Q-graded Hopf quasigroups

TL;DR

This paper introduces Q-graded Hopf quasigroups, a new algebraic structure generalizing existing concepts, and explores their properties, representations, and smash products, contributing to the field of algebraic systems.

Contribution

It defines Q-graded Hopf quasigroups, studies their properties, representations, and smash products, expanding the understanding of algebraic generalizations.

Findings

Defined Q-graded Hopf quasigroups and their properties

Developed the theory of Q-graded Hopf quasimodules and their construction

Analyzed smash products of Q-graded Hopf quasigroups

Abstract

Firstly, we introduce a class of new algebraic systems which generalize Hopf quasigroups and Hopf $\pi-$algebras called $Q$-graded Hopf quasigroups, and research some properties of them. Secondly, we define the representations of $Q$-graded Hopf quasigroups, i.e $Q$-graded Hopf quasimodules, research the construction method and fundamental theorem of them. Thirdly, we research the smash products of $Q$-graded Hopf quasigroups.

Equations

$$p^{-1}(pq)=q,(qp)p^{-1}=q,\forall q\in Q.$$

$$(p^{-1}{)}^{-1}=p,(pq{)}^{-1}=q^{-1}{p}^{-1},\forall p,q\in Q.$$

$$a(bc)=(ab)c,a1=a=1a,\forall p,q,r\in \pi ,a\in A_p,b\in A_q,c\in A_r.$$

$$(ab)\cdot m=a\cdot (b\cdot m),1\cdot m=m,\forall p,q,r\in \pi ,a\in A_p,b\in A_q,m\in M_r.$$

$$f_{pq}(a\cdot m)=a\cdot f_q(m),\forall p,q\in \pi ,a\in A_p,m\in M_q.$$

$$S_p(h_{(1,p)})h_{(2,p)}={ϵ}_p(h)1=h_{(1,p)}{S}_p(h_{(2,p)}),\forall p\in \pi ,h\in H_p.$$

$$R^{12}{R}^{23}=R^{23}{R}^{12},$$

$$(hg{)}_{(1,pq)}\otimes (hg{)}_{(2,pq)}=h_{(1,p)}{g}_{(1,q)}\otimes h_{(2,p)}{g}_{(2,p)},\forall p,q\in Q,h\in H_p,g\in H_q,$$

$${ϵ}_{pq}(hg)={ϵ}_p(h){ϵ}_q(g),\forall p,q\in Q,h\in H_p,g\in H_q.$$

$${\Delta }_e(1)=1\otimes 1,{ϵ}_e(1)=1_k.$$

$$S_p(h_{(1,p)})(h_{(2,p)}g)={ϵ}_p(h)g=h_{(1,p)}(S_p(h_{(2,p)})g),\forall p,q\in Q,h\in H_p,g\in H_q.$$

$$(g{S}_p(h_{(1,p)}))h_{(2,p)}=g{ϵ}_p(h)=(g{h}_{(1,p)})S_p(h_{(2,p)}),\forall p,q\in Q,h\in H_p,g\in H_q.$$

$$h_{(1,p)}(g{h}_{(2,p)})=(h_{(1,p)}g)h_{(2,p)},\forall p,q\in Q,h\in H_p,g\in H_q.$$

$$h_{(1,p)}(h_{(2,p)}g)=(h_{(1,p)}{h}_{(2,p)})g,h(g_{(1,p)}{g}_{(2,p)})=(h{g}_{(1,p)})g_{(2,p)},\forall p,q\in Q,h\in H_p,g\in H_q.$$

$$h_{(1,p)}(g(h_{(2,p)}f))=((h_{(1,p)}g)h_{(2,p)})f,\forall p,q,r\in Q,h\in H_p,g\in H_q,f\in H_r.$$

$$hg=gh,\forall p,q\in Q,h\in H_p,g\in H_q.$$

$${\Delta }_p(h)=h_{(1,p)}\otimes h_{(2,p)}=h_{(2,p)}\otimes h_{(1,p)},\forall p\in Q,h\in H_p.$$

$$S_p(h_{(1,p)})h_{(2,p)}={ϵ}_p(h)1=h_{(1,p)}{S}_p(h_{(2,p)}),\forall p\in Q,h\in H_p.$$

$$S_{pq}(hg)=S_q(g)S_p(h),\forall p,q\in Q,h\in H_p,g\in H_q.$$

$$S_p(h{)}_{(1,p^{-1})}\otimes S_p(h{)}_{(2,p^{-1})}={\Delta }_{p^{-1}}(S_p(h))=S_p(h_{(2,p)})\otimes S_p(h_{(1,p)}),\forall p\in Q,h\in H_p.$$

$$S_e(1)=1.$$

$${ϵ}_{p^{-1}}{S}_p(h)={ϵ}_p(h),\forall p\in Q,h\in H_p.$$

$$S_q(g)S_p(h)=S_q(g_{(1,q)}{ϵ}_q(g_{(2,q)}))S_p(h_{(1,p)}{ϵ}_p(h_{(2,p)}))$$

$$\stackrel{(\text{ref}2b)}{=}{S}_q(g_{(1,q)})S_p(h_{(1,p)}){ϵ}_{pq}(h_{(2,p)}{g}_{(2,q)})$$

$$=S_q(g_{(1,q)})((S_p(h_{(1,p)})(h_{(2,p)}{g}_{(2,q)}{)}_{(1,pq)})S_{pq}((h_{(2,p)}{g}_{(2,p)}{)}_{(2,pq)}))$$

$$=S_q(g_{(1,q)})((S_p(h_{(1,p)})(h_{(2,p)}{g}_{(2,q)}))S_{pq}(h_{(3,p)}{g}_{(3,q)}))$$

$$=S_q(g_{(1,q)(1,q)})((S_p(h_{(1,p)(1,p)})(h_{(1,p)(2,p)}{g}_{(1,q)(2,q)}))S_{pq}(h_{(2,p)}{g}_{(2,q)}))$$

$$\stackrel{(\text{ref}2c)}{=}{S}_q(g_{(1,q)(1,q)})({ϵ}_p(h_{(1,p)})g_{(1,q)(2,q)}{S}_{pq}(h_{(2,p)}{g}_{(2,q)}))$$

$$=S_q(g_{(1,q)(1,q)})(g_{(1,q)(2,q)}{S}_{pq}({ϵ}_p(h_{(1,p)})h_{(2,p)}{g}_{(2,q)}))$$

$$=S_q(g_{(1,q)(1,q)})(g_{(1,q)(2,q)}{S}_{pq}(h{g}_{(2,q)}))$$

$$\stackrel{(\text{ref}2c)}{=}{ϵ}_q(g_{(1,q)})S_{pq}(h{g}_{(2,q)})$$

$$=S_{pq}(h{ϵ}_q(g_{(1,q)})g_{(2,q)})=S_{pq}(hg).$$

$${\Delta }_{p^{-1}}(S_p(h))=S_p(h_{(2,p)}{)}_{(1,p^{-1})}\otimes {ϵ}_p(h_{(1,p)})S_p(h_{(2,p)}{)}_{(2,p^{-1})}$$

$$\stackrel{(\text{ref}2c)}{=}{S}_p(h_{(2,p)}{)}_{(1,p^{-1})}\otimes S_p(h_{(1,p)(1,p)})(h_{(1,p)(2,p)}{S}_p(h_{(2,p)}{)}_{(2,p^{-1})})$$

$$={ϵ}_p(h_{(1,p)(1,p)(2,p)})S(h_{(2,p)}{)}_{(1,p^{-1})}$$

$$\otimes S_p(h_{(1,p)(1,p)(1,p)})(h_{(1,p)(2,p)}{S}_p(h_{(2,p)}{)}_{(2,p^{-1})})$$

Full text

$Q$-graded Hopf quasigroups

**Guodong Shi  Shuanhong Wang111Corresponding author: shuanhwang@seu.edu.cn

**School of Mathematics, Southeast University

Nanjing, Jiangsu 210096, P. R. of China

Abstract. Firstly, we introduce a class of new algebraic systems which generalize Hopf quasigroups and Hopf $\pi-$algebras called $Q$-graded Hopf quasigroups, and research some properties of them. Secondly, we define the representations of $Q$-graded Hopf quasigroups, i.e $Q$-graded Hopf quasimodules, research the construction method and fundamental theorem of them. Thirdly, we research the smash products of $Q$-graded Hopf quasigroups.

Keywords: $Q$-graded Hopf quasigroups; $Q$-graded Hopf quasimodules; Smash products.

Mathematics Subject Classification: 16W30.

Introduction

Hopf quasigroups[5]were introduced by Klim and Majid, there are generalization of Hopf algebras[7] that are not required to be associative. In publications we can find some results on Hopf quasigroup : Breziński research the Hopf modules and the fundamental theorem[1] on Hopf quasigroups, Breziński with Jiao show the smash (co)products[2] and $R-$smash products[3] on Hopf quasigoups, Jiao and Wang give the smash biproducts[4] on quasigroups, Klim and Majid introduce the bicrossproduct[6] on Hopf quasigroups.

On the other hand, Hopf $\pi$-coalgebras[9] were introduced by Virelizier, and then continued by Zunino and Wang. It turms out that many of the classical results in Hopf algebras can be generalized to the Hopf $\pi$-coalgebras, Virelizier gives a generalized version of the Fundamental theorem for Hopf algebras and introduced $\pi-$integrals, Zunino introduced Yetter-Drinfeld modules[16], the Drifeld double, and a generalization of the center constrction of a monoidal category[15], Wang introduces Doi-Hopf modules[11], entwined modules[14], Drinfeld’codouble[12] and coalgebra Galois theory[13] for Hopf $\pi-$coalgebras, and he proves a version of Maschke’theorem[10]. Hopf $\pi$-algebras are dual of Hopf $\pi$-coalgebras[8].

In this paper, we introduce $Q-$graded Hopf quasigroups , they generalize the Hopf quasigroups like Hopf $\pi-$algebras generalize Hopf algebras and they generalize Hopf $\pi-$algebras like Hopf quasigroups generalize Hopf algebras. The paper is organized as follows: In section 1, we recall the definitions of quasigroups, Hopf $\pi-$algebras and Hopf quasigroups. In section 2, we give the definition of $Q-$graded Hopf quasigroups and generalize many classical theorems of Hopf quasigroups to $Q-$graded Hopf quasigroups. In section 3, we define $Q-$graded Hopf quasimodules and show a construct method and fundamental theorem of Hopf module on it. In section 4, we show the smash products on $Q-$graded Hopf quasigroups.

Throughout this article, $\pi$ is a group, $Q$ is a quasigroup and all the vector spaces, tensor products and maps are over a fixed field $k$. For a coalgebra $C$, we will use the Heyneman-Sweedler’s notation $\Delta(c)=c_{1}\otimes c_{2},$ for any $c\in C$ (summation omitted).

1 Preliminaries

Definition 1.1**.**

A quasigroup is a set $Q$ with a product, identity $e$ and with the property that for each $p\in Q$ there is $p^{-1}\in Q$ satisfying

[TABLE]

It is easy to see that in any quasigroup $Q$, one has unique inverses and

[TABLE]

A quasigroup is $flexile$ if $p(qp)=(pq)p$ for all $p,q\in Q$ and $alternative$ if also $p(pq)=(pp)q,~{}p(qq)=(pq)q$ for all $p,q\in Q$. It is called $Moufang$ ir $p(q(pr))=((pq)p)r$ for all $p,q,r\in Q$ or $commutative$ if $pq=qp$ for all $p,q\in Q$ or $associative$ if $p(qr)=(pq)r,$ for all $p,q,r\in Q$. Obviously, associative quasigroup is group.

Remark 1.2**.**

Let $Q$ be a quasigroup, then the following identities are equivalent, $\forall p,q,r\in Q,$ (1) $p(q(pr))=((pq)p)r,$ (2) $((pq)r)q=p(q(rq)),$ (3) $(pq)(rp)=(p(qr))p.$

Definition 1.3**.**

A Hopf quasigroup is a possibly-noassociative but unital algebra $H$ equipped with algebra homomorphisms $\Delta:H\rightarrow H\otimes H,~{}\epsilon:H\rightarrow k$ forming a coassociative coaglebra and a map $S:H\rightarrow H$ such that

$S(h_{1})(h_{2}g)=\epsilon(h)g=h_{1}(S(h_{2})g)$, (gh_{1})S(h_{2})=g\epsilon(h)=(g(S(h_{1}))h_{2}),~{}$$\forall h,g\in H.

Definition 1.4**.**

A $\pi-$graded algebra ($\pi-$algebra) is a family $A=\{A_{p}\}_{p\in\pi}$ of $k-$spaces endowed with a family $m=\{m_{p,q}:A_{p}\otimes A_{q}\rightarrow A_{pq}\}_{p,q\in\pi}$ of $k-$ maps (the $Q-$graded multiplication) and a $k-$linear map $\mu:k\rightarrow A_{e}$ (the unit) such that (we denote $m_{p,q}(a\otimes b)\equiv ab,~{}\mu(1_{k})\equiv 1)$

[TABLE]

Definition 1.5**.**

Let $A$ be a $\pi-$graded algebra, a $\pi-$graded left $A-$module is a family $M=\{M_{p}\}_{p\in\pi}$ of $k-$spaces endowed with a family $\varphi=\{\varphi_{p,q}:A_{p}\otimes M_{q}\rightarrow M_{pq}\}_{p,q\in\pi}$ of $k-$maps such that(we denote $\varphi(a\otimes m)\equiv a\cdot m$)

[TABLE]

A map of $\pi-$graded left $A-$modules is a family $f=\{f_{p}:M_{p}\rightarrow M^{{}^{\prime}}_{p}\}_{p\in\pi}$ of $k-$maps such that

[TABLE]

Definition 1.6**.**

A $\pi-$graded Hopf algebra (Hopf $\pi-$algebra) is a $\pi-$graded algebra $H=\{H_{p},m_{p,q},\mu\}_{p,q\in\pi}$ endowed with a family $S=\{S_{p}:H_{p}\rightarrow H_{p^{-1}}\}_{p\in\pi}$ of $k-$linear maps (the antipode) such that each $H_{p}$ is a coassociative counitary coalgebra with comultiplication $\Delta_{p}$ and counit map $\epsilon_{p}$, and $\{m_{p,q}\}_{p,q\in\pi}$ and $\mu$ are coalgebra maps and (we denote $\Delta_{p}(h)\equiv h_{(1,p)}\otimes h_{(2,p)}$)

[TABLE]

Definition 1.7**.**

Given a $k-$space $M,$ a $k-$map $R:M\otimes M\rightarrow M\otimes M$ is said to be a solution of the Long-equation if

[TABLE]

where $R^{12}=R\otimes I,$ $R^{23}=I\otimes R:M\otimes M\otimes M\rightarrow M\otimes M\otimes M.$

2 $Q$-graded Hopf quasigroups

Definition 2.1**.**

A $Q-$graded Hopf quasigroup $H$ is

(1) A $Q-$graded algebra, i.e a family $H=\{H_{p}\}_{p\in Q}$ of $k-$spaces endowed with a family $m=\{m_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq}\}_{p,q\in Q}$ of $k-$ maps (the $Q-$graded multiplication) and a $k-$linear map $\mu:k\rightarrow H_{e}$ (the unit) such that $h1=h=1h,~{}\forall p\in Q,~{}h\in H_{p}.$

(2) Each $H_{p}$ is a coassociative counitary coalgebra with comultiplication $\Delta_{p}$ and counit $\epsilon_{p}$,

(3) $\{m_{p,q}\}$ are coalgebra maps, i.e

[TABLE]

[TABLE]

(4) $\mu$ is coalgebra map, i.e

[TABLE]

(5) A family $S=\{S_{p}:H_{p}\rightarrow H_{p^{-1}}\}_{p\in Q}$ of $k-$linear maps (the antipode) such that

[TABLE]

[TABLE]

$H$ is a $Q-$graded Hopf quasigroup, then

(1) $H$ is $flexible$ if $Q$ is $flexible$ and

[TABLE]

(2) $H$ is $alternative$ if $H$ is flexible, $Q$ is $alternative$ and

[TABLE]

(3)$H$ is $Moufang$ if $Q$ is $Moufang$ and

[TABLE]

(4) $H$ is $commutative$ if $Q$ is $commutative$ and

[TABLE]

(5) $H$ is $cocommutative$ if

[TABLE]

Now we show an example of $Q-$graded Hopf quasigroups.

Example 2.2**.**

For any quasigroup $Q$, we can construct a $Q-$graded Hopf quasigroup as follows: a family $kQ=\{H_{p}=(kp)\}_{p\in Q}$of $k-$spaces, the mulitiplcation $m_{p,q}(k_{1}p\otimes k_{2}q)=k_{1}k_{2}pq,$ the unitary element $e\in Q$, comultiplication $\Delta_{p}(p)=p\otimes p,$ counit $\epsilon_{p}(p)=1_{k},$ antipode $S_{p}(p)=p^{-1}.$

Proposition 2.3**.**

$H$* is a $Q-$graded Hopf quasigroup, then*

(1)

[TABLE]

(2)

[TABLE]

(3)

[TABLE]

(4)

[TABLE]

(5)

[TABLE]

Proof.

(1) and (4) are obtained by the definition of $Q-$graded Hopf quasigroup.

(2)

[TABLE]

(3)

[TABLE]

(5)

[TABLE]

∎

Proposition 2.4**.**

Let $H$ be a $Q-$graded Hopf quasigroup, if $H$ is $commutative$ or $cocommutative$, then $S_{p^{-1}}S_{p}(h)=h,~{}\forall p\in Q,~{}h\in H_{p}.$

Proof.

If $H$ is commutative,

[TABLE]

If $H$ is cocommutative,

[TABLE]

∎

Proposition 2.5**.**

Let $H$ be a $Moufang$ $Q-$graded Hopf quasigroup such that $S_{p}$ is invertible for any $p\in Q$, then $\forall p,q,r\in Q,~{}h\in H_{p},~{}g\in H_{q},~{}f\in H_{r},$ the following three conditions are equivalent :

(1) $h_{(1,p)}(g(h_{(2,p)}f))=((h_{(1,p)}g)h_{(2,p)})f$,

(2) $((hg_{(1,q)})f)g_{(2,q)}=h(g_{(1,p)}(fg_{(2,q)}))$,

(3) $(h_{(1,p)}g)(fh_{(2,p)})=(h_{(1,p)}(gf))h_{(2,p)}$.

Proof.

$(1)\Rightarrow(2)$

[TABLE]

using $S_{p}$ are invertible and $H$ is $Moufang$, the above equation is equivalent to

[TABLE]

$(2)\Rightarrow(1)$ The proof is similarly to $(1)\Rightarrow(2)$.

$(1)\Rightarrow(3)$

[TABLE]

replacing $g$ with $gf_{(1,r)}$ and $f$ with $S_{r}(f_{(2,r)})$, we have:

[TABLE]

replacing $h$ with $h_{(1,p)}$ and $f$ with $f_{(1,r)}$, multiply on the right by $f_{(2,r)}h_{(2,p)}$, we have:

[TABLE]

[TABLE]

the left side of the above equation equals $(h_{(1,p)}g)(fh_{(2,p)}),$ now consider the right side of the equation: by the coassociativity and $\Delta_{p}$ ia algebra map, the right side equals:

[TABLE]

so,

[TABLE]

$(3)\Rightarrow(1)$ Assume (3) holds, then

[TABLE]

[TABLE]

the left side of above equation equals:

[TABLE]

so, we have:

[TABLE]

replacing $g$ with $gS_{r}(f_{(1,r)})$ and $f$ with $f_{(2,r)}$,

[TABLE]

so,

[TABLE]

replacing $h$ with $h_{(1,p)}$ and $f$ with $S^{-1}_{pr}(h_{(2,p)}f)$, we have:

[TABLE]

∎

Lemma 2.6**.**

Let $H$ be a cocommutative and flexible $Q-$graded Hopf quasigroup, then

[TABLE]

Proof.

[TABLE]

So,

[TABLE]

∎

Therfore we have a notion of left adjoint action of a $Q-$graded Hopf quasigroup $H$ when it is cocommutative and flexible.

Definition 2.7**.**

Let $Q$ be an associative quasigroup and $H$ be a $Q-$graded Hopf quasigroup, an associator $\delta$ be defined by

[TABLE]

Theorem 2.8**.**

Let $Q$ be an associative quasigroup and $H$ be a $Q-$graded Hopf quasigroup,

(1) the associator $\delta$ exists and is uniquely determined as

[TABLE]

(2) $\delta(1,h,g)=\delta(h,1,g)=\delta(h,g,1)=\epsilon_{p}(h)\epsilon_{q}(g)1.$

(3)$\delta(h_{(1,p)},S_{p}(h_{(2,p)}),g)=\delta(S_{p}(h_{(1,p)}),h_{(2,p)},g)=\epsilon_{p}(h)\epsilon_{q}(g)1,$

$\delta(h,g_{(1,q)},S_{q}(g_{(2,q)}))=\delta(h,S_{q}(g_{(1,q)}),g_{(2,q)})=\epsilon_{p}(h)\epsilon_{q}(g)1,$**

$\delta(h_{(1,p)}g_{(1,q)},S_{q}(g_{(2,q)}),S_{p}(h_{(2,p)}))=\delta(S_{p}(h_{(1,p)})S_{q}(g_{(1,q)}),g_{(2,q)},h_{(2,p)})=\epsilon_{p}(h)\epsilon_{q}(g)1,$**

$\delta(S_{p}(h_{(1,p)}),S_{q}(g_{(1,q)}),g_{(2,q)}h_{(2,p)})=\delta(h_{(1,p)},g_{(1,p)},S_{q}(g_{(2,q)})S_{p}(h_{(2,p)}))=\epsilon_{p}(h)\epsilon_{q}(g)1,$**

$\delta(S_{p}(h_{(1,p)}),h_{(2,p)}S_{q}(g_{(1,q)}),g_{(2,q)})=\delta(h_{(1,p)},S_{p}(h_{(2,p)})g_{(1,q)},S_{q}(g_{(2,q)}))=\epsilon_{p}(h)\epsilon_{q}(g)1.$**

Proof.

(1) We suppose $\delta$ exists then applying it to the rebracket, then

[TABLE]

we verify similarily,

[TABLE]

(2)

[TABLE]

the rest are similar.

(3)

[TABLE]

the rest are similar. ∎

3 The first fundamental theorem for $Q-$graded Hopf quasigroups

Definition 3.1**.**

Let $H=\{H_{p}\}_{p\in Q}$ be a (not necessary associative) $Q-$graded algebra and for any $p\in Q$, $H_{p}$ is a (not necessary coassociative) coalgebra $(H_{p},\Delta_{p},\epsilon_{p})$ such that for any $p,q\in Q$, $m_{p,q},~{}\mu$ are maps of coalgebras, then

(1) A family $\beta=\{\beta_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq}\otimes H_{q},\beta_{p,q}(h\otimes g)=hg_{(1,q)}\otimes g_{(2,q)}\}_{p,q\in Q}$ of $k-$maps is called the right $Galois$ map of $H$.

(2) A family $\gamma=\{\gamma_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{pq},\gamma_{p,q}(h\otimes g)=h_{(1,p)}\otimes h_{(2,p)}g\}_{p,q\in Q}$ of $k-$maps is called the left $Galois$ map of $H$.

(3) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq}\otimes H_{q}\}_{p,q\in Q}$ of $k-$maps is almost left $H-$linear if

[TABLE]

(4) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq}\otimes H_{q}\}_{p,q\in Q}$ of $k-$maps is almost right $H-$colinear if

[TABLE]

(5) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq}\otimes H_{q}\}_{p,q\in Q}$ of $k-$maps is right $H-$colinear if

[TABLE]

(6) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{pq}\}_{p,q\in Q}$ of $k-$maps is almost right $H-$linear if

[TABLE]

(7) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{pq}\}_{p,q\in Q}$ of $k-$maps is almost left $H-$colinear if

[TABLE]

(8) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{pq}\}_{p,q\in Q}$ of $k-$maps is left $H-$colinear if

[TABLE]

(9) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq^{-1}}\otimes H_{q}\}_{p,q\in Q}$ of $k-$maps is almost left $H-$linear if

[TABLE]

(10) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq^{-1}}\otimes H_{q}\}_{p,q\in Q}$ of $k-$maps is almost right $H-$colinear if

[TABLE]

(11) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq^{-1}}\otimes H_{q}\}_{p,q\in Q}$ of $k-$maps is right $H-$colinear if

[TABLE]

(12) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{p^{-1}q}\}_{p,q\in Q}$ of $k-$maps is almost right $H-$linear if

[TABLE]

(13) A family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{p^{-1}q}\}_{p,q\in Q}$ of $k-$maps is almost left $H-$colinear if

[TABLE]

(14) A family $(\phi=\{\phi_{p,q}:H_{p}\otimes H_{p}\rightarrow H_{p}\otimes H_{p^{-1}q}\}_{p,q\in Q}$of $k-$maps is left $H-$colinear if

[TABLE]

Lemma 3.2**.**

If for any $p\in Q,$ $\Delta_{p}$ is a coassociative, then the notion of almost $H-$colinearity coincides with that of $H-$colinearity.

Proof.

Suppose a family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{pq}\}_{p,q\in Q}$ of $k-$maps is left $H-$colinear,

[TABLE]

therefore $\phi$ is almost left $H-$colinear. Similarly, suppose a family $\phi=\{\phi_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{pq}\}_{p,q\in Q}$ of $k-$maps is almost left $H-$colinear,

[TABLE]

therefore $\phi$ is left $H-$colinear. Analogously we can get the other results. ∎

Lemma 3.3**.**

Let $H$ be a $Q-$graded Hopf quasigroup, then the right Galois map $\beta$ is almost left $H-$linear and almost right $H-$colinear, the left Galois map $\gamma$ is almost right $H-$linear and almost left $H-$colinear.

Proof.

Since

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

$\beta$ is almost left $H-$linear and almost $H-$colinear. Similarly we can get the result for $\gamma$.

∎

Theorem 3.4**.**

Let $H=\{H_{p}\}_{p\in Q}$ be a (not necessary associative) $Q-$graded algebra and for any $p\in Q$, $H_{p}$ is a (not necessary coassociative) coalgebra $(H_{p},\Delta_{p},\epsilon_{p})$ such that for any $p,q\in Q$, $m_{p,q},~{}\mu$ are maps of coalgebras. Then $H$ is a $Q-$graded Hopf quasigroup $\Longleftrightarrow$

(1) For any $p\in Q$, $\Delta_{p}$ is coassociative .

(2) There is a family $\beta^{\star}=\{\beta^{\star}_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{pq^{-1}}\otimes H_{q}\}_{p,q\in Q}$ of $k-$maps which is almost left $H-$linear and satisfy

[TABLE]

[TABLE]

(3) There is a family $\gamma^{\star}=\{\gamma^{\star}_{p,q}:H_{p}\otimes H_{q}\rightarrow H_{p}\otimes H_{p^{-1}q}\}_{p,q\in Q}$ of $k-$maps which is almost right $H-$linear and satisfy

[TABLE]

[TABLE]

Proof.

$(\Longrightarrow)$ $H$ is a $Q-$graded Hopf quasigroup, by definition, $\forall p\in Q,$ $\Delta_{p}$ is coassociative. We set

[TABLE]

and

[TABLE]

Since

[TABLE]

[TABLE]

$\beta^{-1}$ is almost left $H-$linear .

And we have

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Analogous computation yields the result for $\gamma^{\star}$.

$(\Longleftarrow)$ We denote

[TABLE]

[TABLE]

Since $\beta^{\star}$ and $\gamma^{\star}$ are almost $H-$linear, we have

[TABLE]

[TABLE]

Define two families of $k-$linear maps:

[TABLE]

[TABLE]

Since for any $p\in Q,$ $\Delta_{p}$ is coassociative and $\beta$ is almost right $H-$colinear, by lemma 2.2, $\beta$ is right $H-$colinear, and hence $\beta^{-1}$ is right $H-$colinear. So

[TABLE]

[TABLE]

Applying $I\otimes\epsilon_{q}\otimes I$ to both sides of above equation, we have

[TABLE]

So, we have

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

so we get

[TABLE]

Applying $I\otimes\epsilon_{q}$ to the above equation, we have

[TABLE]

Following similar chain of arguments we can conclude that the the map $S^{\star}$ satisfy

[TABLE]

Finally, we can compute

[TABLE]

[TABLE]

Thus the maps $S$ and $S^{\star}$ coincide, $H$ is a $Q-$graded Hopf quasigroup. ∎

4 $Q-$graded quasimodules and $Q-$graded Hopf quasimodules

Definition 4.1**.**

Let $H$ be a $Q-$graded Hopf quasigroup , a $Q-$graded $H-$quasimodule is a pair $(M,\varphi)$, where $M=\{M_{q}\}_{q\in Q}$ is a family of $k-$spaces and $\varphi=\{\varphi_{p,q}:H_{p}\otimes M_{q}\rightarrow M_{pq}\}_{p,q\in Q}$ ($Q$-graded action) is a family of $k-$maps such that

[TABLE]

[TABLE]

A map of $Q-$graded $H-$quasimodules is a family $f=\{f_{q}:M_{q}\rightarrow N_{q}\}_{q\in Q}$ of $k-$maps such that

[TABLE]

Remark 4.2**.**

$H$* is a $Q-$graded Hopf quasimodule with the $Q-$graded multiplication as $Q-$graded action.*

Example 4.3**.**

Let $Q$ be a quasigroup and $X=\{X_{q}\}_{q\in Q}$ is a set, then we can define a $Q-$graded $kQ-$quasimodule on a family $kX=\{(kX_{q})\}_{q\in Q}$ of $k-$spaces as follows:

[TABLE]

Proposition 4.4**.**

Suppose $M,$ N are two $Q-$graded $H-$quasimodules, then $M\otimes N=\{M_{q}\otimes N_{q}\}_{q\in Q}$ is a $Q-$graded $H-$quasimodule with the diagonal action:

[TABLE]

Proof.

[TABLE]

Similarly, we can proof $S_{p}(h_{(1,p)})\cdot(h_{(2,p)}\cdot(m\otimes n))=\epsilon_{p}(h)(m\otimes n),$ $M\otimes N$ with the diagonal action is a $Q-$graded $H-$quasimodule. ∎

Definition 4.5**.**

Let $H$ be a $Q-$graded Hopf quasigroup, a $Q-$graded $H-$Hopf quasimodule is a $Q-$graded $H-$quasimodule $M=\{M_{q},\varphi_{p,q}\}_{p,q\in Q}$ and $\forall q\in Q,$ $M_{q}$ is coassociative counitary left $H_{q}-$comodule $(M_{q},\rho_{q})$ such that (we denote $\rho_{q}(m)=m_{(-1,q)}\otimes m_{(0,q)}$)

[TABLE]

A map of $Q-$graded $H-$Hopf quasimodules is a map of $Q-$graded $H-$quasimodules $f=\{f_{q}:M_{q}\rightarrow N_{q}\}_{q\in Q}$ such that

[TABLE]

Example 4.6**.**

Let $Q$ be a quasigroup, we can define a $Q-$graded $kQ-$Hopf quasimodule on $Q-$graded $kQ-$quasimodule $kX=\{(kX_{q})\}_{q\in Q}$ by

[TABLE]

Lemma 4.7**.**

Let $M$ be a $Q-$graded $H-Hopf$ quasimodule, then

[TABLE]

[TABLE]

Proof.

[TABLE]

[TABLE]

∎

Theorem 4.8**.**

Let $(M,\varphi,\rho)$ be a $Q-$graded $H-$Hopf quasimodule, we replace the $Q-$graded action $\varphi$ by $\vartriangleright=\{\vartriangleright_{p,q}:H_{p}\otimes M_{q}\rightarrow M_{pq},\vartriangleright_{p,q}(h\otimes m)=(hm_{(-2,q)})\cdot(S_{q}(m_{(-1,q)})\cdot m_{(0,q)})\}_{p,q\in Q}$ (we denote $\vartriangleright_{p,q}(h\otimes m)\equiv h\vartriangleright m$), then $(M,\vartriangleright,\rho)$ is a $Q-$graded $H-$Hopf quasimodule.

Proof.

Obviously,

[TABLE]

Verifying the conditon $(\ref{4a}):$ince

[TABLE]

we have

[TABLE]

and

[TABLE]

[TABLE]

using the above equation,

[TABLE]

replacing $h$ with $S_{p}(h_{(1,p)})$ and $g$ with $h_{(2,p)},$

[TABLE]

Similary, we can get

[TABLE]

So, $(M,\varphi)$ is a $Q-$graded $H-$quasimodule. Now we verify the condition $(\ref{4c})$ as follows:

[TABLE]

∎

Definition 4.9**.**

Let $M$ be a $Q-$graded $H-$Hopf quasimodule, the coinvariant of $M$ on $H$ is a vector space

[TABLE]

Lemma 4.10**.**

Let $M=\{M_{q}\}_{q\in Q}$ be a $Q-$graded $H-$Hopf quasimodule, then $H\otimes M^{coH}=\{H_{q}\otimes M^{coH}\}_{q\in Q}$ is a $Q-$graded $H-$Hopf quasimodule.

Proof.

Defining the $Q-$graded $H-$module action $\varphi$ on $H\otimes M^{coH}$:

[TABLE]

Since

[TABLE]

and

[TABLE]

$(H\otimes M^{coH},\varphi)$ is a $Q-$graded $H-$quasimodule.

For any $q\in Q,$ we define $H_{q}-$comodule coaction on $H_{q}\otimes M^{coH}$:

[TABLE]

Since

[TABLE]

and

[TABLE]

$(H_{q}\otimes M^{coH},\rho_{q})$ is a coassociative counitary $H_{q}-$comodule.

Now we verify the condition $(\ref{4c})$ for $H\otimes M^{coH},$

[TABLE]

So, $H\otimes M^{coH}$ is a $Q-$graded $H-$Hopf quasimodule. ∎

Theorem 4.11**.**

Let $M=\{M_{q}\}_{q\in Q}$ be a $Q-$graded $H-$Hopf quasimodule, then (as $Q-$graded $H-$Hopf quasimodule)

[TABLE]

Proof.

For any $q\in Q,$ we define

[TABLE]

Verify $\sigma=\{\sigma_{q}\}_{q\in Q}$ satisfy the condition (4.2):

[TABLE]

Verify $\sigma=\{\sigma_{q}\}_{q\in Q}$ sastisfy the condtion (4.4)

[TABLE]

So, $\sigma$ is a map of $Q-$graded $H-$quasimodules.

For any $q\in Q,$ we define

[TABLE]

noticing by (4.5), $\sigma_{q}^{-1}(m)\subset H_{q}\otimes M^{coH},~{}\forall q\in Q,~{}m\in M_{q},$ so $\sigma^{-1}$ is wel defined.

Since

[TABLE]

and

[TABLE]

$\sigma$ is an isomorphism of $Q-$graded $H-$Hopf quasimodules, then $M\cong H\otimes M^{coH}$.

∎

5 $Q-$graded Long dimodules

Definition 5.1**.**

Let $H$ be a $Q-$graded Hopf quasigroups, a $Q-$graded Long $H-$dimodule is a triple $(M,\varphi,\rho),$ where $(M,\varphi)$ is a $Q-$graded left $H-$quasimodule, $(M,\rho)=\{(M_{q},\rho_{q,e}):\rho_{q,e}:M_{q}\rightarrow M_{q}\otimes H_{e},\rho_{q,e}(m)\equiv m_{(o,q)\otimes m_{(1,e)}}\}_{q\in Q}$ is a family of coassociative counitary right $H_{e}-$comodules such that

[TABLE]

Example 5.2**.**

Let $Q$ be a quasigroup , we can define a $Q-$graded Long $kQ-$dimodule on $Q-$graded $kQ-$quasimodule $kX$ by

[TABLE]

Lemma 5.3**.**

Let $(M,\varphi,\rho)$ be a $Q-$graded Long $H-$dimodule, then we have

[TABLE]

and

[TABLE]

Proof.

[TABLE]

and

[TABLE]

∎

Proposition 5.4**.**

Let $M$ be a $Q-$graded $H-$quasimodule, then $M\otimes H_{e}=\{M_{q}\otimes H_{e}\}_{q\in Q}$ is a $Q-$graded Long $H-$dimodule by :

[TABLE]

[TABLE]

Proof.

Since

[TABLE]

and

[TABLE]

therefore $M\otimes H_{e}$ is a $Q-$graded $H-$quasimodule. By the coassociativity of $\Delta_{e}$ and the counitality of $\epsilon_{e}$ , it is easy to check $M\otimes H_{e}$ is a right $H_{e}-$comodule. Now we proof the condition $(\ref{5a})$ as follows:

[TABLE]

so, $M\otimes H_{e}$ is a $Q-$graded Long $H-$dimodule.

∎

Proposition 5.5**.**

Let $H$ be a $Q-$graded Hopf quasigroup and $M$ is a right $H_{e}-$comodule, then $H\otimes M=\{H_{q}\otimes M\}_{q\in Q}$ is a $Q-$graded Long $H-$dimodule by

[TABLE]

[TABLE]

Proof.

Since

[TABLE]

$H\otimes M$ is a $Q-$graded left $H-$quasimodule.

Verify $H\otimes M$ is a right $H_{e}-$comodule is easy and be left to reader, we verify condtion $(\ref{5a})$ as follows:

[TABLE]

Then, $H\otimes M$ is a $Q-$graded Long $H-$dimodule.

∎

Corollary 5.6**.**

Let $(M,\varphi)$ be a $Q-$graded left $H-$quasimodule, then $(M,\varphi,\rho)$, then $M$ is a $Q-$graded Long $H-$dimodule by

[TABLE]

Proposition 5.7**.**

Suppose $(M,\varphi_{M},\rho_{M})$ and $(N,\varphi_{N},\rho_{N})$ are two $Q-$graded Long $H-$dimodules, then $\{M_{q}\otimes N_{q}\}_{q\in Q}$ is a $Q-$graded Long $H-$dimodule by

[TABLE]

[TABLE]

Proof.

Since

[TABLE]

and

[TABLE]

similarly, we can get $S_{p}(h_{(1,p)})\cdot(h_{(2,p)}\cdot(m\otimes n))=\epsilon_{p}(h)(m\otimes n),$ so $\{M_{q}\otimes N_{q}\}_{q\in Q}$ is a $Q-$graded $H-$quasimodule.

It is easy to check $\{M_{p}\otimes N_{p}\}_{q\in Q}$ is a family of right $H_{e}-$comodules, now we verify the condition (5.1):

[TABLE]

therefore $\{M_{q}\otimes N_{q}\}_{q\in Q}$ is a $Q-$graded Long $H-$dimodule. ∎

Proposition 5.8**.**

Let $(M,\varphi,\rho)$ be a Long $H-$dimodule. Then, for any $q\in Q$, the map

[TABLE]

is a solution of Long-equation for $M_{q}$.

Proof.

[TABLE]

we have $R_{q}^{12}R_{q}^{23}=R_{q}^{23}R_{q}^{12}~{},\forall q\in Q.$

∎

By a $Q-$graded Long $H-$dimodule, we can construct a family of solution to Long-equation.

6 The smash product for $Q$-graded Hopf quasigroups

Definition 6.1**.**

Let $H$ be a $Q-$graded Hopf quasigroup, a $H-$quasimodule is a pair $(M,\varphi),$ where $M$ is a $k-$space, $\varphi=\{\varphi_{p}:H_{p}\otimes M\rightarrow M\}_{p\in Q}$ is a family of $k-$maps such that,

[TABLE]

[TABLE]

Definition 6.2**.**

Let $H$ be a $Q-$graded Hopf quasigroup, an (not necessarily associative) algebra $A$ is called a $H-$quasimodule algebra if $A$ is a $H-$quasimodule and

[TABLE]

[TABLE]

A coalgebra $C$ is a $H-$quasimodule coalgebra if $C$ is a $H-$quasimodule and

[TABLE]

[TABLE]

A Hopf quasigroup $A$ is called a $H-$quasimodule Hopf quasigroup if it is both a $H-$quasimodule algebra and a $H-$quasimodule coalgebra.

Lemma 6.3**.**

Let $H$ is a $Q-$graded Hopf quasigroup, $A$ is a $H-$quasimodule Hopf quasigroup, then

[TABLE]

Proof.

Since

[TABLE]

and

[TABLE]

we have

[TABLE]

repalcing $h$ with $h_{(2,p)}$ and $a$ with $a_{2},$ and multiplying by $S(h_{(1,p)}\cdot a_{1})$ on the left, we have

[TABLE]

then we get

[TABLE]

∎

Theorem 6.4**.**

Suppose $H$ be a $Q-$graded Hopf quasigroup and $A$ is a $H-$quasimodule Hopf quasigroup such that

[TABLE]

then the following statements are equivalent:

(1) There is a smash product $Q-$graded Hopf quasigroup $A\rtimes H$ built on $A\otimes H=\{A\otimes H_{p}\}_{p\in Q}$ with

(smash product)

[TABLE]

(unit)

[TABLE]

(Coproduct)

[TABLE]

(counit)

[TABLE]

(antipode)

[TABLE]

(2)

[TABLE]

Proof.

(1)$\Longrightarrow(2)$

[TABLE]

Applying $I\otimes\epsilon_{q}$ to both sides of this equation, set $b=g_{(1,q)}\cdot(S_{p}(h)_{(1,p^{-1})}\cdot a_{1})$ and replace $a,~{}g,~{}h$ by $a_{2},~{}g_{(2,q)},~{}h_{(2,q)},$ we obtain

[TABLE]

∎

(2)$\Longrightarrow(1)$

It is easy to check $A\rtimes H$ is a $Q-$graded algebra and $\forall p\in Q,~{}A\rtimes H_{p}$ is a coassocitive counitary coalgebra and $\mu$ is a coalgebra map,

Since

[TABLE]

and

[TABLE]

the condition (2.1) is verified.

Since

[TABLE]

and

[TABLE]

the equation (2.3) is verified.

Similarly, since

[TABLE]

and

[TABLE]

the equation (2.4) is verified, therefore $A\otimes H$ is a $Q-$graded Hopf quasigroup.