Abstract：Let (H,α) be a monoidal Hom-bialgebra and (B,β) be a left (H,α)-Hom-comodule coalgebra. The new monoidal Hom-algebra 
H is constructed with a Hom-twisted product Bσ[H] and a B×H Hom-smash coproduct. Moreover, a sufficient and necessary condition for H to be a monoidal Hom-bialgebra is given. In addition, let (H,α) be a Hom-σ-Hopf algebra with Hom-σ-antipode SH, and a sufficient condition for this new monoidal Hom-bialgebra H with the antipode S defined by S(b×h)=(1B×SH(α-1(b(-1))))·(SB(b(0))×1H) to be a monoidal Hom-Hopf algebra is derived.
Key words：monoidal Hom-Hopf algebra; Hom-twisted product; Hom-smash coproduct
Caenepeel et al.[1] described Hom-structures from the point of view of monoidal categories and introduced monoidal Hom-algebras, monoidal Hom-coalgebras, monoidal Hom-Hopf algebras, etc. In 2013, Chen et al.[2] introduced integrals for monoidal Hom-Hopf algebras and Hom-smash products. Then, Liu et al.[3] constructed a Hom-smash coproduct, which generalized the classical smash coproduct and illustrated the category of Hom-Yetter-Drinfeld modules and proved that the category is a braided monoidal category. In Ref.[4], the authors constructed a Hom-crossed product, which generalizes the classical crossed product introduced in Refs.[5-7]. Can we give a new monoidal Hom-algebra construction of the special Hom-crossed product and the Hom-smash coproduct?
In this paper, we give a positive answer to the question above.We recall the basic definitions concerning monoidal Hom-Hopf algebras, and we find a sufficient and necessary condition for 
H, with the Hom-twisted product Bσ[H] and the Hom-smash coproduct B×H for a comodule coalgebra (B,β) over (H,α), to form a monoidal Hom-bialgebra and derive a sufficient condition for this new bialgebra to be a monoidal Hom-Hopf algebra.
Throughout this paper, we refer the readers to the books of Sweedler[8] for the relevant concepts on the general theory of Hopf algebras.
Let Mk=(Mk,⊗,k,a,l,r) denote the usual monoidal category of k-vector space and linear maps between them. Recall from Ref.[1] that there is the Hom-category 
=(H(Mk), (k, 
), a new monoidal category, associated with Mk as follows:
3) The tensor product is given by (M, μ)⊗(N, ν)=(M⊗N, μ⊗
4) The tensor unit is given by (k, id);
⊗id)⊗ζ-1)=(μ⊗(id⊗
⊗
⊗
).
A unital monoidal Hom-assciative algebra is a vector space A together with an element 1A∈A and linear maps m: A⊗A→A; a⊗b|→ab, α∈Autk(A) such that α(1A)=1A, a1A=1A a=α(a), α(a)(bc)=(ab)α(c),α(ab)=α(a)α(b), for all a, b,c∈A.
A counital monoidal Hom-coassociative coalgebra (C,γ) is a vector space C together with linear maps Δ: C→C⊗C, Δ(c)=c1⊗c2 and ε: C→k such that Δ(γ(c))=γ(c1)⊗γ(c2),γ-1(c1)⊗Δ(c2)=Δ(c1)⊗γ-1(c2),c1ε(c2)=γ-1(c)=c2ε(c1),ε(γ(c))=ε(c), for all c∈C.
A monoidal Hom-bialgebra H=(H,α,m,η,Δ,ε) in Ref.
i.e., for any h, g∈H, Δ(hg)=Δ(h)Δ(g), Δ(1H)=1H⊗1H, ε(hg)=ε(h)ε(g), ε(1H)=1.
S.
). Furthermore, the antipode of monoidal Hom-Hopf algebras has almost all of the properties of antipode of Hopf algebras such as S(hg)=S(g)S(h), S(1H)=1H, Δ(S(h))=S(h2)⊗S(h1),ε∘ S=ε.
S is a monoidal Hom-anti-(co)algebra homomorphism. Since α is bijective and commutes with antipode S, we can also have the inverse α-1 commuting with S, that is, S∘ α-1=α-1∘ S. In the following, we recall the actions and coactions on monoidal Hom-coalgebras.
Now let (C,γ) be a monoidal Hom-coalgebra.
⊗C, ρM(m)=m(-1)⊗m(0) such that ε(m(-1))m(0)=μ-1(m), ΔC(m(1))⊗μ-1(m(0))=γ-1(m(-1))⊗(m(0)(-1)⊗m(0)(0)),ρM(μ(m))=γ(m(-1))⊗μ(m(0)), for all m∈M.
In Ref.[3], a left (H,α)-Hom-comodule coalgebra (B,β) is a monoidal Hom-coalgebra, with a monoidal Hom-bialgebra (H,α) and a left (H,α)-Hom-comodule (B,β), obeying the following axioms: for all b∈B,b(-1)⊗ΔB(b(0))=b1(-1)b2(-1)⊗b1(0)⊗b2(0), b(-1)ε(b(0))=ε(b)1H.
Recall from Ref.[3] that there is the Hom-smash coproduct (B×H,β×α), with a left (H,α)-Hom-comodule (B,β) and the Hom-comultiplication given by Δ(b×h)=(b1×b2(-1)α-1(h1))⊗(β(b2(0))×h2), for all b∈B, h∈H.
Recalled from Ref.[4] that the Hom-crossed product B#σH of B with H is the vector space B⊗H, with a monoidal Hom-Hopf algebra (H,α) and a monoidal Hom-algebra (B,β). H acts weakly on B and σ: H⊗H→B is convolution invertible. The multiplication is given by (a#h)(b#k)=a[(h11β-2(b))σ(h12,α-1(k1))]# α(h2k2), for any h, k∈H and a, b∈B.
HIf B#σH is a Hom-associative algebra with 1⊗1 as an identity element, then we call B#σH a Hom-crossed product. A necessary and sufficient condition for B#σH to be a Hom-crossed product is that σ satisfies the following conditions:
(α(h1)σ(l1,α(k1)))σ(α(h2), l2k2)=
σ(h1, l1)σ(α-1(h2l2),α-1(k))
(1)
σ(h1, l1)(h2l2a)=(α(h1)(l1β-1(a)))σ(α(h2),α(l2))
(2)
σ(1, h)=σ(h, 1)=ε(h)1B
(3)
for all h, k, l∈H and a∈B.
If the action “·” is trivial, that is, ha=εH(h)1Ba, for every h∈H, a∈B, then we write B#σH=Bσ[H]. In the following, we call the Hom-crossed product Bσ[H] a Hom-twisted product, with multiplication:
(a#h)(b#k)=β-1(ab)σ(h1, k1)#α(h2k2)
If the action “·” is trivial, then Eqs.(1) and (2) are, respectively, substituted into
σ(α(l1),α2(k1))σ(h, l2k2)=
σ(h1,l1)σ(α-1(h2l2),α-1(k))
(4)
σ(α-1(h),α-1(l))β(a)=β(a)σ(h, l)
(5)
for all h, k, l∈H and a∈B.
Lemma 1 Let (B,β) be a monoidal Hom-algebra and (H,α) be a monoidal Hom-bialgebra, with a Hom-twisted product Bσ[H] given by
(a#h)(b#k)=β-1(ab)σ(h1, k1)#α(h2k2)
Then we have the following conclusions:
1) The associativity of Bσ[H] is satisfied if and only if Eqs.(4) and (5) hold.
2) 1B#1H is the unit of Bσ[H] if and only if Eq.(3) holds.
3) Bσ[H] is an associative algebra if and only if Eqs.(3) to (5) hold.
Let (H,α) be a monoidal Hom-bialgebra. Let (B,β,ΔB,εB) be a left (H,α)-comodule coalgebra and (B,β, mB,ηB) be a monoidal Hom-algebra. Let σ: H⊗H→B be a linear map. In this section, we derive necessary and sufficient conditions for B⊗H to be a monoidal Hom-bialgebra.
If (B⊗
H.
First, we have the following lemmas:
Lemma 2 Let Bσ[H] be a Hom-twisted product and B×H be a Hom-smash coproduct. The following are equivalent:
β(b1)ε(h)⊗β(b2)=b1σ(α-2(h), b2(-1))⊗β2(b2(0))
(6)
α-1(h)b(-1)⊗β(b(0))=b(-1)α-1(h)⊗β(b(0))
(7)
for all b∈B and h∈H.
Lemma 3 Let Bσ[H] be a Hom-twisted product and B×H be a Hom-smash coproduct. If the identity ρ(1)=1⊗1 holds, then the following are equivalent:
σ(h, l)1⊗σ(h, l)2=σ(h1, l1)⊗σ(h2, l2)
(8)
σ(h1, l1)(-1)α-1(h2l2)⊗β(σ(h1, l1)(0))=
h1l1⊗σ(h2, l2)
(9)
Lemma 4 Let Bσ[H] be a Hom-twisted product and B×H be a Hom-smash coproduct. If (β(a1)⊗α(a2(-1))α-1(h1))⊗β2(a2(0))=(a1σ(α(a2(-1)1),α-1(h1))⊗α2(a2(-1)2)h2)⊗β2(a2(0)) holds, then we have
β(a1)ε(h)⊗β(a2)=a1σ(a2(-1),α-2(h))⊗β2(a2(0))
for all b∈B and h∈H.
All of the above lemmas can be directly checked, so they are left to the readers.
Theorem 1 Let (H,α) be a monoidal Hom-bialgebra over a field k, and suppose that (B,β) is a left (H,α)-Hom-comodule coalgebra and a monoidal Hom-algebra with trivial action “·”. Suppose that (Bσ[H],β⊗α) is a Hom-twisted product with a convolution invertible morphism σ and (B×H,β×α) is a Hom-smash coproduct. Then the following conditions are equivalent:
a.
2) The conditions hold for all a, b∈B, h, l∈H.
① σ is a coalgebra map;
② εB is an algebra map;
③ Δ(ab)=β-1(a1b1)σ(a2(-1), b2(-1))⊗ β(a2(0), b2(0));
④ β(b1)ε(h)⊗β(b2)=b1σ(α-2(h), b2(-1))⊗β2(b2(0));
⑤ α-1(h)b(-1)⊗β(b(0))=b(-1)α-1(h)⊗β(b(0));
⑥ σ(h1, l1)(-1)α-1(h2l2)⊗β(σ(h1, l1)(0))=h1l1⊗σ(h2, l2);
⑦ ΔB(1B)=1B⊗1B;
⑧ (B,β) is a left (H,α)-comodule algebra.
Proof 1)⟹2) follows from the Lemmas 1 to 4, so it remains to show that 2)⟹1). It is easy to check ε ((a×h)(b×k))=ε(a×h)ε(b×k), ε(1B×1H)=ε(1k) and Δ(1B×1H)=(1B×1H)⊗ (1B×1H). In order to prove that Δ((a×h)(b×k))=Δ(a×h)·Δ(b×k), it is enough to show that for every a, b∈B, h, g∈H.
(10)
(11)
(12)
Indeed, we use (10) and (11) to compute:
Δ((a×1H)(b×g))=Δ((a×1H)((β-1(b)×1H)·
(1B×α-1(g))))=
Δ(β-1(ab)×1H)Δ(1B×g)=
Δ(a×1H)(Δ(β-1(b)×1H)Δ(1B×α-1(g)))=
Δ(a×1H)Δ(b×g)
This shows that
Δ((a×1H)(b×g))=Δ(a×1H)Δ(b×g)
(13)
Similarly, we can obtain
Δ((a×h)(1B×g))=Δ(a×h)Δ(1B×g)
(14)
Now, we use Eqs.(12), (13) and (14):
Δ((a×h)(b×g))=Δ(((β-1(a)×1H)·
(1B×α-1(h)))(β-1(b)×1H)(1B×α-1(g))))=
Δ(a×1H)Δ((1B×α-1(h))·
((β-2(b)×1H)(1B×α-2(g)))=
Δ(a×1H)(Δ((1B×α-2(h))·
(β-2(b)×1H))Δ(1B×α-1(g)))=
Δ(a×1H)(Δ(1B×α-1(h))·
(Δ(β-2(b)×1H)Δ(1B×α-2(g))))=
Δ((β-1(a)×1H)(1B×α-1(h)))·
Δ((β-1(b)×1H)(1B×α-1(g)))=
Δ(a×h)Δ(b×g)
Following this, we will show that Eqs.(10) to (12) are true.
By Lemma 4, we see that
Δ(a×1H)Δ(1B×h)=[(a1×a2(-1)α-1(1H))⊗
(β(a2(0))×1H)][(1B×1Hα-1(h1))⊗(β(1B)×h2)]=
(a1σ(α(a2(-1)), h11)×α(α(a2(-1))2h12))⊗
(β(a2(0))σ(1H, h21)×α2(h22))=
(β(a1)×α(a2(-1))h1)⊗(β2(a2(0))×α(h2))=
Δ(β(a)⊗α(h))=Δ((a×1H)(1B×h))
Also, Eq.(10) is proved. We use ③ and ⑧:
Δ(a×1H)Δ(b×1H)=[(a1×α(a2(-1)))·
(b1×α(b2(-1)))]⊗[(β(a2(0))×1H)(β(b2(0))×1H)]=
((ab)1×α((ab)2(-1)))⊗(β((ab)2(0))×1H)=
Δ(ab×1H)=Δ((a×1H)(b×1H))
and (11) is proved. We use ④ and ⑤:
Δ(1B×h)Δ(b×1H)=[(1B×h1)⊗(1B×h2)]·
[(b1×b2(-1)1H)⊗(β(b2(0))×1H)]=
(β(b1)×α(b2(-1))h1)⊗(β2(b2(0))×α(h2))=
Δ(β(b)⊗α(h))=Δ((1B×h)(b×1H))
and (12) is proved.
⊗α) is a monoidal Hom-bialgebra. The proof is completed.
Definition 1 Let (H,α) be a monoidal Hom-bialgebra, (B,β) an monoidal Hom-algebra, σ: H⊗H→B a linear map and SH: H→H a linear map. SH is called a Hom-σ-antipode of (H,α) if
(1⊗α)(σ⊗mH)ΔH⊗H(1⊗SH)Δ(h)=ε(h)(1B×1H)
and
(1⊗α)(σ⊗mH)ΔH⊗H(SH⊗1)Δ(h)=ε(h)(1B×1H)
hold for every h∈H. In this case, we say that (H,α) is a Hom-σ-Hopf algebra.
Proposition 1 Let 
H,β⊗α) be a monoidal Hom-bialgebra.
S(b×h)=(1B×SH(α-1(b(-1))))(SB(b(0))×1H)
for all b∈B and h∈H.
Proof We need to prove S*id=η∘ ε=id*S holds.
(id*S)(b×h)=m(id⊗S)Δ(b×h)=
m(id⊗S)((b1×b2(-1)α-1(h1))⊗(β(b2(0))×h2))=
(b1×b2(-1)α-1(h1))[(1B×SH(b2(0)(-1)α-2(h2))·
(SB(β(b2(0))(0))×1H)]=
(β-1(β-1(b11B)σ((α-1(b2(-1)α-1(h1)))1,
SH(b2(0)(-1)α-2(h2))1))×
α((α-1(b2(-1)α-1(h1)))2)·
SH(b2(0)(-1)α-2(h2))2))·
(SB(β2(b2(0)(0)))×1H)=
(β-1(b1)σ((b2(-1)1α-1(h1))1,
(SH(b2(-1)2α-2(h2)))1)×α((b2(-1)1α-1(h1))2)·
SH(b2(-1)2α-2(h2))2))(SB(β(b2(0)))×1H)=
(b1×1H)(SB(b2(0))×1H)ε(h)=
(1B×1H)ε(b)ε(h)=(1B×1H)ε(b×h)
A similar proof shows that S*id=η∘ ε. This concludes our proof.
References：
[1]Caenepeel S, Goyvaerts I. Monoidal hom-Hopf algebra [J]. Communications in Algebra, 2011, 39(6): 2216-2240. DOI:10.1080/00927872.2010.490800.
[2]Chen Yuanyuan, Wang Zhongwei, Zhang Liangyun. Integrals for monoidal Hom-Hopf algebras and their applications [J]. Journal of Mathematical Physics, 2013, 54(7): 073515-1-073515-22. DOI:10.1063/1.4813447.
[3]Liu Ling, Shen Bingliang. Radford’s biproducets and Yetter-Drinfeld modules for monoidal Hom-Hopf algebras [J]. Journal of Mathematical Physics, 2014, 55(3):031701-1-031701-16. DOI:10.1063/1.4866760.
[4]Lu Daowei, Wang Shuanhong. Crossed product and Galois extension of monoidal Hom-Hopf algebras [EB/OL]. (2014-05-29). http://arxiv.org/abs/1405.7528.
[5]Blattner R, Cohen M, Montgomery S. Crossed product and inner actions of Hopf algebras [J]. Transactions of the American Mathematical Society, 1986, 298(2): 671-711. DOI:10.2307/2000643.
[6]Yukio D, Mitsuhiro T. Cleft comodule algebras for a bialgebra [J]. Communications in Algebra, 1986, 14(5): 801-817. DOI:10.1080/00927878608823337.
[7]Montgomery S. Hopf algebras and their actions on rings [M]. Providence, RI, USA:American Mathematical Society, 1992.
[8]Sweedler M E. Hopf algebras [M]. New York:WA Benjamin Inc, 1969.
Citation：You Miman, Wang Shuanhong．Monoidal Hom-Hopf algebra on Hom-twisted product[J]．Journal of Southeast University (English Edition),2016,32(3):391-394．DOI:10.3969/j.issn.1003-7985.2016.03.022.
DOI：10.3969/j.issn.1003-7985.2016.03.022
摘要：设(H,α)是monoidal Hom-Hopf代数,(B,β)是左(H,α)-Hom-余模余代数．构造了由Hom-扭曲积Bσ[H]和Hom-冲余积B×H构成的新monoidal Hom-代数
并给出了
成为monoidal Hom-双代数的充分必要条件此外, 设(H,α) 是带有Hom-σ-反对极SH的Hom-σ-Hopf 代数, 并找到此monoidal Hom-双代数带有定义为S(b×h)=(1B×SH(α-1(b(-1))))(SB(b(0))×1H)的反对极S成为monoidal Hom-Hopf 代数的充分条件．
关键词：monoidal Hom-Hopf代数; Hom-扭曲积; Hom-冲余积
中图分类号：O153.3
Received：2014-06-11．
Foundation item：s：The National Natural Science Foundation of China (No.11371088, 10871042, 11571173), the Fundamental Research Funds for the Central Universities (No.KYLX15_0105).
Biographies：You Miman (1984—), female, doctor; Wang Shuanhong (corresponding author), male, doctor, professor, shuanhwang@seu.edu.cn.