Free Rota-Baxter systems and a Hopf algebra structure

TL;DR

This paper constructs a linear basis for free Rota-Baxter systems using Gröbner-Shirshov bases and establishes a left counital Hopf algebra structure on them, advancing algebraic understanding.

Contribution

It introduces a new linear basis for free Rota-Baxter systems and defines a Hopf algebra structure on them, combining combinatorial and algebraic techniques.

Findings

Established a linear basis for free Rota-Baxter systems

Defined a left counital Hopf algebra structure on these systems

Applied Gröbner-Shirshov bases method to algebraic structures

Abstract

In this paper, we give a linear basis of a free Rota-Baxter system on a set by using the Gr\"{o}bner-Shirshov bases method and then we obtain a left counital Hopf algebra structure on a free Rota-Baxter system.

Equations

$$R(a)R(b)=R(R(a)b+aS(b)),S(a)S(b)=S(R(a)b+aS(b)).$$

$$P(x)P(y)=P(P(x)y)+P(xP(y))+\lambda P(xy),x,y\in A.$$

$$\Omega =\bigcup _{m=1}^{\infty }{\Omega }_m,$$

$$\Omega (Y)=\bigcup _{m=1}^{\infty }\left\{{\omega }^{(m)}(y_1,y_2,\cdots ,y_m)|y_i\in Y,1\le i\le m,{\omega }^{(m)}\in {\Omega }_m\right\}.$$

$$⟨\Omega ;X{⟩}_n=(X\cup \Omega (⟨\Omega ;X{⟩}_{n-1}){)}^{\ast }.$$

$$⟨\Omega ;X⟩=\bigcup _{n=0}^{\infty }⟨\Omega ;X{⟩}_n.$$

$$u=u_1{u}_2\cdots u_n,n\ge 0,$$

$$(f,g{)}_w=\sum {\alpha }_i{\pi }_i{|}_{s_i},$$

$$(f,g{)}_w\equiv 0\mathrm{mod}(\mathbb{S},w).$$

$$\mathrm{wt}(u)=(\text{mbox}deg(u),\text{mbox}bre(u),u_1,u_2,\cdots ,u_n).$$

$$u{>}_{{}_{\mathrm{Dl}}}v\text{mbox}if\mathrm{wt}(u)>\mathrm{wt}(v)\text{mbox}lexicographically,$$

$$(\omega ,u_i^{\prime })>(\theta ,v_i^{\prime })\text{mbox}lexicographically.$$

$$\mathbb{S}=\left\{\left.\begin{array}[]{ll}R(u)R(v)-R(R(u)v)-R(uS(v))\\ S(u)S(v)-\ S(R(u)v)-S(uS(v))\\ \end{array}\right|u,v\in\langle\{R,S\},X\rangle\right\}$$

$$f(u,v)=R(u)R(v)-R(R(u)v)-R(uS(v)),$$

$$h(u,v)=S(u)S(v)-S(R(u)v)-S(uS(v)),$$

$$(f(u,v),f(v,w){)}_{w_1},w_1=R(u)R(v)R(w),$$

$$(f(\pi {|}_{{}_{R(u)R(v)}},w),f(u,v){)}_{w_2},w_2=R(\pi {|}_{{}_{R(u)R(v)}})R(w),$$

$$f(u,\pi {|}_{{}_{R(v)R(w)}}),f(v,w){)}_{w_3},w_3=R(u)R(\pi {|}_{{}_{R(v)R(w)}}),$$

$$(f(\pi {|}_{{}_{S(u)S(v)}},w),h(u,v){)}_{w_4},w_4=R(\pi {|}_{{}_{S(u)S(v)}})R(w),$$

$$(f(u,\pi {|}_{{}_{S(v)S(w)}}),h(v,w){)}_{w_5},w_5=R(u)R(\pi {|}_{{}_{S(v)S(w)}}),$$

$$(h(u,v),h(v,w){)}_{w_6},w_6=S(u)S(v)S(w),$$

$$(h(\pi {|}_{{}_{S(u)S(v)}},w),h(u,v){)}_{w_7},w_7=S(\pi {|}_{{}_{S(u)S(v)}})S(w),$$

$$(h(u,\pi {|}_{{}_{S(v)S(w)}}),h(v,w){)}_{w_8},w_8=S(u)S(\pi {|}_{{}_{S(v)S(w)}}),$$

$$(h(\pi {|}_{{}_{R(u)R(v)}},w),f(u,v){)}_{w_9},w_9=S(\pi {|}_{{}_{R(u)R(v)}})S(w),$$

$$(h(u,\pi {|}_{{}_{R(v)R(w)}}),f(v,w){)}_{w_{{}_{10}}},w_{{}_{10}}=S(u)S(\pi {|}_{{}_{R(v)R(w)}}).$$

$$(f(u,v),f(v,w){)}_{w_1}$$

$$\mathbb{S}=\left\{\left.\begin{array}[]{ll}R(u)R(v)-R(R(u)v)-R(uS(v))\\ S(u)S(v)-\ S(R(u)v)-S(uS(v))\\ \end{array}\right|u,v\in\langle\{R,S\},X\rangle\right\}.$$

$$\Phi_{\infty}(X)=\left\{w\in\langle\{R,S\};X\rangle\left|\begin{array}[]{ll}w\neq\pi|_{{}_{Q(u)Q(v)}},\pi\in\langle\{R,S\},X\rangle^{\star},\\ Q\in\{R,S\},\ u,\ v\in\langle\{R,S\};X\rangle\\ \end{array}\right.\right\}.$$

$$w\diamond v=\left\{\begin{array}[]{l@{\quad\quad}l}Q(R(\widetilde{w})\diamond\widetilde{v}+\widetilde{w}\diamond S(\widetilde{v})),&{if}\ w=Q(\widetilde{w}),v=Q(\widetilde{v}),Q\in\{R,S\},\\ wv,&{otherwise.}\end{array}\right.$$

$$w\diamond v=w_1{w}_2\cdots w_{t-1}(w_t\diamond v_1)v_2\cdots v_l.$$

Full text

Free Rota-Baxter systems and a Hopf algebra structure111Supported by the NNSF of China (no. 11571121),

the NSF of Guangdong Province (no. 2017A030313002) and the Science and Technology Program of Guangzhou (no. 201707010137).

Jianjun Qiu

 School of Mathematics and Statistics, Lingnan Normal University

Zhanjiang 524048, P. R. China

jianjunqiu@126.com

Yuqun Chen222Corresponding author.

 School of Mathematical Sciences, South China Normal University

Guangzhou 510631, P. R. China

yqchen@scnu.edu.cn

Abstract: In this paper, we give a linear basis of a free Rota-Baxter system on a set by using the Gröbner-Shirshov bases method and then we obtain a left counital Hopf algebra structure on a free Rota-Baxter system.

Key words: Gröbner-Shirshov basis; free Rota-Baxter system; left counital Hopf algebra.

AMS 2010 Subject Classification: 16S15, 13P10, 17A50, 16T25

1 Introduction

A triple $(A;R,S)$ consisting of an associative unitary algebra $A$ over a field $k$ and two $k$-linear operators $R,S:A\rightarrow A$ is called a Rota-Baxter system if, for any $a,b\in A$,

[TABLE]

Rota-Baxter system was introduced by Brzeziński in a recent paper [10], which can be viewed as an extension of the Rota-Baxter algebra of weight [math].

An associative unitary algebra $A$ together with a $k$-linear operator $P:A\rightarrow A$ is called a Rota-Baxter algebra of weight $\lambda$, if

[TABLE]

Here $\lambda$ is a fixed element in the field $k$.

Baxter [3] firstly studied the Rota-Baxter algebras. Some combinatoric properties of Rota-Baxter algebras were studied by Rota [25] and Cartier [13]. The constructions of free Rota-Baxter associative algebras on both commutative and noncommutative cases were given by using different methods, for example, [7, 13, 16, 17, 19, 20, 21, 25].

Gröbner bases and Gröbner-Shirshov bases were invented independently by Shirshov for ideals of free Lie algebras [26] and implicitly free associative algebras [26] (see also [4, 5]), by Hironaka [23] for ideals of the power series algebras (both formal and convergent), and by Buchberger [11] for ideals of the polynomial algebras. Gröbner bases and Gröbner-Shirshov bases theories have been proved to be very useful in different branches of mathematics. See, for example, the books [2, 9, 12], the papers [4, 5], and the surveys [6, 8]. In fact, Gröbner-Shirshov bases theory is a useful tool for constructing free objects for many algebra varieties. We will apply Gröbner-Shirshov bases method to construct free Rota-Baxter system generated by a set.

The Hopf algebra originated from the study of topology and has widely applications on mathematics and physics. See for instance [1, 14]. Many classical Hopf algebras are build from free objects on various context, which include the free associative algebra and the enveloping algebra of Lie algebras. Recently, there are some Hopf algebra structures on other free objects, such as the dendriform algebras [24] and Rota-Baxter algebras [15, 27, 18]. Inspired by the ideas of the above papers, we will establish a left counital Hopf algebra structure on the free object of Rota-Baxter system.

The paper is organized as follows. In Section 2, we give a Gröbner-Shirshov basis of a free Rota-Baxter system and then a linear basis of such algebra is obtained by Composition-Diamond lemma for associative $\Omega$-algebras. In Section 3, by using the construction of free Rota-Baxter system obtained in Section 2, we give a left counital Hopf algebra structure on a free Rota-Baxter system.

2 Free Rota-Baxter systems

2.1 Gröbner-Shirshov

bases for associative $\Omega$-algebras

In this subsection, we review Gröbner-Shirshov bases theory for associative unitary $\Omega$-algebras. For more details, see for instance, [7, 22].

Let

[TABLE]

where $\Omega_{m}$ is a set of $m$-ary operators for any $m\geq 1$. For any set $Y$, let

[TABLE]

Let $X$ be a set. Define $\langle\Omega;X\rangle_{0}=X^{*}$, where $X^{*}$ is the free monoid with the unit $1$ on the set $X$. Assume that we have defined $\langle\Omega;X\rangle_{n-1}$. Define

[TABLE]

Then it is easy to see that $\langle\Omega;X\rangle_{n}\subseteq\langle\Omega;X\rangle_{n+1}$ for any $n\geq 0$. Set

[TABLE]

If $u\in\langle\Omega;X\rangle$, then $u$ is said to be an $\Omega$-word on the set $X$. For any $u\in X\cup\Omega(\langle\Omega;X\rangle)$, $u$ is called prime. Therefore, for any $u\in\langle\Omega;X\rangle$, $u$ can be uniquely expressed in the canonical form

[TABLE]

where each $u_{i}$ is prime. The breath of $u$, denoted by bre($u$), is defined to be the number $n$. By the depth of $u$, denoted by dep($u$), we mean ${\rm dep}(u)=\min\{n|u\in\langle\Omega;X\rangle_{n}\}.$

An associative unitary $\Omega$-algebra over a field $k$ is an associative unitary $k$-algebra $A$ with a set of multilinear operators $\Omega$, where $\Omega=\bigcup_{m=1}^{\infty}\Omega_{m}$ and each $\Omega_{m}$ is a set of $m$-ary multilinear operators on $A$.

Let $k\langle\Omega;X\rangle$ be the linear space spanned by $\langle\Omega;X\rangle$ over the field $k$. Then $k\langle\Omega;X\rangle$ is a free associative unitary $\Omega$-algebra on $X$.

Let $\star\notin X$. By a $\star$-$\Omega$-word on $X$ we mean any expression in $\langle\Omega;X\cup\{\star\}\rangle$ with only one occurrence of $\star$. Let $\langle\Omega;X\rangle^{\star}$ denote the set of all $\star$-$\Omega$-words on $X$. If $\pi$ is a $\star$-$\Omega$-word and $s\in k\langle\Omega;X\rangle$, then we call $\pi|_{s}:=\pi|_{\star\mapsto s}$ an $s$-$\Omega$-word.

Now, we assume that $\langle\Omega;X\rangle$ is equipped with a monomial order $>$. This means that $>$ is a well order on $\langle\Omega;X\rangle$ such that for any $v,w\in\langle\Omega;X\rangle$ and $\pi\in\langle\Omega;X\rangle^{\star}$, if $w>v$, then $\pi|_{w}>\pi|_{v}$.

For any $f\in k\langle\Omega;X\rangle$, let $\bar{f}$ be the leading $\Omega$-word of $f$ with respect to the order $>$. If the coefficient of $\bar{f}$ is $1$, then we call that $f$ is monic. We also call a set $\mathbb{S}\subseteq k\langle\Omega;X\rangle$ monic if each $s\in\mathbb{S}$ is monic.

Let $f,g\in k\langle\Omega;X\rangle$ be monic. Then we define two kinds of compositions.

1. (I)

If $w=\bar{f}a=b\bar{g}$ for some $a,b\in\langle\Omega;X\rangle$ such that bre($w$)$<$bre($\bar{f}$)$+$bre($\bar{g}$), then we call $(f,g)_{w}=fa-bg$ the intersection composition of $f$ and $g$ with respect to the ambiguity $w$. 2. (II)

If $w=\bar{f}=\pi|_{{}_{\bar{g}}}$ for some $\pi\in\langle\Omega;X\rangle^{\star}$, then we call $(f,g)_{w}=f-\pi|_{{}_{g}}$ the inclusion composition of $f$ and $g$ with respect to the ambiguity $w$.

If $\bar{f}=\pi|_{{}_{\bar{g}}}$, then the transformation $f\mapsto f-\alpha\pi|_{{}_{g}}$ is called the Elimination of the Leading Word (ELW) of $f$ by $g$, where $g$ is monic and $\alpha$ is the coefficient of $\bar{f}$.

Let $\mathbb{S}\subseteq k\langle\Omega;X\rangle$ be monic. The composition $(f,g)_{w}$ is called trivial modulo $(\mathbb{S},w)$ if

[TABLE]

where each $\alpha_{i}\in k$, $\pi_{i}\in\langle\Omega;X\rangle^{\star}$, $s_{i}\in\mathbb{S}$ and $\pi_{i}|_{\overline{s_{i}}}<w$. If this is the case, we write

[TABLE]

In general, for any $p$ and $q$, $p\equiv q\ {\rm mod}(\mathbb{S},w)$ means that $p-q=\sum\alpha_{i}\pi_{i}|_{s_{i}},$ where each $\alpha_{i}\in k$, $\pi_{i}\in\langle\Omega;X\rangle^{\star}$, $s_{i}\in\mathbb{S}$ and $\pi_{i}|_{\overline{s_{i}}}<w$.

A monic set $\mathbb{S}$ is called a Gröbner-Shirshov basis in $k\langle\Omega;X\rangle$ if any composition $(f,g)_{w}$ of $f,g\in\mathbb{S}$ is trivial modulo $(\mathbb{S},w)$.

In fact, the proof of Composition-Diamond lemma for associative unitary $\Omega$-algebras is the same as the one for nonunitary $\Omega$-algebras in [7]. See also [22].

Theorem 2.1

([7], Composition-Diamond lemma for associative unitary $\Omega$-algebras)*   Let $\mathbb{S}\subseteq k\langle\Omega;X\rangle$ be monic, $>$ a monomial order on $\langle\Omega;X\rangle$ and $Id(\mathbb{S})$ the $\Omega$-ideal of $k\langle\Omega;X\rangle$ generated by $\mathbb{S}$. Then the following statements are equivalent:*

1. (i)

The set $\mathbb{S}$ is a Gröbner-Shirshov basis in $k\langle\Omega;X\rangle$. 2. (ii)

If $f\in Id(\mathbb{S})$, then $\bar{f}=\pi|_{\overline{s}}$ for some $\pi\in\langle\Omega;X\rangle^{\star}$ and $s\in\mathbb{S}$. 3. (iii)

The set $Irr(\mathbb{S})=\{w\in\langle\Omega;X\rangle|w\neq\pi|_{\overline{s}},\pi\in\langle\Omega;X\rangle^{\star},s\in\mathbb{S}\}$ is a $k$-linear basis of $k\langle\Omega;X|\mathbb{S}\rangle:=k\langle\Omega;X\rangle/Id(\mathbb{S})$.

2.2 Free Rota-Baxter systems

In this subsection, we give a Gröbner-Shirshov basis of a free Rota-Baxter system on a set and then a linear basis of such an algebra is obtained by the Composition-Diamond lemma for associative unitary $\Omega$-algebras.

Let $X$ be a well-ordered set and $\Omega=\{R,S\}$, where both $R$ and $S$ are 1-array operators. Assume that $R>S$. If $u\in\langle\{R,S\},X\rangle$, then we define ${\rm deg}(u)$ to be the number of all occurrences of all $x\in X$ and $\omega\in\Omega$. For example, if $w=x_{1}x_{2}R(R(x_{1})S(x_{2}))$, where $x_{1},x_{2}\in X$, then ${\rm deg}(u)=7$.

If $u=u_{1}u_{2}\cdots u_{n}\in\langle\{R,S\};X\rangle,n\geq 1,$ where each $u_{i}$ is prime, then we let

[TABLE]

Define the Deg-lex order $>_{{}_{{\rm Dl}}}$ on $\langle\{R,S\};X\rangle$ as follows. For any $u=u_{1}u_{2}\cdots u_{n}$, $v=v_{1}v_{2}\cdots v_{m}\in\langle\{R,S\};X\rangle$, where $u_{i},v_{j}$ are prime, define

[TABLE]

where $u_{i}>v_{i}$ if deg($u_{i}$)$>$deg($v_{i}$) or deg($u_{i}$)=deg($v_{i}$) such that one of the following conditions holds:

(a) $u_{i},v_{i}\in X$ and $u_{i}>v_{i}$;

(b) $u_{i}=R(1)$ or $u_{i}=S(1)$ and $v_{i}\in X$;

(c) $u_{i}=\omega(u_{i}^{\prime}),v_{i}=\theta(v_{i}^{\prime})$, $\omega,\theta\in\{R,S\}$ and

[TABLE]

It is easy to see that $>_{{}_{{\rm Dl}}}$ is a monomial order on $\langle\{R,S\};X\rangle$.

Let $k\langle\{R,S\},X\rangle$ be the free associative unitary $\{R,S\}$-algebra generated by the set $X$.

Theorem 2.2

With the order $>_{{}_{{\rm Dl}}}$ on $\langle\{R,S\};X\rangle$, the set

[TABLE]

is a Gröbner-Shirshov basis in $k\langle\{R,S\},X\rangle$.

Proof. Let

[TABLE]

[TABLE]

where $u,\ v\in\langle\{R,S\},X\rangle$. All the possible compositions of $\{R,S\}$-polynomials in $\mathbb{S}$ are listed as below:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We check that all the compositions are trivial. Here, we just check one as an example.

[TABLE]

$\square$

Note that $RS(X)=k\langle\{R,S\},X|\mathbb{S}\rangle$ is a free Rota-Baxter system on the set $X$, where

[TABLE]

Let

[TABLE]

The elements of $\Phi_{\infty}(X)$ are called Rota-Baxter system words.

Theorem 2.3

The set $Irr(\mathbb{S})=\Phi_{\infty}(X)$ is a linear basis of the free Rota-Baxter system $RS(X)$.

Proof. By Theorems 2.1 and 2.2, we can obtain the result. $\square$

By using ELWs, we have the following algorithm, which is an algorithm to compute the product of two Rota-Baxter system words in the free Rota-Baxter system $RS(X)$.

Algorithm 2.4

Let $w,v\in\Phi_{\infty}(X)$. We define $w\diamond v$ by induction on $n={\rm dep}(w)+{\rm dep}(v)$.

1. (a)

If $n=0$, then $w,v\in X^{*}$ and $w\diamond v=wv$; 2. (b)

If $n\geq 1$, there are two cases to consider:

*(i) If ${\rm bre}(w)={\rm bre}(v)=1$, then *

[TABLE]

(ii) If ${\rm bre}(w)>1$ or ${\rm bre}(v)>1$ and assume that $w=w_{1}w_{2}\cdots w_{t}$ and $v=v_{1}v_{2}\cdots v_{l},$ where $u_{i}$ and $v_{j}$ are prime, then

[TABLE]

3 A left counital Hopf algebra structure on free Rota-Baxter system

In this section, similar to the Hopf algebra structure on free Rota-Baxter algebra given by Gao, Guo and Zhang[18], we establish a left counital Hopf algebra structure on the free Rota-Baxter system $RS(X)$.

3.1 A left counital bialgebra structure

In this subsection, we give a left counital bialgebra structure on the free Rota-Baxter system $RS(X)$.

Definition 3.1

([28])* (a)   A left counital coalgebra is a triple $(H,\Delta,\varepsilon)$, where the comultiplication $\Delta:H\rightarrow H\otimes H$ satisfies the coassociativity and the left counit $\varepsilon:H\mapsto k$ satisfies the left counicity: $(\varepsilon\otimes{\rm id})\Delta=\beta_{l},$ where $\beta_{l}(w):RS(X)\rightarrow k\otimes RS(X),w\mapsto 1_{k}\otimes w.$*

(b)   A left counital bialgebra is a quintuple $(H,\mu,u,\Delta,\varepsilon)$, where $(H,\mu,u)$ is a $k$-algebra and $(H,\Delta,\varepsilon)$ is a left counital coalgebra such that $\Delta:H\rightarrow H\otimes H$ and $\varepsilon:H\rightarrow k$ are algebra homomorphisms.

Define $\Delta:RS(X)\rightarrow RS(X)\otimes RS(X)$, where for any $w\in\Phi_{\infty}(X)$, $\Delta(w)$ is defined by induction on ${\rm dep}(w)$:

If ${\rm dep}(w)=0$, then we define

[TABLE]

and

[TABLE]

where $w=x_{1}x_{2}\cdots x_{m},\ m\geq 2$ with each $x_{i}\in X$.

Assume that $\Delta(w)$ has been defined for any $w\in\Phi_{\infty}(X)$ with ${\rm dep}(w)\leq n$. Let $w\in\Phi_{\infty}(X)$ with ${\rm dep}(w)=n+1$. If ${\rm bre}(w)=1$, define

[TABLE]

where $\Delta(\widetilde{w})$ is defined by induction hypothesis. If ${\rm bre}(w)>1$, say $w=w_{1}w_{2}\cdots w_{m}$, where each $w_{i}$ is prime, define

[TABLE]

Define $u:k\rightarrow RS(X),1_{k}\mapsto 1$ and $\varepsilon:RS(X)\rightarrow k$ by

[TABLE]

where $1_{k}$ is the unit of the field $k$.

It is easy to see that $(RS(X),\diamond,u)$ is a unitary $k$-algebra.

Lemma 3.2

For any $w,v\in RS(X)$, we have

[TABLE]

Proof. It is sufficient to prove that $\Delta(w\diamond v)=\Delta(w)\diamond\Delta(v)$ for any $w,v\in\Phi_{\infty}(X)$. Induction on $(m,n)$, where $m={\rm dep}(w)+{\rm dep}(v)$ and $n={\rm bre}(w)+{\rm bre}(v)$.

If $(m,n)=(0,0)$, i.e. $w=v=1$, then $\Delta(w\diamond v)=1\otimes 1=\Delta(w)\diamond\Delta(v)$.

If $(m,n)=(0,1)$, then $w\in X,\ v=1$ or $w=1,\ v\in X$. Thus $\Delta(w\diamond v)=\Delta(w)\diamond\Delta(v)$.

Assume that we have proved $\Delta(w\diamond v)=\Delta(w)\diamond\Delta(v)$ for all $u,v\in\Phi_{\infty}(X)$ with $(m,n)<(p,q)$.

Let $u,v\in\Phi_{\infty}(X)$ with $({\rm dep}(w)+{\rm dep}(v),{\rm bre}(w)+{\rm bre}(v))=(p,q)$ . There are two cases to consider.

Case 1. If $q=1$, i.e. $w=1$ or $v=1$, then by definition, the result is true.

Case 2. If $q=2$, then we have two cases.

(i) If $(w,v)\notin\bigcup_{Q\in\{R,\ S\}}Q(\Phi_{\infty}(X))\times Q(\Phi_{\infty}(X))$, then $w\diamond v=wv\in\Phi_{\infty}(X)$ and the result is true by the definition of $\Delta$.

(ii) If $(w,v)\in\bigcup_{Q\in\{R,\ S\}}Q(\Phi_{\infty}(X))\times Q(\Phi_{\infty}(X))$, then $w=R(w^{\prime})$ and $v=R(v^{\prime})$ or $w=S(w^{\prime})$ and $v=S(v^{\prime})$. Using the Sweedler notation, we can write

[TABLE]

(a) If $w=R(w^{\prime})$ and $v=R(v^{\prime})$, then

[TABLE]

[TABLE]

(b) If $w=S(w^{\prime})$ and $v=S(v^{\prime})$, then

[TABLE]

Case 3. If $q>2$, say $w=w_{1}w_{2}\cdots w_{t}$ and $v=v_{1}v_{2}\cdots v_{l}$, then we have two cases to consider.

(i) If $(w_{t},v_{1})\notin\bigcup_{Q\in\{R,\ S\}}Q(\Phi_{\infty}(X))\times Q(\Phi_{\infty}(X))$, then $w\diamond v=wv\in\Phi_{\infty}(X)$ and the result is true by the definition of $\Delta$.

(ii) If $(w_{t},v_{1})\in\bigcup_{Q\in\{R,\ S\}}Q(\Phi_{\infty}(X))\times Q(\Phi_{\infty}(X))$, then by Case 2 or by induction, we have $\Delta(w_{t}\diamond v_{1})=\Delta(w_{t})\diamond\bigtriangleup(v_{1})$. By the definition of $\diamond$, we have $w_{t}\diamond v_{1}=\sum u_{i}$, where ${\rm bre}(u_{i})=1$. Therefore

[TABLE]

This completes the proof by induction. $\square$

Lemma 3.3

For any $w,v\in RS(X)$, we have $\varepsilon(w\diamond v)=\varepsilon(w)\diamond\varepsilon(v).$

Proof. It is easy to prove the result. $\square$

Dr. Xing Gao has kindly pointed out that $\varepsilon$ is not a right counit in Lemma 3.3.

Lemma 3.4

The triple $(RS(X),\Delta,\varepsilon)$ is a left couital bialgebra.

Proof. It is sufficient to prove the coassociativity and the left counicity for $w\in\Phi_{\infty}(X)$.

We check the coassociativity by induction on $(m,n)$ for $w\in\Phi_{\infty}(X)$, where $m={\rm dep}(w)$ and $n={\rm bre}(w)$.

If $(m,n)=(0,0)$, then $w=1$ and $(\Delta\otimes{\rm id})\Delta(w)=1\otimes 1\otimes 1=({\rm id}\otimes\Delta)\Delta(w)$.

Assume that the result is true for any $w$ with $({\rm dep}(w),{\rm bre}(w))<(p,q)$.

Let $w\in\Phi_{\infty}(X)$ with $({\rm dep}(w),{\rm bre}(w))=(p,q)>(0,0)$. Then we have two cases to consider.

Case 1. If $q=1$, then $w=x\in X$ or $w=R(w^{\prime})$ or $w=S(w^{\prime})$.

Subcase 1. If $w=x\in X$, then

[TABLE]

Subcase 2. If $w=S(w^{\prime})$, then

[TABLE]

Subcase 3. If $w=R(w^{\prime})$, then similar to the Subcase 2, we have

[TABLE]

Case 2. If $q>1$, then we can let $w=w_{1}w_{2}$ with $w_{1},w_{2}\neq 1$. Let us denote

[TABLE]

Then

[TABLE]

Similarly, we have

[TABLE]

By induction hypotheses, we have

[TABLE]

Therefore,

[TABLE]

This completes the proof of coassociativity by induction.

We also use induction on $(m,n)$ for $w\in\Phi_{\infty}(X)$, where $m={\rm dep}(w)$ and $n={\rm bre}(w)$ to check the left counicity conditions:

[TABLE]

where

[TABLE]

If $(m,n)=(0,0)$, then it is easy to see that the result is true.

Assume that the result is true for any $w$ with $(m,n)<(p,q)$.

Let $w\in\Phi_{\infty}(X)$ with $({\rm dep}(w),{\rm bre}(w))=(p,q)>(0,0)$.

Case I. If ${q=\rm bre}(w)=1$, then $w=x\in X$ or $w=R(w^{\prime})$ or $w=S(w^{\prime})$.

Subcase I-1. If $w=x\in X$, then

[TABLE]

Subcase I-2. If $w=S(w^{\prime})$, then

[TABLE]

Subcase I-3. If $w=R(w^{\prime})$, then similar to the Subcase I-2, $(\varepsilon\otimes{\rm id})\Delta(R(w^{\prime}))=\beta_{l}(R(w^{\prime}))$.

Case II. If $q={\rm bre}(w)>1$, then we can let $w=w_{1}w_{2}$ with $w_{1},w_{2}\neq 1$. Let

[TABLE]

By induction hypotheses, we have

[TABLE]

That is

[TABLE]

Thus

[TABLE]

This completes the proof of $(\varepsilon\otimes{\rm id})\Delta(w)=\beta_{l}(w)$ by induction. $\square$

Remark: Note that $\varepsilon$ is not a right counit on $RS(X)$. If $R\neq S$, then

[TABLE]

where

[TABLE]

By Lemmas 3.2-3.4, we have the following theorem.

Theorem 3.5

The quintuple $(RS(X),\diamond,u,\Delta,\varepsilon)$ is a left counital bialgebra.

3.2 A left counital Hopf algebra structure

For a $k$-algebra $(A,\mu,u)$ and a left counital bialgebra coalgebra $(C,\Delta,\varepsilon)$, we define the convolution of two linear maps $f,g\in{\rm Hom}(C,A)$ to be the map $f\ast g\in{\rm Hom}(C,A)$ given by the composition

[TABLE]

Let $(H,\mu,u,\Delta,\varepsilon)$ be a left counital bialgebra. A $k$-linear endomorphism $T$ of $H$ is called a right antipode of $H$ if it is the right inverse of ${\rm id}_{H}$ under the convolution product

[TABLE]

A left counital Hopf algebra is a left counital bialgebra $H$ with an right antipode $T$. For more about (left counital ) Hopf algebra, see for instance [1, 28, 19].

Recall that a left counital bialgebra $(H,\mu,u,\Delta,\varepsilon)$ is called a graded left counital bialgebra if there are a sequence of $k$-vector spaces $H^{(n)},\ n\geq 0,$ such that

1. (a)

$H=\bigoplus_{n=0}^{\infty}H^{(n)}$; 2. (b)

For any $p,q\geq 0$, $H^{(p)}H^{(q)}\subseteq H^{(p+q)}$; 3. (c)

For any $n\geq 0$, $\Delta(H^{(n)})\subseteq\bigoplus_{p+q=n}H^{(p)}\otimes H^{(q)}$.

A graded left counital bialgebra $H=\bigoplus_{n=0}^{\infty}H^{(n)}$ is called connected if $H^{(0)}=k$ and $ker\varepsilon=\bigoplus_{n>0}H^{(n)}$.

Define $H_{RS}^{(n)}$ the $k$-linear space spanned by $\{w\in\Phi_{\infty}(X)|{\rm deg}(w)=n\}$, i.e.

[TABLE]

Then

[TABLE]

Lemma 3.6

Let $p,q\geq 0$. Then

[TABLE]

Proof. We just have to prove that $w\diamond v\in H_{RS}^{(p+q)}$ for any $w,v\in\Phi_{\infty}(X)$ with ${\rm deg}(w)=p$ and ${\rm deg}(v)=q$. Induction on $n=p+q$.

If $n=0$, then $w=v=1$ and it is easy to see that the result is true.

Assume that the result is true for $n=p+q$.

Let $w,v\in\Phi_{\infty}(X)$ with ${\rm deg}(w)+{\rm deg}(v)=p+q+1$. Let

[TABLE]

Case 1. If $(w_{t},v_{1})\notin\bigcup_{Q\in\{R,\ S\}}Q(\Phi_{\infty}(X))\times Q(\Phi_{\infty}(X))$, then $w\diamond v=wv\in\Phi_{\infty}(X)$ and it is easy to see the result is true.

Case 2. If $(w_{t},v_{1})\in\bigcup_{Q\in\{R,\ S\}}Q(\Phi_{\infty}(X))\times Q(\Phi_{\infty}(X))$, then $w_{t}=R(w^{\prime}),v_{1}=R(v^{\prime})$ or $w_{t}=S(w^{\prime}),v_{1}=S(v^{\prime})$.

(i) If $w_{t}=S(w^{\prime}),v_{1}=S(v^{\prime})$, then

[TABLE]

Let $l={\rm deg}(w_{t})+{\rm deg}(v_{1})$. By induction hypotheses, $R(w^{\prime})\diamond v^{\prime},w^{\prime}\diamond S(v^{\prime})\in H_{RS}^{(l-1)}$. Therefore, $w\diamond v\in H_{RS}^{(p+q+1)}$.

(ii) If $w_{t}=R(w^{\prime}),v_{1}=R(v^{\prime})$, then similar to the proof of (i), we have the result is true.

This completes the proof by induction. $\square$

Lemma 3.7

For any $n\geq 0$,

[TABLE]

Proof. For $n=0,1$, it is easy to see that the result is true. Assume that the result is true for $0\leq n\leq m$. We just have to prove that $\Delta(w)\in\bigoplus_{p+q=m+1}H_{RS}^{(p)}\otimes H_{RS}^{(q)}$ for any $w$ with ${\rm deg}(w)=m+1$.

If ${\rm bre}(w)=1$, then we have $w=R(w^{\prime})$ or $w=S(w^{\prime})$.

If $w=S(w^{\prime})$, then by induction hypotheses, we have

[TABLE]

Similarly, if $w=R(w^{\prime})$, then $\Delta(w)\in\bigoplus_{p+q=m+1}H_{RS}^{(p)}\otimes H_{RS}^{(q)}$.

If ${\rm bre}(w)\geq 2$, say $w=uv$ with $u,v\neq 1$, then by induction hypotheses and Lemma 3.6, we have

[TABLE]

This completes the proof by induction. $\square$

By a similar proof of the theorems in ([28, 19]), we have

Lemma 3.8

A connected left counital bialgebra $(H,\mu,u,\Delta,\varepsilon)$ is a left counital Hopf algebra.

By Lemmas 3.6-3.8, we have the following theorem.

Theorem 3.9

The free Rota-Baxter system $RS(X)=\bigoplus_{n=0}^{\infty}H_{RS}^{(n)}$ is a connected left counital bialgebra. It follows that $RS(X)$ is a left counital Hopf algebra.

Acknowledgement We wish to express our thanks to Dr. Xing Gao and Prof. Li Guo for helpful and valuable suggestions and comments.