´alez Rodr´ıguez, Cleft extensions and Galois extensions for Hom-associative algebras, Int. J. Math. 27 (2016) 1650025. [2] S. Caenepeel and I. Goyvaerts, Monoidal Hom–Hopf algebras, Comm. Alg. 39 (2011) 2216–2240. [3] V. G. Drinfeld, Quantum groups, Proc. Internat. Congr. Math. Berkeley 1 (1986) 789–820. [4] S. J. Guo and X. H. Zhang, Separable functors for the category of Doi Hom–Hopf modules, Colloquium Math. 143 (2016) 23–37. [5] J. T. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2006) 314–361. [6] D. W. Lu and S. H. Wang, The Drinfeld double versus the Heisenberg double for Hom–Hopf algebras, J. Algebra Appl. 15 (2016) 1650059. [7] D. W. Lu and S. H. Wang, Yetter–Drinfeld–Long bimodule are modules, Czechoslovak Math. J. 67 (2017) 379–387. [8] A. Makhlouf and F. Panaite, Yetter–Drinfeld modules for Hom-bialgebras, J. Math. Phys. 55 (2014) 013501. [9] A. Makhlouf and F. Panaite, Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfel’d double of a Hom–Hopf algebra, J. Algebra 441 (2015) 314–343. [10] A. Makhlouf and S. D. Silvestrov, Hom-algebra structure, J. Gen. Lie Theory Appl. 2 (2008) 52–64. [11] A. Makhlouf and S. D. Silvestrov, Hom-algebras and Hom-coalgebras, J. Algebra Appl. 9 (2010) 553–589. J. Algebra Appl. Downloaded from www.worldscientific.com [12] F. Panaite and F. V. Oystaeyen, L-R-smash biproducts, double biproducts and a braided category of Yetter–Drinfeld–Long bimodules, Rocky Mountain J. Math. 40 by QUFU NORMAL UNIVERSITY on 10/11/17. For personal use only. (2008) 2013–2024. [13] D. Yau, Hom-quantum groups I: Quasitriangular Hom-bialgebras, J. Phys. A 45 (2012) 065203.

1850133-19