Free differential Lie Rota-Baxter algebras and Gr\"obner-Shirshov bases

TL;DR

This paper develops a Gr"obner-Shirshov bases framework for differential Lie $ ext{Omega}$-algebras and constructs a linear basis for free differential Lie Rota-Baxter algebras, advancing algebraic structure theory.

Contribution

It introduces the Gr"obner-Shirshov bases theory for differential Lie $ ext{Omega}$-algebras and provides a basis for free differential Lie Rota-Baxter algebras, which was previously unknown.

Findings

Established Gr"obner-Shirshov bases for differential Lie $ ext{Omega}$-algebras

Constructed a linear basis for free differential Lie Rota-Baxter algebra

Advanced the understanding of algebraic structures involving Rota-Baxter operators

Abstract

We establish the Gr\"obner-Shirshov bases theory for differential Lie $\Omega$-algebras. As an application, we give a linear basis of a free differential Lie Rota-Baxter algebra on a set.

Equations

$$D(x\cdot y)=D(x)\cdot y+x\cdot D(y)+\lambda D(x)\cdot D(y),x,y\in R.$$

$$P(x)\cdot P(y)=P(x\cdot P(y))+P(P(x)\cdot y)+\lambda P(x\cdot y),x,y\in R.$$

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

$$⟨D,\Omega ;X{⟩}_n=S(\Delta (X\cup \Omega (⟨D,\Omega ;X{⟩}_{n-1}))),$$

$$(D,\Omega ;X{)}_n=(\Delta (X\cup \Omega ((D,\Omega ;X{)}_{n-1})){)}^{\ast \ast }.$$

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

$$D(u)=D(u_1)(u_2\cdots u_t)+u_{1}D(u_2\cdots u_t)+\lambda D(u_1)D(u_2\cdots u_t).$$

$${\omega }^{(m)}:⟨D,\Omega ;X{⟩}^m\to ⟨D,\Omega ;X⟩,(u_1,u_2,\cdots ,u_m)\mapsto {\omega }^{(m)}(u_1,u_2,\cdots ,u_m)$$

$$wt(u)=(deg(u),bre(u),u_1,u_2,\cdots ,u_m).$$

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

$$(\omega ,u_{i1},u_{i2},\cdots ,u_{it})>(\theta ,v_{i1},v_{i2},\cdots ,v_{il})\text{mbox}lexicographically.$$

$$I_n^t=\{(i_1,i_2,\cdots ,i_n)\in \{0,1{\}}^n|i_1+i_2+\cdots +i_n=t\}.$$

$$D(u)$$

$$(f,g{)}_w=lc(D^i(f){)}^{-1}{D}^i(f)a-lc(D^j(g){)}^{-1}b{D}^j(g)$$

$$(f,g{)}_w=lc(D^i(f){)}^{-1}{D}^i(f)-lc(D^j(g){)}^{-1}\pi {|}_{{}_{D^{{}^j}(g)}}$$

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

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

$$Irr(S)=\left\{w\in\langle D,\Omega;X\rangle\left|\begin{array}[]{ll}w\neq\pi|_{{}_{\overline{D^{i}(s)}}},\ s\in S,\ i\geq 0,\ \pi|_{{}_{D^{i}(s)}}\mbox{is }\\ \mbox{a normal differential}\ D^{i}(s)\mbox{-word}\end{array}\right.\right\}$$

$$ALSW(D,\Omega ;X{)}_0:=ALSW(Z_0),$$

$$NLSW(D,\Omega ;X{)}_0:=NLSW(Z_0)=\{[u]|u\in ALSW(D,\Omega ;X{)}_0\}$$

$$ALSW(D,\Omega ;X{)}_{n-1},$$

$$NLSW(D,\Omega ;X{)}_{n-1}:=\{[u]|u\in ALSW(D,\Omega ;X{)}_{n-1}\}.$$

$$ALSW(D,\Omega ;X{)}_n:=ALSW(Z_n).$$

$$[u]:=\left\{\begin{array}[]{ll}u,&if\ u=D^{i}(x),x\in X,\\ D^{i}(\omega^{(m)}([u_{1}],[u_{2}],\cdots,[u_{m}])),&if\ u=D^{i}(\omega^{(m)}(u_{1},u_{2},\cdots,u_{m})).\end{array}\right.$$

$$[u]:=[[u_1][u_2]\cdots [u_t]]$$

$$NLSW(D,\Omega ;X{)}_n:=\{[u]|u\in ALSW(D,\Omega ;X{)}_n\}.$$

$$ALSW(D,\Omega ;X):=\bigcup _{n=0}^{\infty }ALSW(D,\Omega ;X{)}_n,$$

$$NLSW(D,\Omega ;X):=\bigcup _{n=0}^{\infty }NLSW(D,\Omega ;X{)}_n.$$

$$NLSW(D,\Omega ;X)=\{[u]|u\in ALSW(D,\Omega ;X)\}.$$

$$D([xy])=[D(x)y]+[xD(y)]+\lambda [D(x)D(y)],x,y\in L.$$

Full text

Free differential Lie Rota-Baxter algebras and Gröbner-Shirshov bases111Supported by the NNSF of China (11171118, 11571121) and the NSF of Guangdong (2015A030310502).

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: We establish the Gröbner-Shirshov bases theory for differential Lie $\Omega$-algebras. As an application, we give a linear basis of a free differential Lie Rota-Baxter algebra on a set.

Key words: Gröbner-Shirshov basis, Lyndon-Shirshov word, differential Lie Rota-Baxter algebra

AMS 2000 Subject Classification: 16S15, 13P10, 16W99, 17A50

1 Introduction

Let $k$ be a field and $\lambda\in k$. A differential algebra of weight $\lambda$ or a $\lambda$-differential algebra ([29, 23, 19]) is a $k$-algebra $(R,\cdot)$ together with a differential operator (of weight $\lambda$) $D:R\rightarrow R$ satisfying

[TABLE]

The differential algebras were first studied by J.F. Ritt [29] and have developed to be an important branch of mathematics in both theory and applications (see for instance [15, 19, 33]).

A Rota-Baxter algebra of weight $\lambda$ or $\lambda$-Rota-Baxter algebra ([4, 22, 30]) is a $k$-algebra $(R,\cdot)$ together with a Rota-Baxter operator (of weight $\lambda$) $P:R\rightarrow R$ satisfying

[TABLE]

The Rota-Baxter operator on an associative algebra initially appeared in probability [4] and then in combinatorics [30] and quantum field theory [14]. There are a number of studies on associative Rota-Baxter algebras on both commutative and noncommutative case. For more details we refer the reader to [22] and the references given there. The Rota-Baxter operator of weight 0 on a Lie algebra is also called the operator form of the classical Yang-Baxter equation [31]. The Lie Rota-Baxter algebras are closely related with the pre-Lie algebras. Recently, there are many results on Lie Rota-Baxter algebras and related topics (see for instance [2, 3, 21, 26, 28]).

Similarly to the relation between differential operator and integral operator as in the First Fundamental Theorem of Calculus, L. Guo and W. Keigher [23] introduced the notion of differential Rota-Baxter algebra which is a $k$-algebra $R$ together with a differential operator $D$ and a Rota-Baxter operator $P$ such that $DP=Id_{R}$.

As we known, the free objects of various varieties of linear algebras play an important role. Sometimes, it is difficult to give a linear basis of a free algebra, for example, it is an open problem to find a linear basis of a free Jordan algebra. A linear basis of the free differential associative (resp. commutative and associative) Rota-Baxter algebra on a set was given by L. Guo and W. Keigher [23]. In this paper, we apply the Gröbner-Shirshov bases method to construct a free differential Lie Rota-Baxter algebra. Especially, we give a linear basis of a free differential Lie Rota-Baxter algebra on a set.

Gröbner bases and Gröbner-Shirshov bases have been proved to be very useful in different branches of mathematics, which were invented independently by A.I. Shirshov [32], H. Hironaka [24] and B. Buchberger [13] on different types of algebras. For more details on the Gröbner-Shirshov bases and their applications, see for instance the surveys [8, 10], the books [1, 11, 16, 18] and the papers [9, 27, 28, 17, 20].

The $\Omega$-algebra was introduced by A.G. Kurosh [25]. A differential Lie $\Omega$-algebra over a field $k$ is a differential Lie algebra $L$ with a set of multilinear operators $\Omega$ on $L$. It is easy to see that a differential Lie Rota-Baxter algebra is a differential Lie $\Omega$-algebra with a single operator satisfying the Rota-Baxter relation.

The paper is organized as follows. In Section 2, we review the Gröbner-Shirshov bases theory for differential associative $\Omega$-algebras. In Section 3, we firstly construct a free differential Lie $\Omega$-algebra by the differential nonassociative Lyndon-Shirshov $\Omega$-words, which is a generalization of the classical nonassociative Lyndon-Shirshov words. Secondly, we establish the Gröbner-Shirshov bases theory for differential Lie $\Omega$-algebras. In Section 4, we obtain a Gröbner-Shirshov basis of a free $\lambda$-differential Lie Rota-Baxter algebra and then a linear basis of such an algebra is obtained by the Composition-Diamond lemma for differential Lie $\Omega$-algebras.

2 Gröbner-Shirshov

bases for $\lambda$-differential associative $\Omega$-algebras

In this section, we briefly review the Gröbner-Shirshov bases theory for $\lambda$-differential associative $\Omega$-algebras, which can be found in [27].

2.1 Free $\lambda$-differential associative $\Omega$-algebras

Let $D$ be a $1$-ary operator and

[TABLE]

where $\Omega_{m}$ is a set of $m$-ary operators for any $m\geq 1$. For any set $Y$, we define the following notations:

$S(Y)$: the set of all nonempty associative words on $Y$.

$Y^{*}$: the set of all associative words on $Y$ including the empty word $1$.

$Y^{**}$: the set of all nonassociative words on $Y$.

$\Delta(Y):=\bigcup_{m=0}^{\infty}\{D^{m}(y)|y\in Y\},$ where $D^{0}(y)=y,y\in Y$.

$\Omega(Y):=\bigcup_{m=1}^{\infty}\left\{\omega^{(m)}(y_{1},y_{2},\cdots,y_{m})|y_{i}\in Y,1\leq i\leq m,\omega^{(m)}\in\Omega_{m}\right\}.\\$

Let $X$ be a set. Define the differential associative and nonassociative $\Omega$-words on $X$ as follows. For $n=0$, define $\langle D,\Omega;X\rangle_{0}=S(\Delta(X)),\ (D,\Omega;X)_{0}=(\Delta(X))^{**}.$ For $n>0$, define

[TABLE]

[TABLE]

Set

[TABLE]

The elements of $\langle D,\Omega;X\rangle$ (resp. $(D,\Omega;X)$) are called differential associative (resp. nonassociative) $\Omega$-words on $X$. A differential associative $\Omega$-word $u$ is called prime if $u\in\Delta(X\cup\Omega(\langle D,\Omega;X\rangle))$.

Let $k$ be a field and $\lambda\in k$. A $\lambda$-differential associative $\Omega$-algebra over $k$ is a $\lambda$-differential associative $k$-algebra $R$ together with a set of multilinear operators $\Omega$ on $R$.

Let $DA\langle\Omega;X\rangle=k\langle D,\Omega;X\rangle$ be the semigroup algebra of $\langle D,\Omega;X\rangle$. Let $u=u_{1}u_{2}\cdots u_{t}\in\langle D,\Omega;X\rangle$, where each $u_{i}$ is prime. If $t=1$, i.e. $u=D^{i}(u^{\prime})$ for some $i\geq 0,u^{\prime}\in X\cup\Omega(\langle D,\Omega;X\rangle)$, then we define $D(u)=D^{i+1}(u^{\prime}).$ If $t>1$, then we recursively define

[TABLE]

Extend linearly $D$ to $DA\langle\Omega;X\rangle$. For any $\omega^{(m)}\in\Omega_{m}$, define

[TABLE]

and extend it linearly to $DA\langle\Omega;X\rangle^{m}$.

Theorem 2.1

([27]) $(DA\langle\Omega;X\rangle,D,\Omega)$ is a free $\lambda$-differential associative $\Omega$-algebra on the set $X$.

2.2 Composition-Diamond lemma for   $\lambda$-differential   associative $\Omega$-algebras

Let $\star$ is a symbol, which is not in $X$. By a differential $\star$-$\Omega$-word we mean any expression in $\langle D,\Omega;X\cup\{\star\}\rangle$ with only one occurrence of $\star$. The set of all the differential $\star$-$\Omega$-words on $X$ is denoted by $\langle D,\Omega;X\rangle^{\star}$. Let $\pi$ be a differential $\star$-$\Omega$-word and $s\in DA\langle\Omega;X\rangle$. Then we call $\pi|_{s}=\pi|_{\star\mapsto s}$ a differential $s$-word.

Let $deg(u)$ be the number of all occurrences of $x\in X$, $\omega\in\Omega$ and $D$ in $u$. If $u=u_{1}u_{2}\cdots u_{m}$, where $u_{i}$ is prime, then the breath of $u$, denoted by $bre(u)$, is defined to be the number $m$. Define

[TABLE]

Let $X$ and $\Omega$ be well-ordered sets and assume that $\omega>D$ for any $\omega\in\Omega$. We define the Deg-lex order $>_{{}_{Dl}}$ on $\langle D,\Omega;X\rangle$ as follows. For any $u=u_{1}u_{2}\cdots u_{n}$ and $v=v_{1}v_{2}\cdots v_{m}\in\langle D,\Omega;X\rangle$, where $u_{i},v_{j}$ are prime, define

[TABLE]

where if $u_{i}=\omega(u_{i1},u_{i2},\cdots,u_{it}),v_{i}=\theta(v_{i1},v_{i2},\cdots,v_{il})$, $\omega,\theta\in\{D\}\cup\Omega$ and $deg(u_{i})=deg(v_{i})$, then $u_{i}>_{{}_{Dl}}v_{i}$ if

[TABLE]

It is easy to check that $>_{{}_{Dl}}$ is a well order on $\langle D,\Omega;X\rangle$. For any $0\neq f\in DA\langle\Omega;X\rangle$, let $\bar{f}$ be the leading term of $f$ with respect to the order $>_{{}_{Dl}}$. Let us denote $lc(f)$ the coefficient of the leading term $\bar{f}$ of $f$.

For $1\leq t\leq n$, define

[TABLE]

Lemma 2.2

([27]) If $u=u_{1}u_{2}\cdots u_{n}\in\langle D,\Omega;X\rangle$, where each $u_{i}$ is prime, then

[TABLE]

Lemma 2.3

([27]) Let $u=u_{1}u_{2}\cdots u_{n}\in\langle D,\Omega;X\rangle$, where each $u_{i}$ is prime.

1. (a)

If $\lambda=0$, then $\overline{D^{i}(u)}=D^{i}(u_{1})u_{2}\cdots u_{n}$ and $lc(D^{i}(u))=1$. 2. (b)

If $\lambda\neq 0$, then $\overline{D^{i}(u)}=D^{i}(u_{1})D^{i}(u_{2})\cdots D^{i}(u_{n})$ and $lc(D^{i}(u))=\lambda^{(n-1)i}$.

It follows that if $u,v\in\langle D,\Omega;X\rangle$ and $u>_{{}_{Dl}}v$, then $\overline{D(u)}>\overline{D(v)}$.

Proposition 2.4

([27]) For any $u,v\in\langle D,\Omega;X\rangle,\pi\in\langle D,\Omega;X\rangle^{\star}$, if $u>_{{}_{Dl}}v$, then $\overline{\pi|_{u}}>_{{}_{Dl}}\overline{\pi|_{v}}$.

If $\overline{\pi|_{s}}=\pi|_{\overline{s}}$, where $s\in DA\langle\Omega;X\rangle$ and $\pi\in\langle D,\Omega;X\rangle^{\star}$, then $\pi|_{s}$ is called a normal differential $s$-word. Note that not each differential $s$-word is a normal differential $s$-word, for example, if $u=D(x)P(D^{2}(\star))$ and $s=xy$, where $P\in\Omega,\ x,y\in X$, then $\pi|_{s}$ is not a normal differential $s$-word. However, if we take $\pi^{\prime}=D(x_{1})P(\star)$, then $\pi|_{s}=\pi^{\prime}|_{D^{2}(s)}$ and $\pi^{\prime}|_{D^{2}(s)}$ is a normal differential $D^{2}(s)$-word.

Lemma 2.5

([27]) For any differential $s$-word $\pi|_{s}$, there exist $i\geq 0$ and $\pi^{\prime}$ such that $\pi|_{s}=\pi^{\prime}|_{D^{i}(s)}$ and $\pi^{\prime}|_{D^{i}(s)}$ is a normal differential $D^{i}(s)$-word.

Let $f,g\in DA\langle\Omega;X\rangle$. There are two kinds of compositions.

1. (i)

If there exists a $w=\overline{D^{i}(f)}a=b\overline{D^{j}(g)}$ for some $a,b\in\langle D,\Omega;X\rangle$ such that $bre(w)<bre(\bar{f})+bre(\bar{g})$, then we call

[TABLE]

the intersection composition of $f$ and $g$ with respect to the ambiguity $w$. 2. (ii)

If there exists a $\pi\in\langle D,\Omega;X\rangle^{\star}$ such that $w=\overline{D^{i}(f)}=\pi|_{{}_{\overline{D^{{}^{j}}({g})}}}$, where $\pi|_{{}_{D^{{}^{j}}({g})}}$ is a normal differential $D^{{}^{j}}({g})$-word, then we call

[TABLE]

the inclusion composition of $f$ and $g$ with respect to the ambiguity $w$.

Let $S$ be a subset of $DA\langle\Omega;X\rangle$. Then the composition $(f,g)_{w}$ is called trivial modulo $(S,w)$ if

[TABLE]

where each $\alpha_{i}\in k$, $\pi_{i}\in\langle D,\Omega;X\rangle^{\star}$, $s_{i}\in S$, $\pi_{i}|_{{}_{D^{l_{i}}(s_{i})}}$ is a normal differential $D^{l_{i}}(s_{i})$-word and $\pi_{i}|_{{}_{\overline{D^{l_{i}}(s_{i})}}}<_{{}_{Dl}}w$. If this is the case, we write

[TABLE]

In general, for any two polynomials $p$ and $q$, $p\equiv_{ass}q\ \ mod(S,w)$ means that $p-q=\sum\alpha_{i}\pi_{i}|_{{}_{D^{l_{i}}(s_{i})}},$ where each $\alpha_{i}\in k$, $\pi_{i}\in\langle D,\Omega;X\rangle^{\star}$, $s_{i}\in S$, $\pi_{i}|_{{}_{D^{l_{i}}(s_{i})}}$ is a normal differential $D^{l_{i}}(s_{i})$-word and $\pi_{i}|_{{}_{\overline{D^{l_{i}}(s_{i})}}}<_{{}_{Dl}}w$.

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

Theorem 2.6

([27], Composition-Diamond lemma for differential associative $\Omega$-algebras)    Let $S$ be a subset of $DA\langle\Omega;X\rangle$, $Id_{DA}(S)$ the ideal of $DA\langle\Omega;X\rangle$ generated by $S$ and $>_{{}_{Dl}}$ the Deg-lex order on $\langle D,\Omega;X\rangle$ defined as before. Then the following statements are equivalent:

1. (i)

$S$* is a Gröbner-Shirshov basis in $DA\langle\Omega;X\rangle$.* 2. (ii)

$f\in Id_{DA}(S)\Rightarrow\bar{f}=\pi|_{{}_{\overline{D^{i}(s)}}}$* for some $\pi\in\langle D,\Omega;X\rangle^{\star}$, $s\in S$ and $i\geq 0$.* 3. (iii)

The set

[TABLE]

is a linear basis of the differential associative $\Omega$-algebra $DA\langle\Omega;X|S\rangle:=DA\langle\Omega;X\rangle/Id_{DA}(S)$.

3 Gröbner-Shirshov

bases for $\lambda$-differential Lie $\Omega$-algebras

3.1 Lyndon-Shirshov words

In this subsection, we review the concept and some properties of Lyndon-Shirshov words, which can be found in [7, 32].

For any $u\in X^{*}$, let us denote by $deg(u)$ the degree (length) of $u$. Let $>$ be a well order on $X$. Define the lex-order $>_{lex}$ and the deg-lex order $>_{deg-lex}$ on $X^{*}$ with respect to $>$ by:

(i) $1>_{lex}u$ for any nonempty word $u$, and if $u=x_{i}u^{\prime}$ and $v=x_{j}v^{\prime}$, where $x_{i},x_{j}\in X$, then $u>_{lex}v$ if $x_{i}>x_{j}$, or $x_{i}=x_{j}$ and $u^{\prime}>_{lex}v^{\prime}$ by induction.

(ii) $u>_{deg-lex}v$ if $deg(u)>deg(v)$, or $deg(u)=deg(v)$ and $u>_{lex}v$.

A nonempty associative word $w$ is called an associative Lyndon-Shirshov word on $X$, if $w=uv>_{lex}vu$ for any decomposition of $w=uv$, where $1\neq u,v\in X^{*}$.

A nonassociative word $(u)\in X^{**}$ is said to be a nonassociative Lyndon-Shirshov word on $X$ with respect to the lex-order $>_{lex}$, if

• (a)

$u$ is an associative Lyndon-Shirshov word on $X$;

• (b)

if $(u)=((v)(w))$, then both $(v)$ and $(w)$ are nonassociative Lyndon-Shirshov words on $X$;

• (c)

if $(v)=((v_{1})(v_{2}))$, then $v_{2}\leq_{lex}w$.

Let $ALSW(X)$ (resp. $NLSW(X))$ denote the set of all the associative (resp. nonassociative) Lyndon-Shirshov words on $X$ with respect to the lex-order $>_{lex}$. It is well known that for any $u\in ALSW(X)$, there exists a unique Shirshov standard bracketing way $[u]$ (see for instance [7]) on $u$ such that $[u]\in NLSW(X)$. Then $NLSW(X)=\{[u]|u\in ALSW(X)\}.$

Let $k\langle X\rangle$ be the free associative algebra on $X$ over a field $k$ and $Lie(X)$ be the Lie subalgebra of $k\langle X\rangle$ generated by $X$ under the Lie bracket $(uv)=uv-vu$. It is well known that $Lie(X)$ is a free Lie algebra on the set $X$ and $NLSW(X)$ is a linear basis of $Lie(X)$.

3.2 Differential Lyndon-Shirshov $\Omega$-words

Let $>_{{}_{Dl}}$ be the Deg-lex order on $\langle D,\Omega;X\rangle$ and $\succ$ the restriction of $>_{{}_{Dl}}$ on $\Delta(X\cup\Omega(\langle D,\Omega;X\rangle))$. Define the differential Lyndon-Shirshov $\Omega$-words on the set $X$ as follows.

For $n=0$, let $Z_{0}:=\Delta(X)$. Define

[TABLE]

[TABLE]

with respect to the lex-order $\succ_{lex}$ on $(Z_{0})^{*}$, where $[u]$ is the Shirshov standard bracketing way on $u$.

Assume that we have defined

[TABLE]

[TABLE]

Let $Z_{n}:=\Delta(X\cup\Omega(ALSW(D,\Omega;X)_{n-1}))$. Define

[TABLE]

with respect to the lex-order $\succ_{lex}$ on $Z_{n}^{*}$. For any $u\in Z_{n}$, define the bracketing way on $u$ by

[TABLE]

Let $[Z_{n}]:=\{[u]|u\in Z_{n}\}.$ Thus, the order $\succ$ on $Z_{n}$ induces an order on $[Z_{n}]$ by $[u]\succ[v]$ if $u\succ v$ for any $u,v\in Z_{n}$. For any $u=u_{1}u_{2}\cdots u_{t}\in ALSW(D,\Omega;X)_{n}$, where each $u_{i}\in Z_{n}$, we define

[TABLE]

the Shirshov standard bracketing way on the word $[u_{1}][u_{2}]\cdots[u_{t}]$, which means that $[u]$ is a nonassociative Lyndon-Shirshov word on the set $\{[u_{1}],[u_{2}],\cdots,[u_{t}]\}$. Define

[TABLE]

It is easy to see that $NLSW(D,\Omega;X)_{n}=NLSW([Z_{n}])$ with respect to the lex-order $\succ_{lex}$ on $[Z_{n}]^{*}$.

Set

[TABLE]

[TABLE]

Then, we have

[TABLE]

The elements of $ALSW(D,\Omega;X)$ (resp. $NLSW(D,\Omega;X)$) are called the differential associative (resp. nonassociative) Lyndon-Shirshov $\Omega$-words on the set $X$.

3.3 Free $\lambda$-differential Lie $\Omega$-algebras

In this subsection, we prove that the set $NLSW(D,\Omega;X)$ of all differential nonassociative Lyndon-Shirshov $\Omega$-words on $X$ forms a linear basis of the free $\lambda$-differential Lie $\Omega$-algebra on $X$.

A $\lambda$-differential Lie algebra is a Lie algebra $L$ with a linear operator $D:L\rightarrow L$ satisfying the differential relation

[TABLE]

A $\lambda$-differential Lie $\Omega$-algebra is a $\lambda$-differential Lie algebra $L$ with a set of multilinear operators $\Omega$ on $L$.

Let $(R,\cdot,D,\Omega)$ is a $\lambda$-differential associative $\Omega$-algebra. Then it is easy to check that $(R,[,],D,\Omega)$ is a $\lambda$-differential Lie $\Omega$-algebra under the Lie bracket $[a,a^{\prime}]=a\cdot a^{\prime}-a^{\prime}\cdot a,\ a,a^{\prime}\in R.$

Let $DLie(\Omega;X)$ be the $\lambda$-differential Lie $\Omega$-subalgebra of $DA\langle\Omega;X\rangle$ generated by $X$ under the Lie bracket $(uv)=uv-vu.$

Similar to the proofs of Lemma 2.6 and Theorem 2.8 in [28], we have the following results.

Lemma 3.1

If $u\in ALSW(D,\Omega;X)$, then $\overline{[u]}=u$ with respect to the order $>_{{}_{Dl}}$ on $\langle D,\Omega;X\rangle$.

Theorem 3.2

$DLie(\Omega;X)$* is a free $\lambda$-differential Lie $\Omega$-algebra on the set $X$ and $NLSW(D,\Omega;X)$ is a linear basis of $DLie(\Omega;X)$.*

3.4 Composition-Diamond lemma for differential Lie $\Omega$-algebras

In this subsection, we establish the Composition-Diamond lemma for differential Lie $\Omega$-algebras.

Lemma 3.3

Let $\pi\in\langle D,\Omega;X\rangle^{\star}$ and $v,\pi|_{v}\in ALSW(D,\Omega;X)$. Then there is a $\pi^{\prime}\in\langle D,\Omega;X\rangle^{\star}$ and $c\in\langle D,\Omega;X\rangle$ such that

[TABLE]

where $c$ may be empty. Let

[TABLE]

where $c=c_{1}c_{2}\cdots c_{m}$ with each $c_{i}\in ALSW(D,\Omega;X)$ and $c_{t}\preceq_{lex}c_{t+1}$. Then,

[TABLE]

where each $\alpha_{i}\in k$ and $\pi_{i}|_{v}<_{{}_{Dl}}\pi|_{v}$. It follows that $\overline{[\pi|_{v}]_{v}}=\pi|_{v}$ with respect to the order $>_{{}_{Dl}}$.

Proof. The proof is the same as the one of Lemma 3.2 in [28]. $\square$

Let $0\neq f\in DLie(\Omega;X)\subseteq DA\langle\Omega;X\rangle$. If $\pi|_{\bar{f}}\in ALSW(D,\Omega;X)$, then we call

[TABLE]

a special normal differential $f$-word.

Corollary 3.4

Let $f\in DLie(\Omega;X)$ and $\pi|_{\bar{f}}\in ALSW(D,\Omega;X)$. Then

[TABLE]

where each $\alpha_{i}\in k$ and $\pi_{i}|_{\bar{f}}<_{{}_{Dl}}\pi|_{\bar{f}}$.

Let $f,g\in DLie(\Omega;X)$. There are two kinds of compositions.

1. (i)

If there exists a $w=\overline{D^{i}(f)}a=b\overline{D^{j}(g)}$ for some $a,b\in\langle D,\Omega;X\rangle$ such that $bre(w)<bre(\bar{f})+bre(\bar{g})$, then we call

[TABLE]

the intersection composition of $f$ and $g$ with respect to the ambiguity $w$. 2. (ii)

If there exists a $\pi\in\langle D,\Omega;X\rangle^{\star}$ such that $w=\overline{D^{i}(f)}=\pi|_{{}_{\overline{D^{{}^{j}}({g})}}}$, where $\pi|_{{}_{D^{{}^{j}}({g})}}$ is a normal differential $D^{{}^{j}}({g})$-word, then we call

[TABLE]

the inclusion composition of $f$ and $g$ with respect to the ambiguity $w$.

If $S$ is a subset of $DLie(\Omega;X)$, then the composition $\langle f,g\rangle_{w}$ is called trivial modulo $(S,w)$ if

[TABLE]

where each $\alpha_{i}\in k,\ s_{i}\in S$, $[\pi_{i}|_{{}_{D^{l_{i}}(s_{i})}}]_{{}_{\overline{D^{l_{i}}(s_{i})}}}$ is a special normal differential $D^{l_{i}}(s_{i})$-word and $\pi_{i}|_{{}_{\overline{D^{l_{i}}(s_{i})}}}<_{{}_{Dl}}w$. If this is the case, then we write

[TABLE]

In general, for any two polynomials $p$ and $q$, $p\equiv q\ \ mod(S,w)$ means that $p-q=\sum\alpha_{i}[\pi_{i}|_{{}_{D^{l_{i}}(s_{i})}}]_{{}_{\overline{D^{l_{i}}(s_{i})}}},$ where each $\alpha_{i}\in k$, $\pi_{i}\in\langle D,\Omega;X\rangle^{\star}$, $s_{i}\in S$, $[\pi_{i}|_{{}_{D^{l_{i}}(s_{i})}}]_{{}_{\overline{D^{l_{i}}(s_{i})}}}$ is a normal differential $D^{l_{i}}(s_{i})$-word and $\pi_{i}|_{{}_{\overline{D^{l_{i}}(s_{i})}}}<_{{}_{Dl}}w$.

Definition 3.5

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

Lemma 3.6

Let $f,g\in DLie(\Omega;X)$. Then

[TABLE]

Proof. If $\langle f,g\rangle_{w}$ and $(f,g)_{w}$ are compositions of intersection, where $w=\overline{D^{i}(f)}a=b\overline{D^{j}(g)}$, then

[TABLE]

where $a_{i}\overline{D^{i}(f)}a_{i}^{\prime},\ b_{j}\overline{D^{j}(g)}b_{j}^{\prime}<_{{}_{Dl}}w$. It follows that

[TABLE]

If $\langle f,g\rangle_{w}$ and $(f,g)_{w}$ are compositions of inclusion, where $w=\bar{f}=\pi|_{{}_{\overline{D^{j}(g)}}}$, then

[TABLE]

where $\pi_{i}|_{{}_{\overline{D^{j}(g)}}}<_{{}_{Dl}}w$. It follows that

[TABLE]

The proof is complete. $\square$

Lemma 3.7

Let $S\subset DLie(\Omega;X)\subset DA(\Omega;X)$. Then the following two statements are equivalent:

1. (i)

$S$* is a Gröbner-Shirshov basis in $DLie(\Omega;X)$,* 2. (ii)

$S$* is a Gröbner-Shirshov basis in $DA\langle\Omega;X\rangle$.*

Proof. $(i)\Longrightarrow(ii)$. Suppose that $S$ is a Gröbner-Shirshov basis in $DLie(\Omega;X)$. Then, for any composition $\langle f,g\rangle_{w}$, we have

[TABLE]

where each $\alpha_{i}\in k,\ s_{i}\in S$, $\pi_{i}|_{{}_{\overline{D^{l_{i}}(s_{i})}}}<_{{}_{Dl}}w$ . By Corollary 3.4, we have

[TABLE]

where each $\beta_{t}\in k,\ s_{t}\in S$, $\pi_{t}|_{{}_{\overline{D^{l_{t}}(s_{t})}}}<_{{}_{Dl}}w$. Therefore, by Lemma 3.6, we can obtain that

[TABLE]

Thus, $S$ is a Gröbner-Shirshov basis in $DA\langle\Omega;X\rangle$.

$(ii)\Longrightarrow(i)$. Assume that $S$ is a Gröbner-Shirshov basis in $DA\langle\Omega;X\rangle$. Then, for any composition $\langle f,g\rangle_{w}$ in $S$, we have $\langle f,g\rangle_{w}\in DLie(\Omega;X)$ and $\langle f,g\rangle_{w}\in Id_{DA}(S)$. By Theorem 2.6, $\overline{\langle f,g\rangle_{w}}=\pi_{1}|_{{}_{\overline{D^{i_{1}}(s_{1})}}}\in ALSW(D,\Omega;X)$. Let

[TABLE]

where $\alpha_{1}$ is the coefficient of $\overline{\langle f,g\rangle_{w}}$. Then, $\overline{h_{1}}<_{{}_{Dl}}\overline{\langle f,g\rangle_{w}}$, $h_{1}\in Id_{DA}(S)$ and $h_{1}\in DLie(\Omega;X)$. Now, the result follows from induction on $\overline{\langle f,g\rangle_{w}}$. $\square$

Lemma 3.8

Let $S\subset DLie(\Omega;X)$ and

[TABLE]

Then, for any $h\in DLie(\Omega;X)$, $h$ can be expressed by

[TABLE]

where each $\alpha_{i},\ \beta_{j}\in k,\ u_{i}\in ALSW(D,\Omega;X),u_{i}\leq_{Dl}\bar{h}$ and $s_{j}\in S$, $\pi_{j}|_{{}_{\overline{D^{l_{j}}(s_{j})}}}\leq_{Dl}\bar{h}$.

Proof. By induction on $\overline{h}$, we can obtain the result. $\square$

The following theorem is the Composition-Diamond lemma for differential Lie $\Omega$-algebras. It is a generalization of Shirshov’s Composition lemma for Lie algebras [32], which was specialized to associative algebras by L.A. Bokut [6], see also G.M. Bergman [5] and B. Buchberger [12, 13].

Theorem 3.9

(Composition-Diamond lemma for differential Lie $\Omega$-algebras) Let $S\subset DLie(\Omega;X)$ be a nonempty set and $Id_{DLie}(S)$ the ideal of $DLie(\Omega;X)$ generated by $S$. Then the following statements are equivalent:

1. (I)

$S$* is a Gröbner-Shirshov basis in $DLie(\Omega;X)$.* 2. (II)

$f\in Id_{DLie}(S)\Rightarrow\bar{f}=\pi|_{{}_{\overline{D^{i}(s)}}}\in ALSW(D,\Omega;X)$* for some $s\in S$, $\pi\in\langle D,\Omega;X\rangle^{\star}$ and $i\geq 0$.* 3. (III)

The set

[TABLE]

is a linear basis of the $\lambda$-differential Lie $\Omega$-algebras $DLie(\Omega;X|S)$.

Proof. $(I)\Longrightarrow(II)$. Since $f\in Id_{DLie}(S)\subseteq Id_{DA}(S)$, by Lemma 3.7 and Theorem 2.6, we have $\bar{f}=\pi|_{\overline{D^{i}(s)}}$ for some $s\in S$, $\pi\in\langle D,\Omega;X\rangle^{\star}$ and $i\geq 0$.

$(II)\Longrightarrow(III)$.  Suppose that $\sum\alpha_{i}[u_{i}]=0$ in $DLie(\Omega;X|S)$, where each $[u_{i}]\in Irr(S)$ and $u_{i}>_{{}_{Dl}}u_{i+1}$. That is, $\sum\alpha_{i}[u_{i}]\in{Id_{DLie}(S)}.$ Then each $\alpha_{i}$ must be 0. Otherwise, say $\alpha_{1}\neq 0$, since $\overline{\sum\alpha_{i}[u_{i}]}=u_{1}$ and by (II), we have $[u_{1}]\in Irr(S)$, a contradiction. Therefore, $Irr(S)$ is linear independent. By Lemma 3.8, $Irr(S)$ is a linear basis of $DLie(\Omega;X|S)=DLie(\Omega;X)/Id_{DLie}(S)$.

$(III)\Longrightarrow(I)$.  For any composition $\langle f,g\rangle_{w}$ with $f,g\in{S}$, we have $\langle f,g\rangle_{w}\in{Id_{DLie}(S)}$. Then, by (III) and by Lemma 3.8,

[TABLE]

where each $\beta_{j}\in k,\ \pi|_{{}_{\overline{D^{l_{j}}(s_{j})}}}<_{{}_{Dl}}w$. This proves that $S$ is a Gröbner-Shirshov basis in $DLie(\Omega;X)$. $\square$

4 Free $\lambda$-differential Rota-Baxter Lie algebras

In this section, by using Theorem 3.9 we give a Gröbner-Shirshov basis of a free $\lambda$-differential Rota-Baxter Lie algebra on a set $X$ and then a linear basis of such an algebra is obtained.

4.1 Gröbner-Shirshov bases for free $\lambda$-differential Lie Rota-Baxter algebras

Let $k$ be a field and $\lambda\in k$. A differential Lie Rota-Baxter algebra of weight $\lambda$, called also $\lambda$-differential Lie Rota-Baxter algebra, is a Lie algebra $L$ with two linear operators $P,D:L\rightarrow L$ such that for any $x,y\in L$,

1. (a)

(Rota-Baxter relation) $[P(x)P(y)]=P([xP(y)])+P([P(x)y])+\lambda P([xy]);$ 2. (b)

(differential relation) $D([xy])=[D(x)y]+[xD(y)]+\lambda[D(x)D(y)]$; 3. (c)

(section relation) $D(P(x))=x$.

It is easy to see that any $\lambda$-differential Lie Rota-Baxter algebra is a $\lambda$-differential Lie $\{P\}$-algebra satisfying the relations (a) and (c).

Let $DLie(\{P\};X)$ be the free $\lambda$-differential Lie $\{P\}$-algebra on the set $X$ and write

[TABLE]

[TABLE]

where $u,v\in ALSW(D,\{P\};X)$. Set

[TABLE]

It is clear that $DRBL(X):=DLie(\{P\};X|S)$ is a free $\lambda$-differential Lie Rota-Baxter algebra on $X$.

For any $f\in DLie(\{P\};X)$, let us denote $r(f):=f-lc(f)[\overline{f}]$.

Lemma 4.1

The set $S_{1}:=\{D(P([u]))-[u]|u\in ALSW(D,\{P\};X)\}$ is a Gröbner-Shirshov basis in $DLie(\{P\};X)$.

Proof. It is easy to check that $S_{1}$ is a Gröbner-Shirshov basis in $DLie(\{P\};X)$. $\square$

Lemma 4.2

Let $u,v\in ALSW(D,\{P\};X)$ and $u>_{{}_{Dl}}v$.

1. (a)

If $\lambda\neq 0$ and $j>0$, then

[TABLE] 2. (b)

If $\lambda=0$ and $j>0$, then

[TABLE]

Proof. $(a)$ The proof is by induction on $j$. For $j=1$, we have

[TABLE]

Assume that the result is true for $j-1,j\geq 2$, i.e.

[TABLE]

where each $\alpha_{i}\in k$, $s_{i}\in S_{1}$, $\pi_{i}|_{{}_{\overline{D^{t_{i}}(s_{i})}}}<_{{}_{Dl}}D^{j-1}(P(u))D^{j-1}(P(v))$. Since $S_{1}$ is a Gröbner-Shirshov basis in $DLie(\{P\};X)$,

[TABLE]

where each $\beta_{l}\in k,s_{l}\in S_{1}$, $[\sigma_{l}|_{{}_{D^{n_{l}}(s_{l})}}]_{{}_{\overline{D^{n_{l}}(s_{l})}}}$ is a special normal differential $D^{k_{l}}(s_{l})$-word. By Lemma 2.3,

[TABLE]

Thus, we have

[TABLE]

$(b)$ The proof is similar to Case (a). $\square$

Theorem 4.3

With the order $>_{{}_{Dl}}$ on $\langle D,\{P\};X\rangle$ defined as before, the set $S$ is a Gröbner-Shirshov basis in $DLie(\{P\};X)$.

Proof. There are two cases $\lambda\neq 0$ and $\lambda=0$ to consider.

Case 1. For $\lambda\neq 0$, all possible compositions of the polynomials in $S$ are list as below:

[TABLE]

We check that all the compositions in $S$ are trivial. Here, we just check one composition as example.

If $j>0$, then by Lemma 4.2, we have

[TABLE]

If $j=0$, then

[TABLE]

Case 2. For $\lambda=0$, all possible compositions of the polynomials in $S$ are list as below:

[TABLE]

where $i,j\geq 0$. We check that all the compositions in $S$ are trivial. The proof is similar to Case 1. $\square$

4.2 A linear basis of a free $\lambda$-differential Lie Rota-Baxter algebra

In this subsection, by Theorems 3.9 and 4.3, we obtain a linear basis of the free $\lambda$-differential Lie Rota-Baxter algebra on the set $X$.

For $n=0$, define $\langle\{P\};\Delta(X)\rangle_{0}:=S(\Delta(X))$ and $(\{P\};\Delta(X))_{0}:=(\Delta(X))^{**}.$ For $n>0$, define

[TABLE]

[TABLE]

Set

[TABLE]

Let $\star$ is a symbol, which is not in $X$. By a $\star$-$\Omega$-word on $\Delta(X)$, we mean any expression in $\langle\{P\};\Delta(X)\cup\{\star\}\rangle$ with only one occurrence of $\star$. We will denote by $\langle\{P\};\Delta(X)\rangle^{\star}$ the set of all the $\star$-$\Omega$-words on $\Delta(X)$. Let $\pi$ be a $\star$-$\Omega$-word on $\Delta(X)$ and $u\in\langle\{P\};\Delta(X)\rangle$. Let us denote $\pi|_{u}=\pi|_{\star\mapsto u},$ i.e. $\star$ is replaced by $u$.

It is easy to see that $\langle\{P\};\Delta(X)\rangle\subseteq\langle D,\{P\};X\rangle$. We also use the order $>_{{}_{Dl}}$ on $\langle\{P\};\Delta(X)\rangle$ and $\succ$ on $\Delta(X)\cup P(\langle\{P\};\Delta(X)\rangle)$.

For $n=0$, let $Y_{0}=\Delta(X)$. Define

[TABLE]

[TABLE]

with respect to the lex-order $\succ_{lex}$.

Assume that we have defined

[TABLE]

[TABLE]

Let $Y_{n}:=\Delta(X)\cup P(ALSW(\{P\};\Delta(X))_{n-1})$. Define

[TABLE]

For any $u\in Y_{n}$, define the bracketing way on $u$ by

[TABLE]

Let $[Y_{n}]:=\{[u]|u\in Y_{n}\}.$ Therefore, the order $\succ$ on $Y_{n}$ induces an order on $[Y_{n}]$ by $[u]\succ[v]$ if $u\succ v$ for any $u,v\in Y_{n}$. For any $u=u_{1}u_{2}\cdots u_{m}\in ALSW(\{P\};\Delta(X))_{n}$, where each $u_{i}\in Y_{n}$, let us denote

[TABLE]

the nonassociative Lyndon-Shirshov word on $\{[u_{1}],[u_{2}],\cdots,[u_{m}]\}$ with respect to the lex-order $\succ_{lex}$.

Define

[TABLE]

It is easy to see that $NLSW(\{P\};\Delta(X))_{n}=NLSW([Y_{n}])$. Define

[TABLE]

[TABLE]

Therefore,

[TABLE]

By Theorems 3.9 and 4.3, we have the following theorem.

Theorem 4.4

The set

[TABLE]

is a linear basis of the free $\lambda$-differential Lie Rota-Baxter algebra $DRBL(X)$ on $X$.