Post-Lie Algebra Structures on the Witt Algebra

Abstract

In this paper, we characterize the graded post-Lie algebra structures and a class of shifting post-Lie algebra structures on the Witt algebra. As an application, the homogeneous Rota–Baxter operators and a class of non-homogeneous Rota–Baxter operators of weight 1 on the Witt algebra are studied.

1 Introduction and Preliminaries

In recent years, post-Lie algebras have aroused the interest of a great many authors, see [1, 3,4,5, 8, 9, 14, 15, 20]. Post-Lie algebras were introduced around 2007 by Vallette in [21], who found the structure in a purely operadic manner as the Koszul dual of a commutative trialgebra. Moreover, Vallette in [21] proved that post-Lie algebras have the important algebraic property of being Koszul. As pointed out by Hans Z. Munthe-Kaas in [14], post-Lie algebras also arise naturally from the differential geometry of homogeneous spaces and Klein geometries, topics that are closely related to Cartan’s method of moving frames. In addition, post-Lie algebras also turned up in relations with Lie groups (see [5, 14]), classical Yang–Baxter equation (see [1]), Hopf algebra and classical r-matrices (see [10]) and Rota–Baxter operators (see [11]).

One of the most important problems in the study of post-Lie algebras is to find the post-Lie algebra structures on the (given) Lie algebras. In [15], the authors determined all post-Lie algebra structures on $sl(2, \mathbb {C})$ of special linear Lie algebra of order 2, and in [20], by using the Gröbner basis the package in computer algebra software Maple, they studied the post-Lie algebra structures on the solvable Lie algebra $t(2, \mathbb {C})$ (the Lie algebra of $2\times 2$ upper triangular matrices) and gave all 65 types of these post-Lie algebra structures.

It may be useful and interesting for characterizing the post-Lie algebra structures on some important Lie algebras. As we see, most of the studies on post-Lie algebras have been focused on the finite-dimensional cases. It is natural to consider the infinite-dimensional cases. In particular, the authors in [11] obtained a class of induced post-Lie algebra structures on the Witt algebra using the homogeneous Rota–Baxter operators. As a matter of fact, the Rota–Baxter operators were originally defined on associative algebras by G. Baxter to solve an analytic formula in probability in [2] and then developed by the Rota school in [16]. These operators have showed up in many areas in mathematics and mathematical physics (see [6, 11, 12, 18] and the references therein).

Recall that the Witt algebra W is an important infinite-dimensional Lie algebra with the $\mathbb {C}$-basis $\{L_m| m\in \mathbb {Z} \}$ and the Lie brackets are defined by

$$\begin{aligned}{}[L_m,L_n]=(m-n)L_{m+n}. \end{aligned}$$

The Witt algebra occurs in the study of conformal field theory and plays an important role in many areas of mathematics and physics. Based on this background, we will study the post-Lie algebra structures on W in this paper. Below we will denote by $\mathbb {C}$ and $\mathbb {Z}$ the complex number field and the set of integer numbers, respectively. For a fixed integer k, let $\mathbb {Z}_{>k}=\{t\in \mathbb {Z}|t>k\}$, $\mathbb {Z}_{<k}=\{t\in \mathbb {Z}|t<k\}$, $\mathbb {Z}_{\geqslant k}=\{t\in \mathbb {Z}|t\geqslant k\}$ and $\mathbb {Z}_{\leqslant k}=\{t\in \mathbb {Z}|t\leqslant k\}$. We also assume that the field in this paper always is the complex number field $\mathbb {C}$ since the Witt algebra is defined over $\mathbb {C}$. We now turn to the definition of post-Lie algebra following reference [21].

Definition 1.1

A post-Lie algebra $(V, \circ , [ , ])$ is a vector space V over a field k equipped with two k-bilinear products $x\circ y$ and [x, y] and satisfies (V, [, ]) which is a Lie algebra and

$$\begin{aligned}{}[x, y] \circ z= & {} x \circ (y \circ z)-y \circ (x \circ z)-<x,y> \circ z, \end{aligned}$$

(1.1)

$$\begin{aligned} x\circ [y, z]= & {} [x\circ y, z] + [y, x \circ z] \end{aligned}$$

(1.2)

for all $x, y, z \in V$, where $<x,y>=x\circ y-y\circ x$. We also say that $(V, \circ , [ , ])$ is a post-Lie algebra structure on the Lie algebra (V, [, ]). If a post-Lie algebra $(V, \circ , [ , ])$ satisfies $x\circ y=y\circ x$ for all $x,y\in V$, then it is called a commutative post-Lie algebra.

Definition 1.2

Suppose that (L, [, ]) is a Lie algebra. Two post-Lie algebras $(L, [,], \circ _1)$ and $(L, [,], \circ _2)$ on the Lie algebra L are said to be isomorphic if there is an automorphism $\tau$ of the Lie algebra (L, [, ]) such that $\tau (x\circ _1 y)=\tau (x)\circ _2 \tau (y)$ for all $x,y\in L$.

It is not difficult to verify the following proposition.

Proposition 1.3

Let $(V, \circ , [, ])$ be a post-Lie algebra defined by Definition 1.1. Then the following product

$$\begin{aligned} \{x, y\} \triangleq <x,y> + [x, y], \end{aligned}$$

(1.3)

induces an another Lie algebra structure on V, where $<x,y>=x\circ y-y\circ x$. Furthermore, if two post-Lie algebras $(V, \circ _1, [, ])$ and $(V, \circ _2, [, ])$ on the same Lie algebra (V, [, ]) are isomorphic, then the two induced Lie algebras $(V, \{, \}_1)$ and $(V, \{, \}_2)$ are isomorphic.

Remark 1.4

The left multiplications of the post-Lie algebra $(V,[,], \circ )$ are denoted by $\mathcal {L}(x)$, i.e., we have $\mathcal {L}(x)(y) = x\circ y$ for all $x,y\in V$. By (1.2), we see that all operators $\mathcal {L}(x)$ are Lie algebra derivations of Lie algebra (V, [, ]).

Our results can be briefly summarized as follows. In Sect. 2, we classify the graded post-Lie algebra structures on the Witt algebra W, and then, we obtain the induced graded Lie algebras. In Sect. 3, we classify a class of shifting post-Lie algebra structures on the Witt algebra W and give the induced non-graded Lie algebras. In Sect. 4, we first recall the other definitions of post-Lie algebra, and then, the modules over some Lie algebras are given. In Sect. 5, we give the induced Rota–Baxter operators of weight 1 from the post-Lie algebras on W.

2 The Graded Post-Lie Algebra Structure On the Witt Algebra

Recently, the author in [17] proved that any commutative post-Lie algebra structure on the Witt algebra W is trivial (namely, $x\circ y=0$ for all $x,y\in W$). We now will dedicate to the study for the non-commutative cases. Since the Witt algebra is graded, it is also natural to suppose first that the algebras should be graded. Hence, in this section, we mainly consider the graded post-Lie algebra structure on the Witt algebra W. Namely, we assume that it satisfies

$$\begin{aligned} L_m\circ L_n=\phi (m,n)L_{m+n}, \end{aligned}$$

(2.1)

for all $m,n\in \mathbb {Z}$, where $\phi$ is a complex-valued function on $\mathbb {Z}\times \mathbb {Z}$.

Lemma 2.1

There exists a graded post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (2.1) if and only if there is a complex-valued function f on $\mathbb {Z}$ such that

$$\begin{aligned}&\phi (m,n)=(m-n)f(m), \end{aligned}$$

(2.2)

$$\begin{aligned}&(m-n)(f(m+n)+f(m)f(m+n)+f(n)f(m+n)-f(m)f(n))=0.\qquad \qquad \end{aligned}$$

(2.3)

Proof

The “if” part is easy to check. Next we prove the “only if” part. By Remark 1.4, $\mathcal {L}(x)$ is a derivation of W. It is well known that every derivation of Witt algebra is inner (see [22]). So we have

$$\begin{aligned} L_m\circ L_n=\mathcal {L}(L_m)(L_n)=\mathrm{{ad}} (\psi (L_m)) L_n=[\psi (L_m),L_n] \end{aligned}$$

(2.4)

for some linear map $\psi$ from W into itself. Denote by $\psi (L_m)=\sum _{i\in \mathbb {Z}} k_i^{(m)}L_i$, where $k_i^{(m)}\in \mathbb {C}$ for any $i\in \mathbb {Z}$. Then we have by (2.4) that $L_m\circ L_n=\sum _{i\in \mathbb {Z}} (i-n)k_i^{(m)}L_{n+i}.$ This and (2.1) together yield that $(i-n)k_i^{(m)}=0$ for any $i\in \mathbb {Z}\setminus \{m\}$ and $\phi (m,n)=(m-n)k_m^{(m)}$. It follows that $k_i^{(m)}=0$ for any $i\ne m$. Let $f(m)=k_m^{(m)}$ be a complex-valued function on $\mathbb {Z}$, which implies (2.2). Next, by (1.1) with a simple computation we obtain (2.3). $\square$

Let $(\mathcal {P}(\phi _i), \circ _i), i=1,2$ be two algebras on the same space as Witt algebra $W=\mathcal {P}(\phi _i)$ equipped with k-bilinear products $x\circ _i y$ such that $L_m\circ _i L_n=\phi _i(m,n)L_{m+n}$ for all $m,n\in \mathbb {Z}$, where $\phi _i, i=1,2$ are two complex-valued functions on $\mathbb {Z}\times \mathbb {Z}$. Furthermore, let $\tau : \mathcal {P}(\phi _1)\rightarrow \mathcal {P}(\phi _2)$ be a linear map given by $\tau (L_m)=-L_{-m}$ for all $m\in \mathbb {Z}$. Clearly, $\tau$ is a Lie automorphism of the Witt algebra (W, [, ]). Furthermore, we have the following result.

Proposition 2.2

Let $(\mathcal {P}(\phi _i), \circ _i), i=1,2$ be two algebras and $\tau : \mathcal {P}(\phi _1)\rightarrow \mathcal {P}(\phi _2)$ be a map defined as above. Suppose that $\mathcal {P}(\phi _1)$ is a post-Lie algebra. Then $\mathcal {P}(\phi _2)$ is a post-Lie algebra and $\tau$ is an isomorphism from $\mathcal {P}(\phi _1)$ to $\mathcal {P}(\phi _2)$ if and only if $\phi _2(m,n)=-\phi _1(-m,-n)$.

Proof

Suppose that $\mathcal {P}(\phi _2)$ is a post-Lie algebra and $\tau$ is an isomorphism from $\mathcal {P}(\phi _1)$ to $\mathcal {P}(\phi _2)$. Then we have $\tau (L_m \circ _1 L_n)=-\phi _1(m,n)L_{-(m+n)}$ and $\tau (L_m)\circ _2 \tau (L_n)=L_{-m}\circ _2 L_{-n}=\phi _2(-m,-n)$. By $\tau (L_m \circ _1 L_n)=\tau (L_m)\circ _2 \tau (L_n)$, it follows that $\phi _2(m,n)=-\phi _1(-m,-n)$.

Conversely, suppose that $\phi _2(m,n)=-\phi _1(-m,-n)$ for all $m,n\in \mathbb {Z}$. Notice that $\mathcal {P}(\phi _1)$ is a post-Lie algebra; by Lemma 2.1 we know there is a complex-valued function $f_1$ on $\mathbb {Z}$ such that

$$\begin{aligned}&\phi _1(m,n)=(m-n)f_1(m), \end{aligned}$$

(2.5)

$$\begin{aligned}&(m-n)(f_1(m+n)+f_1(m)f_1(m+n)+f_1(n)f_1(m+n)-f_1(m)f_1(n))=0,\nonumber \\ \end{aligned}$$

(2.6)

for all $m,n\in \mathbb {Z}$. By (2.5), we have $\phi _2(m,n)=-\phi _1(-m,-n)=-(n-m)f_1(-m)$. Let $f_2(m)=f_1(-m)$ where $f_2$ is a complex-valued function on $\mathbb {Z}$, then it follows that

$$\begin{aligned} \phi _2(m,n)=(m-n)f_2(m). \end{aligned}$$

(2.7)

Furthermore, by (2.6) and $f_2(m)=f_1(-m)$ we have that

$$\begin{aligned} (m-n)(f_2(m+n)+f_1(m)f_2(m+n)+f_2(n)f_2(m+n)-f_2(m)f_2(n))=0.\qquad \end{aligned}$$

(2.8)

Lemma 2.1 with (2.7) and (2.8) tells us that $\mathcal {P}(\phi _2)$ is a post-Lie algebra. The remainder is to prove that $\tau$ is an isomorphism. But one has

$$\begin{aligned} \tau (L_m \circ _1 L_n)=-\phi _1(m,n)L_{-(m+n)}=\phi _2(-m,-n)L_{-(m+n)}=\tau (L_m) \circ _2 \tau (L_m), \end{aligned}$$

which completes the proof. $\square$

For a complex-valued function f on $\mathbb {Z}$, we denote I, J by

$$\begin{aligned} I=\{m\in \mathbb {Z}| f(m)=0\}, \ \ J=\{m\in \mathbb {Z}| f(m)=-1\}. \end{aligned}$$

Lemma 2.3

There exists a graded post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (2.1) if and only if there is a complex-valued function f on $\mathbb {Z}$ such that (2.2) holds and it satisfies

• (i) $m\in I\cup J$ for all $m\ne 0$ and

• (ii) $m,n\in I\Rightarrow m+n\in I$ and $m+n\in J\Rightarrow m,n\in J$ for $m\ne n$.

Proof

The “if” part is easy to check. Next we prove the “only if” part. By Lemma 2.1, there is a complex-valued function f on $\mathbb {Z}$ satisfying (2.2) and (2.3). Let $n=0$ in (2.3), we have $m(f(m)+f(m)^2)=0.$ Thus, for $m\ne 0$, one has $f(m)=0$ or $f(m)=-1$. This implies the conclusion of (i) holds. Now, we chose a pair of $m, n\in \mathbb {Z}$ with $m\ne n$; then, it can be obtained by (2.3) that

$$\begin{aligned} f(m+n)+f(m)f(m+n)+f(n)f(m+n)-f(m)f(n)=0. \end{aligned}$$

It easy to see by the above equation that the conclusion of (ii) holds. $\square$

Our main result of this section is the following theorem.

Theorem 2.4

A graded post-Lie algebra structure satisfying (2.1) on the Witt algebra W must be one of the following types.

($\mathcal {P}_1$):

$L_m \circ _1 L_n=0$ for all $m,n\in \mathbb {Z}$;

($\mathcal {P}_2$):

$L_m \circ _2 L_n=(n-m)L_{m+n}$ for all $m,n\in \mathbb {Z}$;

($\mathcal {P}_3^a$):

$L_m \circ _3 L_n={\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{}\quad m> 0,\\ -naL_n, &{}\quad m=0, \\ 0, &{}\quad m<0; \end{array}\right. }$

($\mathcal {P}_4^a$):

$L_m \circ _4 L_n={\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{}\quad m< 0,\\ -naL_n, &{} \quad m=0, \\ 0, &{}\quad m>0; \end{array}\right. }$

($\mathcal {P}_5$):

$L_m \circ _5 L_n={\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{}\quad m\geqslant 2,\\ 0, &{}\quad m\leqslant 1; \end{array}\right. }$

($\mathcal {P}_6$):

$L_m \circ _6 L_n={\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{}\quad m\geqslant 1,\\ 0, &{}\quad m\leqslant 2; \end{array}\right. }$

($\mathcal {P}_7$):

$L_m \circ _7 L_n={\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{}\quad m\leqslant -2,\\ 0, &{}\quad m\geqslant -1; \end{array}\right. }$

($\mathcal {P}_8$):

$L_m \circ _8 L_n={\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{}\quad m\geqslant -1,\\ 0, &{}\quad m\leqslant -2 \end{array}\right. }$

where $a\in \mathbb {C}$. Conversely, the above types are all the graded post-Lie algebra structures satisfying (2.1) on the Witt algebra W. Furthermore, the post-Lie algebras $\mathcal {P}_3^a$, $\mathcal {P}_5$ and $\mathcal {P}_6$ are isomorphic to the post-Lie algebras $\mathcal {P}_4^a$, $\mathcal {P}_7$ and $\mathcal {P}_8$, respectively, and other post-Lie algebras are not mutually isomorphic.

Proof

Suppose that $(W, [,], \circ )$ is a post-Lie algebra structure satisfying (2.1) on the Witt algebra W. By Lemma 2.3, there is a complex-valued function f on $\mathbb {Z}$ such that (2.2) holds and it satisfies (i) and (ii) in Lemma 2.3. We discuss the cases of $f(1),f(-1),f(2)$ and $f(-2)$. Lemma 2.3 (i) implies that $f(1),f(-1),f(2),f(-2)\in \{-1,0\}$, and so that $2^4=16$ cases might appear. By a simple discussion, it can be seen that the 8 cases listed in Tabular 1 are true. We here should point out that Theorem 2.22 of [11] has given another method to prove it since they have the same condition as (2.3). Thus, by Lemma 2.1, the graded post-Lie algebra structure on the Witt algebra W must be one of the above 8 types. Conversely, every type of the above 8 cases means that there is a complex-valued function f on $\mathbb {Z}$ such that (2.2) holds, and the conclusions (i) and (ii) of Lemma 2.3 are easily verified. Thus, they are all the graded post-Lie algebra structures on the Witt algebra W by Lemma 2.3.

Finally, by Proposition 2.2 we know that the post-Lie algebras $\mathcal {P}_3^a$, $\mathcal {P}_5$ and $\mathcal {P}_6$ are isomorphic to the post-Lie algebras $\mathcal {P}_4^a$, $\mathcal {P}_7$ and $\mathcal {P}_8$, respectively. We claim that $\mathcal {P}_3^{a_1}$ is not isomorphic to $\mathcal {P}_3^{a_2}$ when $a_1\ne a_2$. If not, then there a linear bijective map $\tau : \mathcal {P}_3^{a_1}\rightarrow \mathcal {P}_3^{a_2}$ as the isomorphism of post-Lie algebras. According to Definition 1.2, $\tau$ first is an automorphism of the Lie algebra (W, [, ]). It follows by [7] that $\tau (L_m) = \epsilon c^mL_{\epsilon m}$ for all $m\in \mathbb {Z}$, where $c\in \mathbb {C}$ with $c\ne 0$ and $\epsilon \in \{\pm 1\}$. This, together with the definitions of $\mathcal {P}_3^{a}$, yields that $a_1=a_2$. This proves the claim above. Similarly, we have $\mathcal {P}_4^{a_1}$ is not isomorphic to $\mathcal {P}_4^{a_2}$ when $a_1\ne a_2$. Clearly, the other post-Lie algebras are not mutually isomorphic. The proof is completed. $\square$

From Theorem 2.4 and Proposition 1.3, we now are able to give some Lie algebras on the space with $\mathbb {C}$-basis $\{L_i|i\in \mathbb {Z} \}$ as follows.

Proposition 2.5

Up to isomorphism, the post-Lie algebras in Theorem 2.4 give rise to the following Lie algebras under the bracket {,} defined in (1.3):

($\mathcal {LP}_1$)::

$\{L_m, L_n\}_1=(m-n)L_{m+n} \quad \hbox {for all} \,\,m,n\in \mathbb {Z}$;

($\mathcal {LP}_3^a$)::

$\{L_m , L_n\}_3= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{}\quad m,n> 0, \\ (m-n)L_{m+n}, &{}\quad m,n<0, \\ -naL_n, &{}\quad m=0, n> 0,\\ -n(a+1)L_n &{}\quad m=0, n<0,\\ 0,&{}\quad \text {otherwise (unless}\ n=0 ) \end{array}\right. }$

($\mathcal {LP}_5$)::

$\{L_m , L_n\}_5= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{}\quad m,n\geqslant 2, \\ (m-n)L_{m+n}, &{}\quad m,n\leqslant 1, \\ 0,&{}\quad \text {otherwise} \end{array}\right. }$

where $a \in \mathbb { C}$.

Proof

Theorem 2.4 tells us that, up to isomorphism, there are five types of graded post-Lie algebra structures satisfying (2.1) on the Witt algebra, that is, $\mathcal {P}_{1}$, $\mathcal {P}_{2}$, $\mathcal {P}^a_{3}$, $\mathcal {P}_{5}$ and $\mathcal {P}_{6}$. By Proposition 1.3 and a simple computation, we can obtain five types of Lie algebras denoted by $\mathcal {LP}_{1}$, $\mathcal {LP}_{2}$, $\mathcal {LP}^a_{3}$, $\mathcal {LP}_{5}$ and $\mathcal {LP}_{6}$, respectively. It is easy to verify that the Lie algebras $\mathcal {LP}_1$, $\mathcal {LP}_5$ are isomorphic to the Lie algebras $\mathcal {LP}_2$, $\mathcal {LP}_6$, respectively, through the linear transformation $L_m\rightarrow -L_m$. The conclusions are easily deducible. $\square$

3 A Class of Shifting Post-Lie Algebra Structures on the Witt Algebra

In this section, we mainly consider a class of non-graded post-Lie algebra structures on the Witt algebra (W, [, ]). Namely, we assume that

$$\begin{aligned} L_m\circ L_n=\phi (m,n)L_{m+n}+\varrho (m,n)L_{m+n+\nu } \end{aligned}$$

(3.1)

for all $m,n\in \mathbb {Z}$, where $\phi$ and $\varrho$ are complex-valued functions on $\mathbb {Z}\times \mathbb {Z}$ with $\varrho \ne 0$, and $\nu$ is a fixed nonzero integer. The motivation of studying this class of non-graded post-Lie algebra structures is inspired by [19], in which a class of non-graded left-symmetric algebraic structures on the Witt algebra has been considered. As in the theory of groups and in the theory of graded Lie algebras, we shall call such non-graded post-Lie algebra a shifting post-Lie algebra since it has a shifting item. Our results show that the characterization of all non-graded post-Lie algebra structures on the Witt algebra seems difficult.

Lemma 3.1

There exists a shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1) if and only if there are complex-valued functions f and g on $\mathbb {Z}$ such that

$$\begin{aligned}&\phi (m,n)=(m-n)f(m), \end{aligned}$$

(3.2)

$$\begin{aligned}&\varrho (m,n)=(m-n+\nu )g(m), \end{aligned}$$

(3.3)

$$\begin{aligned}&(m-n)(f(m+n)+f(m)f(m+n)+f(n)f(m+n)-f(m)f(n))=0,\qquad \end{aligned}$$

(3.4)

$$\begin{aligned}&(m-n)g(m)g(n)=((m-n+\nu )g(m)+(m-n-\nu )g(n))g(m+n+\nu ),\qquad \quad \end{aligned}$$

(3.5)

$$\begin{aligned}&(m-n)(f(m)+f(n)+1)g(m+n)\nonumber \\&\quad =(n-m+\nu )(f(m+n+\nu )-f(m))g(n)\nonumber \\&\qquad +(n-m-\nu )(f(m+n+\nu )-f(n))g(m). \end{aligned}$$

(3.6)

Proof

The conclusion can be obtained by a similar proof as of Lemma 2.1. $\square$

Let $(\mathcal {P}(\phi _i,\varrho _i,\nu _i), \circ _i), i=1,2$ be two algebras on the same space as Witt algebra $W=\mathcal {P}(\phi _i,\varrho _i,\nu _i)$ equipped with k-bilinear products $x\circ _i y$ such that $L_m\circ _i L_n=\phi _i(m,n)L_{m+n}+\varrho _i(m,n)L_{m+n+\nu _i}$ for all $m,n\in \mathbb {Z}$, where $\phi _i,\varrho _i, i=1,2$ are complex-valued functions on $\mathbb {Z}\times \mathbb {Z}$ and $\nu _i$ are nonzero integers. Furthermore, let $\tau : \mathcal {P}(\phi _1,\varrho _1,\nu _1)\rightarrow \mathcal {P}(\phi _2,\varrho _2,\nu _2)$ be a linear map given by $\tau (L_m)=-L_{-m}$ for all $m\in \mathbb {Z}$.

Proposition 3.2

Let $\mathcal {P}(\phi _i,\varrho _i,\nu _i), i=1,2$ be two algebras and $\tau : \mathcal {P}(\phi _1,\varrho _1,\nu _1)\rightarrow \mathcal {P}(\phi _2,\varrho _2,\nu _2)$ be a map defined as above. Suppose that $\mathcal {P}(\phi _1,\varrho _1,\nu _1)$ is a post-Lie algebra. Then $\mathcal {P}(\phi _2, \varrho _2,\nu _2 )$ is a post-Lie algebra and $\tau$ is an isomorphism from $\mathcal {P}(\phi _1, \varrho _1,\nu _1)$ to $\mathcal {P}(\phi _2, \varrho _2,\nu _2 )$ if and only if

$$\begin{aligned} \phi _2(m,n)=-\phi _1(-m,-n), \ \ \varrho _2 (m,n)=-\varrho _1(-m,-n), \ \ \nu _1=-\nu _2. \end{aligned}$$

(3.7)

Proof

The conclusion can be obtained by a similar proof as of Proposition 2.2. $\square$

Since a shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1) is entirely determined by an integer $\nu$ and two complex-valued functions $\phi$ and $\varrho$ on $\mathbb {Z}\times \mathbb {Z}$. According to (3.2) and (3.3) in Lemma 3.1, the classicization of such post-Lie algebras is dependent on the integer $\nu$ and two complex-valued functions f and g on $\mathbb {Z}$. It will be proved that the following Table 2 gives all cases of $\nu , f$ and g, where b is any nonzero complex number.

Table 1 Values of f(n)

Table 2 Values of f(n), g(n) and $\nu$

The above conclusion will be proved by some propositions as follows. First, notice that (3.2) and (3.5) in Lemma 3.1, from Lemma 2.1 and Theorem 2.4 we have the following lemma.

Lemma 3.3

Suppose that $(W, [,], \circ )$ is a shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1). Then $\phi$ satisfies (2.2) and f must be determined by one of the cases $\mathcal {P}_1,\mathcal {P}_2$, $\mathcal {P}_3^a,\mathcal {P}_4^a$, $\mathcal {P}_4$-$\mathcal {P}_8$ in Table 1.

Taking $n=-\nu$ in (3.5), the following equation is often used in our proof.

$$\begin{aligned} \nu g(m)g(-\nu )=(m+2\nu )g(m)g(m). \end{aligned}$$

(3.8)

Proposition 3.4

If f takes the form determined by $\mathcal {P}_1$ or $\mathcal {P}_2$ in Table 1, then $g(\mathbb {Z})=0$. In this case, there is no any shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1).

Proof

If f takes the form determined by $\mathcal {P}_1$, then $f(m)=0$ for all $m\in \mathbb {Z}$. Thus, by (3.6) we have $(m-n)g(m+n)=0$. From this, we deduce that $g(\mathbb {Z})=0$. The case in which f takes the form determined by $\mathcal {P}_2$ is similar. $\square$

Proposition 3.5

Suppose that f takes the form determined by $\mathcal {P}_3^a$ in Table 1, i.e., $f(\mathbb {Z}_{>0})=-1, f(\mathbb {Z}_{<0})=0$ and $f(0)=a$ for some $a\in \mathbb {C}$. Then the shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1) is determined by $\mathcal {NP}^{b,\nu }_1$ or $\mathcal {NP}^{b,\nu }_2$ in Table 2.

Proof

The proof is divided into the following: Assertions 3.5.1, 3.5.2, 3.5.3.

Assertion 3.5.1

• (i) When $\nu >0$, we have $g(\mathbb {Z}_{\geqslant 3})=0$ and $g(\mathbb {Z}_{\leqslant \min \{-3, -1-\nu \}})=0$;

• (ii) When $\nu <0$, we have $g(\mathbb {Z}_{\geqslant \max \{3, 1-\nu \}})=g(\mathbb {Z}_{\leqslant -3})=0$.

For all $m,n\in \mathbb {Z}$ such that $\{m,n, m+n+\nu \} \subset \mathbb {Z}_{>0}$ or $\{m,n, m+n+\nu \} \subset \mathbb {Z}_{<0}$, by (3.6), it follows that $(m-n)g(m+n)=0$. The results are easy to check.

Assertion 3.5.2

If $\nu \geqslant 3$ or $\nu \leqslant -3$, then $g(\mathbb {Z})=0$.

Case I$\nu \geqslant 3$. By Assertion 3.5.1 (i), $g(\mathbb {Z}_{\geqslant 3})=g(\mathbb {Z}_{\leqslant -1-\nu })=0$. Let $m=-n=1$ and $m=-n=2$ in (3.6), respectively, one can deduce $g(1)=g(2)=0$. Note that $-2\nu <-1-\nu$, one has $g(-2\nu )=0$. This, together with $m=-2\nu$ and $n=0$ in (3.5), gives $0=-3\nu g(0)g(-\nu )$, so that $g(0)g(-\nu )=0$. From this, by letting $m=-\nu$ and $n=0$ in (3.5), we obtain $-2\nu g(0)g(0)=0$. In other words, $g(0)=0$. In order to prove that $g(\mathbb {Z})=0$, it is enough to show that $g(m)=0$ for all $m\in \{-1,-2, \cdots , -\nu \}$. Because $-2\nu +1<-\nu -1$ and $-2\nu +2<-\nu -1$, one has $g(-2\nu +1)=g(-2\nu +2)=0$. This, together with $m=-2\nu +1, n=-1$ and $m=-2\nu +2, n=-2$ in (3.5), respectively, gives

$$\begin{aligned} (-3\nu +2)g(-1)g(-\nu )=(-3\nu +4)g(-2)g(-\nu )=0. \end{aligned}$$

We see that $-3\nu +2\ne 0$ and $-3\nu +4\ne 0$ since $\nu \geqslant 3$. This, together with the above equation, yields that $g(-1)g(-\nu )=g(-2)g(-\nu )=0$. Next, using (3.8) for $m=-1$ and $m=-2$, respectively, we deduce $(-1+2\nu )g^2(-1)=(-2+2\nu )g^2(-2)=0$. Since $-1+2\nu \ne 0$ and $-2+2\nu \ne 0$, one has $g(-1)=g(-2)=0$. If we let $m=-1$ and $n=-k$ in (3.6) where $k\geqslant 2$, it follows that

$$\begin{aligned} (k-1)g(-1-k)=(1-k+\nu )f(-1-k+\nu )g(-k). \end{aligned}$$

(3.9)

By letting $k=2$ in (3.9), we have $g(-3)=0$. Again, by letting $k=3$ in (3.9), one has $g(-4)=0$. In turn, we obtain by (3.9) that $g(\mathbb {Z}_{<0})=0$; therefore, the conclusion is proved.

Case II$\nu \leqslant -3$. Using the similar method as of Case I, one also can obtain that $g(\mathbb {Z})=0$.

Assertion 3.5.3

f, g and $\tau$ must be determined by $\mathcal {NP}^{b,\nu }_1$ or $\mathcal {NP}^{b,\nu }_2$ in Table 2.

By Assertion 3.5.2, $g(\mathbb {Z})=0$ if $\nu \notin \{1,2,-1,-2\}$. Since $g\ne 0$, then $\nu \in \{1,2,-1,-2\}$.

Case 1$\nu =1$. From Assertion 3.5.1 (i), $g(\mathbb {Z}_{\geqslant 3})=g(\mathbb {Z}_{\leqslant -3})=0$. Next, we discuss the images f(k) for $k\in \{-1,-2,0,1,2\}$. Taking $m=-n=1$ and $m=-n=2$ in (3.6), respectively, one has $g(1)=g(2)=0$. If we let $n=0$ in (3.5), one has

$$\begin{aligned} mg(m)g(0)=((m+1)g(m)+(m-1)g(0))g(m+1). \end{aligned}$$

(3.10)

Taking $m=-1, -2$ and $-3$ in (3.10), respectively, by $g(-3)=0$, we have that

$$\begin{aligned} g(-1)g(0)=2g(0)g(0), \ 2g(-2)g(0)=3g(-1)g(0), \ g(0)g(-2)=0, \end{aligned}$$

which implies that $g^2(0)=0$. Thus, $g(0)=0$. If we let $m=-2, n=1$ and $m=-2, n=0$ in (3.6), respectively, then one has that $(f(0)+1)g(-2)=0$ and $2(f(0)+1)g(-2)=f(0)g(-2)$. This yields that $g(-2)=0$. Let $m=-1$ and $n=0$ in (3.6), then we have $(f(0)+1)g(-1)=0$. The fact that $g\ne 0$ means that $g(-1)=-b\ne 0$ for some $b\in \mathbb {C}$. At the same time, we must have that $f(0)+1=0$, namely, $f(0)=a=-1$. By the above part of the analysis and discussion, we see that

$$\begin{aligned} f(\mathbb {Z}_{\geqslant 0})=-1, \ \ f(\mathbb {Z}_{<0})=0,\ g(-1)=-b, \ g(\mathbb {Z}\setminus \{-1\})=0 \end{aligned}$$

for some nonzero $b\in \mathbb {C}$. This is of the type $\mathcal {NP}^{b,1}_1$ in Table 2.

Case 2$\nu =2$. By Assertion 3.5.1 (i), $g(\mathbb {Z}_{\geqslant 3})=g(\mathbb {Z}_{\leqslant -3})=0$. Thus, we only discuss the images f(k) for $k\in \{-1,-2,0,1,2\}$. Taking $m=-n=1$ and $m=-n=2$ in (3.6), respectively, it follows that $g(1)=g(2)=0$. Let $m=-1, n=-2$ and $m=-3, n=-1$ in (3.5), respectively, we get $2g(-1)g(-2)=3g(-1)g(-1)$ and $g(-1)g(-2)=0$, which yields that $g^2(-1)=0$. Hence, $g(-1)=0$. We again obtain $g(0)=0$ by taking $m=-1$ and $n=0$ in (3.6). Finally, it follows that $(f(0)+1)g(-2)=0$ by taking $m=-2$ and $n=0$ in (3.6). Since $g\ne 0$, hence $f(0)=a=-1$ and $g(-2)=-b\ne 0$ for some nonzero $b\in \mathbb {C}$. It proves that

$$\begin{aligned} f(\mathbb {Z}_{\geqslant 0})=-1, \ \ f(\mathbb {Z}_{<0})=0,\ g(-2)=-b, \ g(\mathbb {Z}\setminus \{-2\})=0, \end{aligned}$$

which is of the type $\mathcal {NP}^{b,2}_1$ in Table 2.

Case 3$\nu =-1$. By a similar discussion as of Case 1, we obtain the type $\mathcal {NP}^{b,-1}_2$ in Table 2.

Case 4$\nu =-2$. By a similar discussion as of Case 2, we obtain the type $\mathcal {NP}^{b,-2}_2$ in Table 2. $\square$

Proposition 3.6

Suppose that f takes the form determined by $\mathcal {P}_5$ in Table 1, i.e., $f(\mathbb {Z}_{\geqslant 2})=-1, f(\mathbb {Z}_{\leqslant 1})=0$. Then the shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1) is determined by $\mathcal {NP}^{b,\nu }_3$, $\mathcal {NP}^{b,\nu }_4$ or $\mathcal {NP}^{b,\nu }_5$ in Table 2.

Proof

The proof is divided into the following: Assertions 3.6.1, 3.6.2 and 3.6.3.

Assertion 3.6.1

• (i) When $\nu >0$, we have $g(\mathbb {Z}_{\geqslant 5})=g(\mathbb {Z}_{\leqslant 1-\nu })=0$;

• (ii) When $\nu <0$, we have $g(\mathbb {Z}_{ \geqslant \max \{5, 2-\nu })=g(\mathbb {Z}_{\leqslant 1})=0$.

For any $m,n\in \mathbb {Z}$ with $\{m,n, m+n+\nu \} \subset \mathbb {Z}_{\geqslant 2}$ or $\{m,n, m+n+\nu \} \subset \mathbb {Z}_{\leqslant 1}$, one has by (3.6) that $(m-n)g(m+n)=0$. The assertion are easy to check.

Assertion 3.6.2

If $\nu \geqslant 1$ or $\nu \leqslant -5$, then $g(\mathbb {Z})=0$.

Case I$\nu \geqslant 1$. By Assertion 3.6.1 (i), $g(\mathbb {Z}_{\geqslant 5})=g(\mathbb {Z}_{\leqslant 1-\nu })=0$. Taking $n=0$ in (3.6), then

$$\begin{aligned} (m(f(m)+1)-(m+\nu )f(m+\nu ))g(m)=(\nu -m)(f(m+\nu )-f(m))g(0) \end{aligned}$$

(3.11)

for all $m\in \mathbb {Z}$. If we let $m\in \{2,3,4\}$, then $f(m)=f(m+\nu )=-1$, and by (3.11), we have $(m+\nu )g(m)=0$. This implies $g(2)=g(3)=g(4)=0$. Take $m=1$ and $n=-1-\nu$ in (3.5); then by $g(-1-\nu )=0$,

$$\begin{aligned} (2+2\nu )g(1)g(0)=0. \end{aligned}$$

(3.12)

If we let $m=1$ in (3.11), it follows that $-\nu g(1)=(1-\nu ) g(0)$. This, together with (3.12), gives $g(1)=g(0)=0$ for the case $\nu >1$. But for the case $\nu =1$, we also have $g(1)=0$ since $-\nu g(1)=(1-\nu ) g(0)$ and $g(0)=0$ since $g(\mathbb {Z}_{\leqslant 1-\nu })=0$. In order to prove the conclusion, we only need to show that $g(k)=0$ for all $k\in \{2-\nu , 3-\nu , \cdots , -2,-1\}$. For such k, we let $k=s-\nu$, where $2\leqslant s<\nu$. Note that $k<0$, so $f(k)=0$ and $f(k+\nu )=f(s)=-1$. Using (3.11) with $m=k$, we have by $g(0)=0$ that $(k-(k+\nu ))g(k)=0$. In other words, $g(k)=0$, as desired.

Case II$\nu \leqslant -5$. By Assertion 3.6.1 (ii), $g(\mathbb {Z}_{\geqslant 2-\nu })=g(\mathbb {Z}_{\leqslant 1})=0$. From this, we only need to prove that $g(m)=0$ for every $m\in \{2,3,\cdots , 1-\nu \}$. For $m\in \mathbb {Z}$ with $1<m<2-\nu$, it follows that $m+2\nu<2+\nu <0$. Taking $m=2$ and $n=-2-\nu$ in (3.5) and (3.6), respectively, we obtain that

$$\begin{aligned}&(4+\nu )g(2)g(-2-\nu )=0, \end{aligned}$$

(3.13)

$$\begin{aligned}&(4+\nu )g(-\nu )=(4+2\nu )g(2)+4g(-2-\nu ). \end{aligned}$$

(3.14)

Note that $4+\nu \ne 0$, so (3.13) tells us that $g(2)g(-2-\nu )=0$. This, together with (3.14), gives that

$$\begin{aligned}&(4+\nu )g(2)g(-\nu )=(4+2\nu )g^2(2), \end{aligned}$$

(3.15)

$$\begin{aligned}&(4+\nu )g(-2-\nu )g(-\nu )=4g^2(-2-\nu ). \end{aligned}$$

(3.16)

By letting $m=2$ and $m=-2-\nu$ in (3.8), respectively, we have that

$$\begin{aligned}&\nu g(2)g(-\nu )=(2+2\nu )g^2(2), \end{aligned}$$

(3.17)

$$\begin{aligned}&\nu g(-2-\nu )g(-\nu )=(\nu -2)g^2(-2-\nu ). \end{aligned}$$

(3.18)

It follows by (3.15) and (3.17) that

$$\begin{aligned} \begin{pmatrix} 4+\nu &{} 4+2\nu \\ \nu &{} 2+2\nu \end{pmatrix} \begin{pmatrix} g(2)g(-\nu )\\ -g^2(2) \end{pmatrix}=0, \end{aligned}$$

and by (3.16) and (3.18) that

$$\begin{aligned} \begin{pmatrix} 4+\nu &{} 4\\ \nu &{}\nu -2 \end{pmatrix} \begin{pmatrix} g(-2-\nu )g(-\nu )\\ -g^2(-2-\nu )) \end{pmatrix}=0. \end{aligned}$$

Since $\nu \leqslant -5$, we deduce

$$\begin{aligned} \det \begin{pmatrix} 4+\nu &{} 4+2\nu \\ \nu &{} 2+2\nu \end{pmatrix}=8+6\nu \ne 0, \ \ \det \begin{pmatrix} 4+\nu &{} 4\\ \nu &{}\nu -2 \end{pmatrix}=(\nu -4)(\nu +2)\ne 0, \end{aligned}$$

which yields that $-g^2(2)=-g^2(-2-\nu )=0$, and then, $g(2)=g(-2-\nu )=0$. Thus, from (3.14) we have $(4+\nu )g(-\nu )=(4+2\nu )g(2)+4g(-2-\nu )=0$. Note that $4+\nu \ne 0$, so $g(-\nu )=0$. This and (3.8) together imply that $(m+2\nu )g^2(m)=0$. For $m\in \mathbb {Z}$ with $1<m<2-\nu$, we have $m+2\nu<2+\nu <0$. Therefore, $g^2(m)=0$ and thereby $g(m)=0$ for every $m\in \{2,3,\cdots , 1-\nu \}$. The proof is completed.

Assertion 3.6.3

f, g and $\tau$ must be determined by $\mathcal {NP}^{b,\nu }_3$, $\mathcal {NP}^{b,\nu }_4$ or $\mathcal {NP}^{b,\nu }_5$ in Table 2.

By Assertion 3.6.2, $g(\mathbb {Z})=0$ if $\nu \notin \{-1, -2,-3,-4\}$. Since $g\ne 0$, then $\nu \in \{-1, -2,-3,-4\}$.

Case 1$\nu =-1$. From Assertion 3.6.1 (ii), $g(\mathbb {Z}_{\geqslant 5})=g(\mathbb {Z}_{\leqslant 1})=0$. Next, we discuss the images f(k) for $k\in \{2,3,4\}$. Taking $m=0, n=3$ and $m=0, n=4$ in (3.6), respectively, one has that $g(3)=g(4)=0$. In this case, $g(2)=-b\ne 0$ for some $b\in \mathbb {C}$. Thus, it summarizes as

$$\begin{aligned} f(\mathbb {Z}_{\geqslant 2})=-1, \ \ f(\mathbb {Z}_{\leqslant 1})=0,\ g(2)=-b, \ g(\mathbb {Z}\setminus \{2\})=0. \end{aligned}$$

It gives the type $\mathcal {NP}_4^{b,-1}$ in Table 2.

Case 2$\nu =-2$. By Assertion 3.6.1 (ii) we have $g(\mathbb {Z}_{\geqslant 5})=g(\mathbb {Z}_{\leqslant 1})=0$. Thus, we only discuss the images f(k) for $k\in \{2,3,4\}$. Taking $m=0$ and $n=4$ in (3.6), it follows that $g(4)=0$. If we let $m=3$ and $\nu =-2$ in (3.8), we get $g(3)g(3)=2g(2)g(3)$. This tells us that if $g(2)=0$, then $g(3)=0$. Since $g\ne 0$, it must be $g(2)=-b\ne 0$. In this case, either $g(3)=0$ which gives the type $\mathcal {NP}_3^{b,-2}$ in Table 2, or $g(3)=2g(2)=-2b\ne 0$ which gives the type $\mathcal {NP}_5^{b,-2}$ in Table 2.

Case 3$\nu =-3$. By Assertion 3.6.1 (ii), $g(\mathbb {Z}_{\geqslant 5})=g(\mathbb {Z}_{\leqslant 1})=0$. Thus, we only discuss the images f(k) for $k\in \{2,3,4\}$. Taking $m=2, n=5$ and $m=2, n=4$ in (3.6), respectively, one has that

$$\begin{aligned} g(2)g(4)=0, \ \ g(3)g(4)=5g(2)g(3). \end{aligned}$$

(3.19)

If we let $\nu =-3$ and $m=2,4$ in (3.8), respectively, we get

$$\begin{aligned} 3g(2)g(3)=4g(2)g(2), \ \ 3g(2)g(3)=2g(4)g(4). \end{aligned}$$

(3.20)

Combining (3.19) with (3.20), it implies that $g^2(4)=10g^2(2)$ and $g(2)g(4)=0$. This means that $g(2)=g(4)=0$. Since $g\ne 0$, we have $g(3)=-b\ne 0$ for some $b\in \mathbb {C}\setminus \{0\}$. It is easy to check that this is just the type $\mathcal {NP}_3^{b,-3}$ in Table 2.

Case 4$\nu =-4$. By Assertion 3.6.1 (ii), we have $g(\mathbb {Z}_{\geqslant 6})=g(\mathbb {Z}_{\leqslant 1})=0$. Thus, we only discuss the images f(k) for $k\in \{2,3,4,5\}$. If we let $\nu =-4$ and $m=2,3,5$ in (3.8), respectively, we get

$$\begin{aligned} 4g(2)g(4)=2g(2)g(2), \ \ 4g(3)g(4)=5g(3)g(3), \ \ 4g(4)g(5)=3g(5)g(5). \end{aligned}$$

(3.21)

Taking $m=2$ and $n=6$ in (3.5), it follows that $g(2)g(4)=0$. This and (3.21) together imply that $g^2(2)=0$ and so that $g(2)=0$. If we let $m=3, n=6$ and $m=3, n=5$ in (3.5), respectively, we obtain that $g(3)g(5)=0$ and $g(4)g(5)=3g(3)g(4)$. This yields that $g(4)g(5)=g(3)g(4)=0$. This and (3.21) together imply that $g^2(3)=g^2(5)=0$, which gives $g(3)=g(5)=0$. Since $g\ne 0$, we have $g(4)=-b\ne 0$ for some $b\in \mathbb {C}\setminus \{0\}$. It is easy to check that this is just the type $\mathcal {NP}_3^{b,-4}$ in Table 2. $\square$

Proposition 3.7

Suppose that f takes the form determined by $\mathcal {P}_6$ in Table 1. Then the shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1) is determined by $\mathcal {NP}^{b,\nu }_6$, $\mathcal {NP}^{b,\nu }_7$ or $\mathcal {NP}^{b,\nu }_8$ in Table 2.

Proof

The conclusion can be obtained by a similar proof as of Proposition 3.6. $\square$

Next, by Proposition 3.2 and similar proofs as of Propositions 3.5, 3.6 and 3.7, one has the following three propositions.

Proposition 3.8

Suppose that f takes the form determined by $\mathcal {P}_4^a$ in Table 1. Then the shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1) is determined by $\mathcal {MP}^{b,\nu }_1$ or $\mathcal {MP}^{b,\nu }_2$ in Table 2.

Proposition 3.9

Suppose that f takes the form determined by $\mathcal {P}_7$ in Table 1. Then the shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1) is determined by $\mathcal {MP}^{b,\nu }_3$, $\mathcal {MP}^{b,\nu }_4$ or $\mathcal {MP}^{b,\nu }_5$ in Table 2.

Proposition 3.10

Suppose that f takes the form determined by $\mathcal {P}_8$ in Table 1. Then the shifting post-Lie algebra structure on the Witt algebra (W, [, ]) satisfying (3.1) is determined by $\mathcal {MP}^{b,\nu }_6$, $\mathcal {MP}^{b,\nu }_7$ or $\mathcal {MP}^{b,\nu }_8$ in Table 2.

Our main result in this section is the following theorem.

Theorem 3.11

A shifting post-Lie algebra structure satisfying (3.1) on the Witt algebra W must be one of the following types.

$$\begin{aligned}&(\mathcal {NP}^{b,\nu }_1): \nu =1{\textit{ or }}2,\\&\quad L_m \circ _1 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m\geqslant 0,\\ nbL_{n}, &{} m=-\nu , \\ 0, &{} m<0,m\ne -\nu ; \end{array}\right. }\\&(\mathcal {NP}^{b,\nu }_2): \nu =-1{\textit{ or }}-2,\\&\quad L_m \circ _2 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m> 0, m\ne -\nu ,\\ (n+\nu )L_{n-\nu }+nbL_{n}, \ \ &{} m=-\nu ,\\ 0, &{} m\leqslant 0; \end{array}\right. }\\&(\mathcal {NP}^{b,\nu }_3): \nu =-2, -3{\textit{ or }}-4,\\&\quad L_m \circ _3 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m\geqslant 2, \ m\ne -\nu \\ (n+\nu )L_{n-\nu }+nbL_{n}, \ \ &{} m=-\nu ,\\ 0, &{} m\leqslant 1; \end{array}\right. }\\&(\mathcal {NP}^{b,\nu }_4): L_m \circ _4 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m\geqslant 3,\\ (n-2)L_{n+2}+b(n-1)L_{n+1}, \ \ &{} m=2,\\ 0, &{} m\leqslant 1; \end{array}\right. }\\&(\mathcal {NP}^{b,\nu }_5): L_m \circ _5 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m\geqslant 4,\\ (n-2)L_{n+2}+nbL_{n}, \ \ &{} m= 2,\\ (n-3)L_{n+3}+2(n-1)bL_{n+1}, \ \ &{} m=3,\\ 0, &{} m\leqslant 1; \end{array}\right. }\\&(\mathcal {NP}^{b,\nu }_6): \nu =-2, -3{\textit{ or }}-4,\\&\quad L_m \circ _6 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{} m\leqslant 1,\\ nbL_n, \ \ &{} m=-\nu ,\\ 0, \ \ &{} m\geqslant 2, m\ne -\nu ; \end{array}\right. }\\&(\mathcal {NP}^{b,\nu }_7): L_m \circ _7 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{} m\leqslant 1,\\ (n-1)bL_{n+1}, \ \ &{} m=2,\\ 0, \ \ &{} m\geqslant 3; \end{array}\right. }\\&(\mathcal {NP}^{b,\nu }_8): L_m \circ _8 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{} m\leqslant 1,\\ nbL_{n}, \ \ &{} m=2,\\ 2(n-1)bL_{n+1}, \ \ &{} m=3,\\ 0, \ \ &{} m\geqslant 4; \end{array}\right. }\\&(\mathcal {MP}^{b,\nu }_1):] \nu =-1{\textit{ or }}-2,\\&\quad L_m \circ _9 L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m\leqslant 0,\\ nbL_{n}, &{} m=-\nu ,\\ 0, &{} m>0,m\ne -\nu ; \end{array}\right. }\\&(\mathcal {MP}^{b,\nu }_2): \nu =1 \,\,or\,\; 2,\\&\quad L_m \circ _{10} L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m< 0, m\ne -\nu ,\\ (n+\nu )L_{n-\nu }+nbL_{n}, \ \ &{} m=-\nu ,\\ 0, &{} m\geqslant 0; \end{array}\right. }\\&(\mathcal {MP}^{b,\nu }_3): \nu =2, 3{\textit{ or }}4,\\&\quad L_m \circ _{11} L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m\leqslant -2, \ m\ne -\nu \\ (n+\nu )L_{n-\nu }+nbL_{n}, \ \ &{} m=-\nu ,\\ 0, &{} m\geqslant -1; \end{array}\right. }\\&(\mathcal {MP}^{b,\nu }_4): L_m \circ _{12} L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m\leqslant -3,\\ (n+2)L_{n-2}+b(n+1)L_{n-1}, \ \ &{} m=-2,\\ 0, &{} m\geqslant -1; \end{array}\right. }\\&(\mathcal {MP}^{b,\nu }_5): L_m \circ _{13} L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m\leqslant -4,\\ (n+2)L_{n-2}+nbL_{n}, \ \ &{} m= -2,\\ (n+3)L_{n-3}+2(n+1)bL_{n-1}, \ \ &{} m=-3,\\ 0, &{} m\geqslant -1; \end{array}\right. }\\&(\mathcal {MP}^{b,\nu }_6): \nu =2, 3{\textit{ or }}4,\\&\quad L_m \circ _{14} L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{} m\geqslant -1,\\ nbL_n, \ \ &{} m=-\nu ,\\ 0, \ \ &{} m\leqslant -2, m\ne -\nu ; \end{array}\right. }\\&(\mathcal {MP}^{b,\nu }_7): L_m \circ _{15} L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{} m\geqslant -1,\\ (n+1)bL_{n-1}, \ \ &{} m=-2,\\ 0, \ \ &{} m\leqslant -3; \end{array}\right. }\\&(\mathcal {MP}^{b,\nu }_8): L_m \circ _{16} L_n= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, &{} m\geqslant -1,\\ nbL_{n}, \ \ &{} m=-2,\\ 2(n+1)bL_{n-1}, \ \ &{} m=-3,\\ 0, \ \ &{} m\leqslant -4 \end{array}\right. } \end{aligned}$$

where b is a nonzero number. Conversely, the above types are all shifting post-Lie algebra structures satisfying (3.1) on the Witt algebra W. Furthermore, the post-Lie algebras $\mathcal {NP}^{b,\nu }_i$ are isomorphic to the post-Lie algebras $\mathcal {MP}^{b,-\nu }_i$, $i=1, 2, \cdots , 8$, respectively, and other post-Lie algebras are not mutually isomorphic.

Proof

Suppose that $(W, [,], \circ )$ is a class of shifting post-Lie algebra structures satisfying (3.1) on the Witt algebra W. By Propositions 3.4–3.10, there are complex-valued functions f and g on $\mathbb {Z}$ such that one of the 16 cases in Table 2 holds. Thus, by Lemma 3.1 we know that the shifting post-Lie algebra structure must be one of the above 16 types. Conversely, every type of the above 16 cases means that there are complex-valued functions f and g on $\mathbb {Z}$ such that (3.2) and (3.3) hold and Eqs. (3.4)–(3.6) are easily verified. Thus, by Lemma 3.1 we see that all they are the shifting post-Lie algebra structure satisfying (3.1) on the Witt algebra W. Next, by Proposition 3.2 we know that the post-Lie algebras $\mathcal {NP}^{b,\nu }_i$ are isomorphic to the post-Lie algebras $\mathcal {MP}^{b,-\nu }_i$, $i=1, 2, \cdots , 8$, respectively. Finally, by a similar proof as of Theorem 2.4, one can see the other post-Lie algebras are not mutually isomorphic. $\square$

Proposition 3.12

Up to isomorphism, the post-Lie algebras in Theorem 3.11 give rise to the following Lie algebras under the bracket {, } defined in (1.3):

$$\begin{aligned}&(\mathcal {LNP}^{b,\nu }_1): \nu =1{\textit{ or }}\nu =2,\\&\quad \{L_m , L_n\}_1= {\left\{ \begin{array}{ll} (n-m)L_{m+n}, \ \ &{} m,n\geqslant 0, \\ (m-n)L_{m+n}, \ \ &{} m,n<0, m,n\ne -\nu \\ nbL_n, &{} m=-\nu , n\geqslant 0,\\ nbL_n-(n+\nu )L_{n-\nu } &{} m=-\nu , n<0, n\ne -\nu ,\\ 0,&{} \text {otherwise (unless}\ n=-\nu ); \end{array}\right. }\\&(\mathcal {LNP}^{b,\nu }_3): \nu =-2,-3{\textit{ or }}-4,\\&\quad \{L_m , L_n\}_3= {\left\{ \begin{array}{ll} (n-m)L_{m+n} \ \ &{} m,n \geqslant 2, m,n\ne -\nu , \\ (m-n)L_{m+n}, &{} m,n\leqslant 1, \\ nbL_{n}, &{} m=-\nu , n\leqslant 1,\\ nbL_n+(n+\nu )L_{n-\nu } &{} m=-\nu , n\geqslant 2, n\ne -\nu ,\\ 0,&{} \text {otherwise (unless}\ n=-\nu ); \end{array}\right. }\\&(\mathcal {LNP}^{b,\nu }_4): \{L_m , L_n\}_4= {\left\{ \begin{array}{ll} (n-m)L_{m+n} \ \ &{} m,n \geqslant 3, \\ (m-n)L_{m+n}, &{} m,n\leqslant 1, \\ (n-1)bL_{n+1}, &{} m=2, n\leqslant 1,\\ (n-1)bL_{n+1}+(n-2)L_{n+2} &{} m=2, n\geqslant 3,\\ 0,&{} \text {otherwise (unless}\ n=2 ); \end{array}\right. }\\&(\mathcal {LNP}^{b,\nu }_5): \{L_m , L_n\}_5= {\left\{ \begin{array}{ll} (n-m)L_{m+n} \ \ &{} m,n \geqslant 4, \\ (m-n)L_{m+n}, &{} m,n\leqslant 1, \\ L_5+bL_3, &{} m=2, n=3,\\ nbL_n, &{} m=2, n\leqslant 1,\\ 2(n-1)bL_{n+1}, &{} m=3, n\leqslant 1,\\ nbL_n+(n-2)L_{n+2} &{} m=2, n\geqslant 4,\\ 2(n-1)bL_{n+1}+(n-3)L_{n+3} &{} m=3, n\geqslant 4,\\ 0,&{} \text {otherwise (unless}\ n=2,3); \end{array}\right. } \end{aligned}$$

where b is a nonzero number.

Proof

The conclusion can be obtained by a similar proof as of Proposition 2.5. $\square$

4 Another Understanding of the Post-Lie Algebra Structures

We should see that there were two different definitions of post-Lie algebra, that is, post-Lie algebra structure on a Lie algebra and post-Lie algebra structure on pairs of Lie algebras. The former is studied as above. Now we recall the latter as follows.

Definition 4.1

[3, 4] Let $(V_1,[,])$ and $(V_2, \{,\})$ be a pair of Lie algebras on the same linear space $V=V_1=V_2$ over a field k. A post-Lie algebra structure on the pair $(V_1, V_2)$ is a k-bilinear product $x\circ y$ on V satisfying the following identities:

$$\begin{aligned} <x,y>= & {} \{x,y\}-[x,y], \\ \{x, y\} \circ z= & {} x \circ (y \circ z)-y \circ (x \circ z), \\ x\circ [y, z]= & {} [x\circ y, z] + [y, x \circ z] \end{aligned}$$

for all $x, y, z \in V$, where $<x,y>=x\circ y-y\circ x$. We also say that $(V_1, V_2, \circ , [ , ], \{,\})$ is a post-Lie algebra.

Inspired by the above definitions, here we would like to give another definition of a post-Lie algebra as follows. Below we will see that the three definitions of post-Lie algebra are equivalent.

Definition 4.2

A post-Lie algebra $(V, \circ , \{ , \})$ is a vector space V over a field k equipped with two k-bilinear products $x\circ y$ and $\{x, y\}$ and satisfies $(V, \{, \})$ which is a Lie algebra and

$$\begin{aligned}&\{x, y\} \circ z = x \circ (y \circ z)-y \circ (x \circ z), \end{aligned}$$

(4.1)

$$\begin{aligned}&x\circ \{y, z\}- \{x\circ y, z\}- \{y, x \circ z\} \nonumber \\&\quad = x\circ<y, z>-<x\circ y, z> - <y, x \circ z> \end{aligned}$$

(4.2)

for all $x, y, z \in V$, where $<x,y>=x\circ y-y\circ x$. We also say that $(V, \circ , \{ , \})$ is a post-Lie algebra structure on Lie algebra $(V, \{, \})$.

In addition, similar to Proposition 1.3, we have that:

Proposition 4.3

A post-Lie algebra $(V, \circ , \{, \})$ defined by Definition 4.2 with the following bracket

$$\begin{aligned}{}[x, y] \triangleq \{x, y\}-<x,y> \end{aligned}$$

defines an another Lie algebra structure on V, where $<x,y>=x\circ y-y\circ x$.

Proposition 4.4

Definitions 1.1, 4.1 and 4.2 of post-Lie algebra are equivalent.

Proof

If $(V, \circ , [, ])$ is a post-Lie algebra defined by Definition 1.1, then by Proposition 1.3 we know that under the Lie bracket $\{x, y\} =<x,y> + [x, y]$, V admits a new Lie algebra structure. Obviously, $(V, \circ , \{, \})$ satisfies (4.1) and (4.2). Conversely, when $(V, \circ , \{, \})$ is a post-Lie algebra defined by Definition 4.2, then by Proposition 4.3 we know that under the Lie bracket $[x, y] =\{x,y\} - <x, y>$, V also admits a new Lie algebra structure. We are able to verify that $(V, \circ , [, ])$ satisfies (1.1) and (1.2). This tell us that whether $(V, \circ , [, ])$ or $(V, \circ , \{, \})$ all are connotations of two Lie algebra structures, which satisfy the conditions of Definitions 4.1. On the other hand, a post-Lie algebra structure on the pair $(V_1, V_2)$ defined by Definitions 4.1 implies $(V_1, \circ , [, ])$ satisfies (1.1) and (1.2) or $(V_2, \circ , \{, \})$ satisfies (4.1) and (4.2). $\square$

Remark 4.5

Recall that the left multiplications of the algebra $A = (V,\circ )$ are denoted by $\mathcal {L}(x)$, i.e., we have $\mathcal {L}(x)(y) = x\circ y$ for all $x,y\in V$. Clearly, by (4.1) we see that the map $\mathcal {L}: V \rightarrow End(V )$ given by $x\mapsto \mathcal {L}(x)$ is a linear representation of the Lie algebra $(V,\{, \})$ when $(V,\circ , \{,\})$ is the post-Lie algebra defined by Definition 4.2.

It can be seen that the study of post-Lie algebra structures on pairs of Lie algebras given by Definition 4.1 is divided into two directions: either when $(V_1, [,])$ is a given Lie algebra, to determine the product $\circ$, or when $(V_2, \{,\})$ is a given Lie algebra, to determine the product $\circ$. By Proposition 4.4, the first direction is characterizing of the post-Lie algebra structures $(V_1, [,], \circ )$ on the Lie algebra $(V_1, [,])$ given by Definition 1.1, and another direction is characterizing of the post-Lie algebra structure $(V_2, \{,\}, \circ )$ on the Lie algebra $(V_2, \{,\})$ given by Definition 4.2. For the first case in which $V_1=W$ is the Witt algebra, the graded or some shifting post-Lie algebra structures are studied in Sects. 2 and 3. The problem of another case in which $V_2=W$ is the Witt algebra should be interesting. We are not going to discuss this problem here. But, inspired by the results of [13, 19], we may give two non-trivial examples as follows.

Example 4.1

The following cases give two class of graded post-Lie algebra structures on W satisfying (2.1) given by Definition 4.2.

$$\begin{aligned} \phi (m,n)=-\frac{(\alpha +n+\alpha \epsilon m)(1+\epsilon n)}{1+\epsilon (m+n)}, \end{aligned}$$

for all $m,n\in {\mathbb {Z}}$, where $\alpha , \epsilon \in \mathbb {C}$ satisfy $\epsilon =0$ or $\epsilon ^{-1}\not \in \mathbb {Z}$, or

$$\begin{aligned} \phi (m,n)= \left\{ \begin{array}{ll} -n-t, &{}\quad {\text{ if }}\ m+n+t\ne 0,\\ \frac{(n+t)(n+t-\beta )}{\beta -t},&{}\quad {\text{ if }}\ m+n+t= 0,\end{array}\right. \end{aligned}$$

for all $m,n\in {\mathbb {Z}}$, where $\beta \in \mathbb {C}$ and $t\in \mathbb {Z}$ satisfy $\beta \ne t$.

Example 4.2

Let $\phi (m,n)=-(n+\alpha )$ and $\varrho (m,n)=\mu$ in (3.1), where $\alpha , \mu \in \mathbb {C}$. This is a shifting post-Lie algebra structure on the Witt algebra given by Definition 4.2 as follows.

$$\begin{aligned} L_m\circ L_n=-(n+\alpha )L_{m+n}+\mu L_{m+n+\nu }. \end{aligned}$$

By Propositions 2.5 and 3.12, we find 7 classes of Lie algebras up to isomorphism. They are $\mathcal {LP}_{1}$, $\mathcal {LP}^a_{3}$, $\mathcal {LP}_{5}$, $\mathcal {LNP}^{b,\nu }_1$, $\mathcal {LNP}^{b,\nu }_3$, $\mathcal {LNP}^{b,\nu }_4$ and $\mathcal {LNP}^{b,\nu }_5$. Note that Remark 4.5 tells us that the post-Lie $(L, \circ , \{,\})$ admits a module structure of Lie algebra $(L, \{,\})$. Thus, by Theorems 2.4 and 3.11, we have the following results on module structures of some Lie algebras.

Proposition 4.6

Let $\mathcal {V}=\mathrm{Span}_{\mathbb {C}}\{v_i|i\in \mathbb {Z}\}$ be a complex linear vector space. Then $\mathcal {V}$ becomes a module over some Lie algebras under the following actions.

• (1) For the Lie algebra $\mathcal {LP}_{1}$: $L_m . v_n=0, \ \text{ for } \text{ all } m,n\in \mathbb {Z}$;

• (2) For the Lie algebra $\mathcal {LP}^a_{3}$:

  $$\begin{aligned} L_m . v_n={\left\{ \begin{array}{ll} (n-m)v_{m+n}, &{} m> 0,\\ -nav_n, &{} m=0, \\ 0, &{} m<0; \end{array}\right. } \end{aligned}$$

• (3) For the Lie algebra $\mathcal {LP}_5$:

  $$\begin{aligned} L_m . v_n={\left\{ \begin{array}{ll} (n-m)v_{m+n}, &{} m\geqslant 2,\\ 0, &{} m\leqslant 1; \end{array}\right. } \end{aligned}$$

• (4) For the Lie algebra $\mathcal {LNP}^{b,\nu }_1$: $\nu =1$ or 2,

  $$\begin{aligned} L_m . v_n= {\left\{ \begin{array}{ll} (n-m)v_{m+n}, \ \ &{} m\geqslant 0,\\ nbv_{n}, &{} m=-\nu , \\ 0, &{} m<0,m\ne -\nu ; \end{array}\right. } \end{aligned}$$

• (5) For the Lie algebra $\mathcal {LNP}^{b,\nu }_3$: $\nu =-2, -3$ or $-4$,

  $$\begin{aligned} L_m . v_n= {\left\{ \begin{array}{ll} (n-m)v_{m+n}, \ \ &{} m\geqslant 2, \ m\ne -\nu \\ (n+\nu )L_{n-\nu }+nbL_{n}, \ \ &{} m=-\nu ,\\ 0, &{} m\leqslant 1; \end{array}\right. } \end{aligned}$$

• (6) For the Lie algebra $\mathcal {NP}^{b,\nu }_4$:

  $$\begin{aligned} L_m. v_n= {\left\{ \begin{array}{ll} (n-m)v_{m+n}, \ \ &{} m\geqslant 3,\\ (n-2)v_{n+2}+b(n-1)v_{n+1}, \ \ &{} m=2,\\ 0, &{} m\leqslant 1; \end{array}\right. } \end{aligned}$$

• (7) For the Lie algebra $\mathcal {NP}^{b,\nu }_5$:

  $$\begin{aligned} L_m . v_n= {\left\{ \begin{array}{ll} (n-m)v_{m+n}, \ \ &{} m\geqslant 4,\\ (n-2)v_{n+2}+nbv_{n}, \ \ &{} m= 2,\\ (n-3)v_{n+3}+2(n-1)bv_{n+1}, \ \ &{} m=3,\\ 0, &{} m\leqslant 1, \end{array}\right. } \end{aligned}$$

where $a,b\in \mathbb {C}$ with $b\ne 0$.

5 Application to Rota–Baxter Operators

Now let us recall the definition of Rota–Baxter operator.

Definition 5.1

Let L be a complex Lie algebra. A Rota–Baxter operator of weight $\lambda \in \mathbb {C}$ is a linear map $R: L\rightarrow L$ satisfying

$$\begin{aligned}{}[R(x),R(y)]=R([R(x),y]+[x,R(y)])+\lambda R([x,y]), \text{ for } \text{ all } x,y\in L. \end{aligned}$$

(5.1)

Note that if R is a Rota–Baxter operator of weight $\lambda \ne 0$, then $\lambda ^{-1}R$ is a Rota–Baxter operator of weight 1. Therefore, one only needs to consider Rota–Baxter operators of weight 0 and 1.

Lemma 5.2

[1] Let L be a complex Lie algebra and $R : L\rightarrow L$ a Rota–Baxter operator of weight 1. Define a product $x\circ y$ = [R(x), y] on L. Then $(L, [, ], \circ )$ is a post-Lie algebra given by Definition 1.1.

Below we will characterize the Rota–Baxter operator $R: W\rightarrow W$ of weight 1 satisfying

$$\begin{aligned} R(L_m)=f(m)L_m, \text{ for } \text{ all } m\in \mathbb {Z} \end{aligned}$$

(5.2)

or

$$\begin{aligned} R(L_m)=f(m)L_m+g(m)L_{m+\nu }, \text{ for } \text{ all } m\in \mathbb {Z}, \end{aligned}$$

(5.3)

where $\nu$ is a nonzero integer and f, g are complex-valued functions on $\mathbb {Z}$ with $g\ne 0$. Using our conclusions on post-Lie algebra, we will obtain the following two theorems.

Theorem 5.3

A homogeneous Rota–Baxter operator of weight 1 satisfying (5.2) on the Witt algebra W must be one of the following types

$$\begin{aligned}&(\mathcal {R}_1): R(L_m)=0, \text{ for } \text{ all } m\in \mathbb {Z};\\&(\mathcal {R}_2): R(L_m)=-L_{m}, \forall m\in \mathbb {Z};\\&(\mathcal {R}^a_3): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m>0, \\ aL_{0}, \ \ &{} m=0,\\ 0, &{} m< 0; \end{array}\right. }\\&(\mathcal {R}^a_4): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m< 0,\\ aL_{0}, \ \ &{} m=0,\\ 0, &{} m>0; \end{array}\right. }\\&(\mathcal {R}_5): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\geqslant 2,\\ 0, \ \ &{} m\leqslant 1; \end{array}\right. }\\&(\mathcal {R}_6): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\leqslant 1,\\ 0, \ \ &{} m\geqslant 2; \end{array}\right. }\\&(\mathcal {R}_7): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\geqslant -1,\\ 0, \ \ &{} m\leqslant -2; \end{array}\right. }\\&(\mathcal {R}_8): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\leqslant -2,\\ 0, \ \ &{} m\geqslant -1, \end{array}\right. } \end{aligned}$$

where $a\in \mathbb {C}$.

Proof

In view of Lemma 5.2, if we define a product $x\circ y = [R(x), y]$ on W, then $(W, [,], \circ )$ is a post-Lie algebra. By (5.2) we have

$$\begin{aligned} L_m\circ L_n=[R(L_m), L_n]=(m-n)f(m)L_{m+n}, \text{ for } \text{ all } m,n\in \mathbb {Z}. \end{aligned}$$

This means that $(W, [,], \circ )$ is a graded post-Lie algebra structure satisfying (2.1) on W with $\phi (m,n)=(m-n)f(m)$. By Theorem 2.4, we see that f must be one of the eight cases listed in Table 1, which can be get the eight forms of R one by one. On the other hand, it is easy to verify that every form of R listed in the above is a Rota–Baxter operator of weight 1 satisfying (5.2). The conclusion is proved. $\square$

Theorem 5.4

A non-homogeneous Rota–Baxter operator of weight 1 satisfying (5.3) on the Witt algebra W must be one of the following types

$$\begin{aligned}&(\mathcal {NR}^b_1): \nu =1{\textit{ or }}2,\\&\quad R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\geqslant 0,\\ -bL_{0}, &{} m=-\nu , \\ 0, &{} m<0,m\ne -\nu ; \end{array}\right. }\\&(\mathcal {NR}^b_2): \nu =-1{\textit{ or }}-2,\\&\quad R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m> 0, m\ne -\nu ,\\ -L_{-\nu }-bL_{0}, \ \ &{} m=-\nu ,\\ 0, &{} m\leqslant 0; \end{array}\right. }\\&(\mathcal {NR}^b_3): \nu =-2, -3{\textit{ or }}-4,\\&\quad R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\geqslant 2, \ m\ne -\nu \\ -L_{-\nu }-bL_{0}, \ \ &{} m=-\nu ,\\ 0, &{} m\leqslant 1; \end{array}\right. }\\&(\mathcal {NR}^b_4): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\geqslant 3,\\ -L_{2}-bL_{1}, \ \ &{} m=2,\\ 0, &{} m\leqslant 1; \end{array}\right. }\\&(\mathcal {NR}^b_5): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\geqslant 4,\\ -L_{2}-bL_{0}, \ \ &{} m= 2,\\ -L_{3}-2bL_{1}, \ \ &{} m=3,\\ 0, &{} m\leqslant 1; \end{array}\right. }\\&(\mathcal {NR}^b_6): \nu =-2, -3{\textit{ or }}-4,\\&\quad R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\leqslant 1,\\ -bL_0, \ \ &{} m=-\nu ,\\ 0, \ \ &{} m\geqslant 2, m\ne -\nu ; \end{array}\right. }\\&(\mathcal {NR}^b_7): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\leqslant 1,\\ -bL_{1}, \ \ &{} m=2,\\ 0, \ \ &{} m\geqslant 3; \end{array}\right. }\\&(\mathcal {NR}^b_8): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\leqslant 1,\\ -bL_{0}, \ \ &{} m=2,\\ -2bL_{1}, \ \ &{} m=3,\\ 0, \ \ &{} m\geqslant 4; \end{array}\right. }\\&(\mathcal {MR}^b_1): \nu =-1{\textit{ or }}-2,\\&\quad R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\leqslant 0,\\ -bL_{0}, &{} m=-\nu ,\\ 0, &{} m>0,m\ne -\nu ; \end{array}\right. }\\&(\mathcal {MR}^b_2): \nu =1{\textit{ or }}2,\\&\quad R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m< 0, m\ne -\nu ,\\ -L_{-\nu }-bL_{0}, \ \ &{} m=-\nu ,\\ 0, &{} m\geqslant 0; \end{array}\right. }\\&(\mathcal {MR}^b_3): \nu =2, 3{\textit{ or }}4,\\&\quad R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\leqslant -2, \ m\ne -\nu \\ -L_{-\nu }-bL_{0}, \ \ &{} m=-\nu ,\\ 0, &{} m\geqslant -1; \end{array}\right. }\\&(\mathcal {MR}^b_4): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\leqslant -3,\\ -L_{-2}-bL_{-1}, \ \ &{} m=-2,\\ 0, &{} m\geqslant -1; \end{array}\right. }\\&(\mathcal {MR}^b_5): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, \ \ &{} m\leqslant -4,\\ -L_{-2}-bL_{0}, \ \ &{} m= -2,\\ -L_{-3}-2bL_{-1}, \ \ &{} m=-3,\\ 0, &{} m\geqslant -1; \end{array}\right. }\\&(\mathcal {MR}^b_6): \nu =2, 3{\textit{ or }}4,\\&\quad R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\geqslant -1,\\ -bL_0, \ \ &{} m=-\nu ,\\ 0, \ \ &{} m\leqslant -2, m\ne -\nu ; \end{array}\right. }\\&(\mathcal {MR}^b_7): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\geqslant -1,\\ -bL_{-1}, \ \ &{} m=-2,\\ 0, \ \ &{} m\leqslant -3; \end{array}\right. }\\&(\mathcal {MR}^b_8): R(L_m)= {\left\{ \begin{array}{ll} -L_{m}, &{} m\geqslant -1,\\ -bL_{0}, \ \ &{} m=-2,\\ -2bL_{-1}, \ \ &{} m=-3,\\ 0, \ \ &{} m\leqslant -4, \end{array}\right. } \end{aligned}$$

where b is a non-zero number. Conversely, the above operators are all the non-homogeneous Rota–Baxter operators of weight 1 satisfying (5.3) on the Witt algebra W.

Proof

In view of Lemma 5.2, if we define a product $x\circ y = [R(x), y]$ on W, then $(W, [,], \circ )$ is a post-Lie algebra. By (5.3) we have

$$\begin{aligned} L_m\circ L_n=[R(L_m), L_n]= & {} (m-n)f(m)L_{m+n}\\&+(m-n+\nu )g(m)L_{m+n+\nu }, \text{ for } \text{ all } m,n\in \mathbb {Z}. \end{aligned}$$

This means that $(W, [,], \circ )$ is a shifting post-Lie algebra structure satisfying (2.4) on W with $\phi (m,n)=(m-n)f(m)$ and $\varrho (m,n)=(m-n+\nu )g(m)$. By Theorem 3.11, we see that f, g and $\nu$ must be one of the 16 cases listed in Table 2, which can get 16 forms of R one by one. On the other hand, it is easy to verify that every form of R listed in the above is a Rota–Baxter operator of weight 1 satisfying (5.3). The proof is completed. $\square$

Remark 5.5

The Rota–Baxter operators given by Theorem 5.3 just are the all homogeneous Rota–Baxter operators of weight 1 described in [11]. But the Rota–Baxter operators given by Theorem 5.4 are new and non-homogeneous.

Remark 5.6

The Rota–Baxter operators on the Witt algebra W can be given a class of solutions of the classical Yang–Baxter equation (CYBE) on $W\ltimes _{\mathrm{ad}^*} W^*$. The details can be found in [11], which discuss the homogeneous case. Along the same lines, we can also consider the non-homogeneous case by use of Theorem 5.4. It is not discussed here.