Poincare-Birkhoff-Witt theorem for pre-Lie and postLie algebras

TL;DR

This paper extends the Poincare-Birkhoff-Witt theorem to pre-Lie and postLie algebras by constructing their universal enveloping algebras and establishing PBW pairs, confirming known results with new proofs.

Contribution

It constructs universal enveloping algebras for pre-Lie and postLie algebras and proves they form PBW pairs, providing new proofs for existing results.

Findings

Established PBW pairs for pre-Lie and postLie algebras

Constructed universal enveloping preassociative and postassociative algebras

Reproved known PBW result for pre-Lie algebras

Abstract

We construct the universal enveloping preassociative and postassociative algebra for a pre-Lie and a postLie algebra respectively. We show that the pairs $(\mathrm{preLie},\mathrm{preAs})$ and $(\mathrm{postLie},\mathrm{postAs})$ are Poincare-Birkhoff-Witt-pairs, for the first one it's a reproof of the result of V. Dotsenko and P. Tamaroff.

Equations

$$\begin{array}{c}(x_1\succ x_2+x_1\prec x_2)\succ x_3=x_1\succ (x_2\succ x_3),(x_1\succ x_2)\prec x_3=x_1\succ (x_2\prec x_3),\\ x_1\prec (x_2\succ x_3+x_2\prec x_3)=(x_1\prec x_2)\prec x_3.\end{array}$$

$$(x\cdot y)\cdot z-x\cdot (y\cdot z)-(y\cdot x)\cdot z+y\cdot (x\cdot z)=[y,x]\cdot z,x\cdot [y,z]=[x\cdot y,z]+[y,x\cdot z].$$

$$R(x)R(y)=R(R(x)y+xR(y)+\lambda xy),x,y\in A.$$

$$a\succ b=P(a)b-bP(a),a\prec b=aP(b)-P(b)a(a\cdot b=\lambda ab-\lambda ba)$$

$$(l+1)yx^{l}=\sum\limits_{i=2}^{l+1}(-1)^{i}\binom{l+1}{i}[y,x^{(i-1)}]x^{l+1-i}+\big{(}yx^{l}+xyx^{l-1}+\ldots+x^{l}y)$$

$$R(b_{1})\ldots R(b_{t})-R\bigg{(}\sum\limits_{i=1}^{t}R(b_{1})\ldots R(b_{i-1})\hat{R}(b_{i})R(b_{i+1})\ldots R(b_{t})\\ +\lambda\sum\limits_{1\leq i_{1}<i_{2}\leq t}R(b_{1})\ldots\hat{R}(b_{i})\ldots\hat{R}(b_{j})\ldots R(b_{t})+\ldots\\ +\lambda^{k-1}\sum\limits_{1\leq i_{1}<i_{2}<\ldots<i_{k}\leq t}R(b_{1})\ldots\hat{R}(b_{i_{1}})\ldots\hat{R}(b_{i_{k}})\ldots R(b_{t})+\ldots+\lambda^{t-1}b_{1}\ldots b_{t}\bigg{)}=0,$$

$$R(R(b_{1})R(b_{2})\ldots R(b_{l-1})b_{l}R(b_{l})^{k}R(b_{l+k+1}))\\ \allowdisplaybreaks=\frac{1}{k+1}R(b_{1})R(b_{2})\ldots R(b_{l-1})R(b_{l})^{k+1}R(b_{l+k+1})\\ +\frac{1}{k+1}R\bigg{(}\sum\limits_{i=2}^{k+1}(-1)^{i}\binom{k+1}{i}R(b_{1})R(b_{2})\ldots R(b_{l-1})[b_{l},R(b_{l})^{(i-1)}]R(b_{l})^{k+1-i}R(b_{l+k+1})\\ -\sum\limits_{1\leq i\leq l-1,\,i=l+k+1}R(b_{1})\ldots R(b_{i-1})\hat{R}(b_{i})R(b_{i+1})\ldots R(b_{t})\\ -\lambda\sum\limits_{1\leq i_{1}<i_{2}\leq l+k+1}R(b_{1})\ldots\hat{R}(b_{i})\ldots\hat{R}(b_{j})\ldots R(b_{l+k+1})-\ldots\\ -\lambda^{s-1}\sum\limits_{1\leq i_{1}<\ldots<i_{s}\leq l+k+1}R(b_{1})\ldots\hat{R}(b_{i_{1}})\ldots\hat{R}(b_{i_{s}})\ldots R(b_{l+k+1})\\ -\ldots-\lambda^{l+k}b_{1}\ldots b_{l+k+1}\bigg{)},$$

$$x\succ y=R(x)y,x\prec y=xR(y)(x\cdot y=\lambda xy)$$

$$xy-yx-[x,y],x>y,x,y\in X,$$

$$R(a)R(b)-R(R(a)b+aR(b)+\lambda ab),$$

$$R(R(z_1){\vec{x}}_{q_1}R(z_2)\dots R(z_s){\vec{x}}_{q_s}{x}_{\beta ,r}{x}_{\beta ,r+1}^{k}R(z_{s+1}))-\Delta ,$$

$$b_1=z_1,b_2=\widetilde{x_{q_1}^1},b_3=\widetilde{x_{q_1}^2},\dots ,b_{l_1+1}=\widetilde{x_{q_1}^{l_1}},b_{l_1+2}=z_2,b_{l_1+3}=\widetilde{x_{q_2}^1},\dots ,$$

$$\text{allowdisplaybreaks}{b}_{l_1+\dots +l_{s-1}+s}=z_s,b_{l_1+\dots +l_{s-1}+s+1}=\widetilde{x_{q_l}^1},\dots ,b_{l_1+\dots +l_s+s}=\widetilde{x_{q_s}^{l_s}},$$

$$b_{l_1+\dots +l_s+s+1}=\dots =b_{l_1+\dots +l_s+s+k+1}=x_{\beta ,r},b_{l_1+\dots +l_s+s+k+2}=z_{s+1}.$$

$$s\ge 1,k,r\ge 0,a,b\in RS(X),z_2,\dots ,z_s\in RS(X)\setminus S(X),$$

$${\vec{x}}_{q_1},\dots ,{\vec{x}}_{q_{s-1}}\in S(X\setminus X_0),{\vec{x}}_{q_s}\in S(X\setminus X_0)\cup \varnothing$$

$$w=R(R(z_1){\vec{x}}_{q_1}R(z_2)\dots R(z_s){\vec{x}}_{q_s}{x}_{\beta ,r}{x}_{\beta ,r+1}^{k}R(z_{s+1}))$$

$${\vec{x}}_{q_j}={\vec{x}}_{q_j}^{\prime }xy{\vec{x}}_{q_j}^{\prime \prime },x>y,1\le j\le s.$$

$$w\mathop{\equiv}\limits^{\eqref{UnivRel}}w^{\prime}+w^{\prime\prime}\\ \mathop{\equiv}\limits^{\eqref{LongRel1}}\frac{1}{k+1}\big{(}R(z_{1})\ldots R(z_{j})\vec{x}_{q_{j}}|_{xy\to yx+[x,y]}R(z_{j+1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})\\ +R(\Sigma_{w^{\prime}})+R(\Sigma_{w^{\prime\prime}})\big{)},$$

$$w\mathop{\equiv}\limits^{\eqref{LongRel1}}\frac{1}{k+1}\big{(}R(z_{1})\ldots R(z_{j})\vec{x}_{q_{j}}R(z_{j+1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})+R(\Sigma_{w})\big{)}\\ {\mathop{\equiv}\limits^{\eqref{UnivRel}}}\frac{1}{k+1}\big{(}R(z_{1})\ldots R(z_{j})\vec{x}_{q_{j}}|_{xy\to yx+[x,y]}R(z_{j+1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})+R(\Sigma_{w})\big{)}.$$

$$\frac{1}{k+1}\sum _{s\in S}R({\alpha }_s([\tilde{x},y]+[x,\tilde{y}]+\lambda [\tilde{x},\tilde{y}]-R^{-1}([x,y])){\beta }_s)$$

$$[x,y]=[R(\tilde{x}),R(\tilde{y})]=R([\tilde{x},y]+[x,\tilde{y}]+\lambda [\tilde{x},\tilde{y}])$$

$$b_1=z_1,b_2={x_{q_1}^1}^{\prime },b_3={x_{q_1}^2}^{\prime },\dots ,b_{l_1+1}={x_{q_1}^{l_1}}^{\prime },b_{l_1+2}=z_2,b_{l_1+3}={x_{q_2}^1}^{\prime },\dots ,$$

$$b_{l_1+\dots +l_{s-1}+s}=z_s,b_{l_1+\dots +l_{s-1}+s+1}={x_{q_l}^1}^{\prime },\dots ,b_{l_1+\dots +l_s+s}={x_{q_s}^{l_s}}^{\prime },b_{l_1+\dots +l_s+s+1}=z_{s+1},$$

$$K=\{k_1,k_2,\dots ,k_p\mid k_1<k_2<\dots <k_p\}\subset \{1,2,\dots ,l_s\}.$$

$$A(z_1,{\vec{x}}_{q_1},z_2,\dots ,z_s,{\vec{x}}_{q_s},z_{s+1})\equiv 0\mathrm{mod}(S,w)$$

$${\vec{x}}_{q_s}={\vec{x}}_{q_s,0}{x}_{\beta ,r}^p+\sum _{i=1}^p{x}_{q_s}^1{x}_{q_s}^2\dots x_{q_s}^{k_i-1}\dots [x_{q_s}^{k_i},w_i]x_{\beta ,r}^{p-i},$$

$$[x_{q_s}^{k_i},w_i]=\sum _{j=1}^{l_s-k_i}{w}_i^1\dots w_i^{j-1}[x_{q_s}^{k_i},w_i^j]w_i^{j+1}\dots w_i^{l_s-k_i}.$$

$$A(z_1,{\vec{x}}_{q_1},z_2,\dots ,z_s,{\vec{x}}_{q_s},z_{s+1})\equiv \sum _{i=1}^p\sum _{j=1}^{l_s-k_i}A(z_1,{\vec{x}}_{q_1},z_2,\dots ,z_s,{\vec{x}}_{q_s}(i,j),z_{s+1})\mathrm{mod}(S,w),$$

$$R(w_{0})R(b)\mathop{\equiv}\limits^{\eqref{almostRB}}R(R(w_{0})b+w_{0}R(b)+\lambda w_{0}b)\\ \mathop{\equiv}\limits^{\eqref{LongRel1}}\frac{1}{k+1}R\big{(}R(z_{1})\vec{x}_{q_{1}}R(z_{2})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})b+R(\Sigma_{w})b\big{)}\\ +R\big{(}R(z_{1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r}x_{\beta,r+1}^{k}R(z_{s+1})R(b)+\lambda R(z_{1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r}x_{\beta,r+1}^{k}R(z_{s+1})b\big{)}\\ \mathop{\equiv}\limits^{\eqref{LongRel1}}\lambda R\big{(}R(z_{1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r}x_{\beta,r+1}^{k}R(z_{s+1})b\big{)}\\ +\frac{1}{k+1}\big{(}R\big{(}R(z_{1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})b+R(\Sigma_{w})b\big{)}\\ +R(z_{1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r}x_{\beta,r+1}^{k}R(z_{s+1})R(b)+R(\Sigma_{w|_{R(z_{s+1})\to R(z_{s+1})R(b)}})\big{)},$$

$$R(w_{0})R(b)\mathop{\equiv}\limits^{\eqref{LongRel1}}\frac{1}{k+1}\big{(}R(z_{1})\vec{x}_{q_{1}}R(z_{2})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})R(b)+R(\Sigma_{w})R(b)\big{)}\\ \mathop{\equiv}\limits^{\eqref{almostRB}}\frac{1}{k+1}\big{(}R(z_{1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})R(b)+R(R(\Sigma_{w})b+\Sigma_{w}R(b)+\lambda\Sigma_{w}b)\big{)}.$$

$$u=R\big{(}\lambda(k+1)R(z_{1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})b+R(z_{1})\ldots R(z_{s})\vec{x}_{q_{s}}x_{\beta,r+1}^{k+1}R(z_{s+1})b\big{)}\\ +R(\Sigma_{w|_{R(z_{s+1})\to R(z_{s+1})R(b)}})-R(\Sigma_{w}R(b)+\lambda\Sigma_{w}b).$$

$$\lambda {\Sigma }_{w}b=\lambda {\Sigma }_w^{\prime }R(z_{s+1})b+\lambda {\Sigma }_w^{\prime \prime }{z}_{s+1}b,$$

Full text

Keywords: Rota—Baxter operator, Gröbner—Shirshov basis, Poincaré—Birkhoff—Witt pair of varieties, pre-Lie algebra, postLie algebra, preassociative algebra (dendriform algebra), postassociative algebra.

1 Introduction

In 1960s, pre-Lie algebras appeared independently in affine geometry (E. Vinberg [43]; J.-L. Koszul [30]), and ring theory (M. Gerstenhaber [17]). Arising from diverse areas, pre-Lie algebras are known under different names like Vinberg algebras, Koszul algebras, left- or right-symmetric algebras (LSAs or RSAs), Gerstenhaber algebras. Pre-Lie algebras satisfy an identity $(x_{1}x_{2})x_{3}-x_{1}(x_{2}x_{3})=(x_{2}x_{1})x_{3}-x_{2}(x_{1}x_{3})$. See [7, 36] for surveys on pre-Lie algebras.

In 2001, J.-L. Loday [32] defined the dendriform (di)algebra (preassociative algebra) as a vector space endowed with two bilinear operations $\succ,\prec$ satisfying

[TABLE]

In 1995, J.-L. Loday also defined [31] Zinbiel algebra (precommutative algebra), on which the identity $(x_{1}x_{2}+x_{2}x_{1})x_{3}=x_{1}(x_{2}x_{3})$ holds. Every preassociative algebra with the identity $x\succ y=y\prec x$ is a precommutative algebra ($x_{1}x_{2}=x_{1}\succ x_{2}$) and under the product $x\cdot y=x\succ y-y\prec x$ is a pre-Lie algebra.

In 2004, dendriform trialgebra (postassociative algebra) was introduced [35], i.e., an algebra with bilinear operations $\prec,\succ,\cdot$ satisfying seven certain axioms. A space $A$ with two bilinear operations $[,]$ and $\cdot$ is called a post-Lie algebra (B. Vallette, 2007 [42]) if $[,]$ is a Lie bracket and the next identities hold

[TABLE]

In last dozen years, an amount of articles devoted to post-Lie algebras in different areas is arisen [8, 16, 38].

Let us explain the choice of terminology. Given a binary quadratic operad $\mathcal{P}$, the defining identities for pre- and post-$\mathcal{P}$-algebras were found in [2]. One can define the operad of pre- and post-$\mathcal{P}$-algebras as $\mathcal{P}\bullet\mathrm{pre}\mathrm{Lie}$ and $\mathcal{P}\bullet\mathrm{post}\mathrm{Lie}$ respectively. Here $\mathrm{pre}\mathrm{Lie}$ and $\mathrm{post}\mathrm{Lie}$ denote the operads (varieties) of pre-Lie algebras and postLie algebras respectively, $\bullet$ denotes the black Manin product of operads [18]. By pre- or postalgebra we will mean pre- or post-$\mathcal{P}$-algebra for some operad $\mathcal{P}$.

Before stating the main problem of the work we introduce a very useful tool to deal with pre- and postalgebras, so called Rot—Baxter operators, and the notion of a Poincaré—Birkhoff—Witt pair.

A linear operator $R$ defined on an algebra $A$ over a field $\Bbbk$ is called a Rota—Baxter operator (RB-operator, for short) of a weight $\lambda\in\Bbbk$ if it satisfies the relation

[TABLE]

In this case, an algebra $A$ is called Rota—Baxter algebra (RB-algebra).

G. Baxter defined the notion of what is now called Rota—Baxter operator on a (commutative) algebra in 1960 [4], solving an analytic problem. The relation (1) with $\lambda=0$ appeared as a generalization of integration by parts formula. G.-C. Rota [40], P. Cartier [9] and others studied different combinatorial properties of RB-operators and RB-algebras. In 1980s, the deep connection between Lie RB-algebras and Yang—Baxter equation was found [5, 41]. More about Rota—Baxter algebras see in the monograph of L. Guo [27].

In 2000, M. Aguiar [1] stated that an associative algebra with a given Rota—Baxter operator $R$ of weight zero under the operations $a\succ b=R(a)b$, $a\prec b=aR(b)$ is a preassociative algebra. In 2002, K. Ebrahimi-Fard [14] showed that an associative RB-algebra of nonzero weight $\lambda$ under the same two products $\succ$, $\prec$ and the third operation $a\cdot b=\lambda ab$ is a postassociative algebra. The analogue of the Aguiar construction for the pair of pre-Lie algebras and Lie RB-algebras of weight zero was stated in 2000 by M. Aguiar [1] and by I.Z. Golubchik, V.V. Sokolov [19]. In 2010 [3], this construction for the pair of post-Lie algebras and Lie RB-algebras of nonzero weight was extended.

In 2013 [2], the construction of M. Aguiar and K. Ebrahimi-Fard was generalized for the case of arbitrary variety.

In 2008, the notion of universal enveloping RB-algebras of pre- and postassociative algebras was introduced [15]. In [15], it was also proved that the universal enveloping of a free pre- or postassociative algebra is free.

In 2010, with the help of Gröbner—Shirshov bases [6], Yu. Chen and Q. Mo proved that every preassociative algebra over a field of characteristic zero injectively embeds into its universal enveloping RB-algebra [11].

In 2013 [25], given a variety $\mathrm{Var}$, it was proved that every pre-$\mathrm{Var}$-algebra (post-$\mathrm{Var}$-algebra) injectively embeds into its universal enveloping $\mathrm{Var}$-RB-algebra of weight $\lambda=0$ ($\lambda\neq 0$). Further, author constructed universal enveloping RB-algebra for a given pre- or postalgebra in commutative [20], associative [21], and Lie [22] cases. In the associative case it gave an answer to the question of L. Guo [27, p. 148].

The classical Poincaré—Birkhoff—Witt theorem states that given a Lie algebra $g$ with a linear basis $\{x_{i}\mid i\in I\}$, where $I$ is a well-ordered set, the monomials $x_{i_{1}}\ldots x_{i_{n}}$ with $i_{1}\leq\ldots\leq i_{n}$ form a linear basis for the universal enveloping associative algebra $U(g)$. As a consequence, we get that the linear basis of the algebra $U(g)$ does not depend on the product in the Lie algebra $g$. Such relationships between two varieties $\mathcal{V},\mathcal{W}$ in the case when there exists a functor $\phi\colon\mathcal{V}\to\mathcal{W}$ associating to every algebra $A\in\mathcal{V}$ an algebra $\phi(A)\in\mathcal{W}$ by changing multiplication in $A$ was generalized by I. Shestakov and A.A. Mikhalev in the term Poincaré—Birkhoff—Witt (PBW-) pair, see details in [37].

Now let us formulate the main problem to which the work is devoted. In advance, $\mathrm{pre}\mathrm{As}$ and $\mathrm{post}\mathrm{As}$ denote the varieties of pre- and postassociative algebras respectively.

Problem 1. a) Prove that every pre-Lie (postLie) algebra injectively embeds into its universal enveloping preassociative (postassociative) algebra.

b) Clarify if the pairs $(\mathrm{pre}\mathrm{As},\mathrm{pre}\mathrm{Lie})$ and $(\mathrm{post}\mathrm{As},\mathrm{post}\mathrm{Lie})$ are PBW-ones.

c) Construct the universal enveloping preassociative (postassociative) algebra for a given pre-Lie (postLie) algebra.

For pre-Lie algebras, Problem 1b and special version of Problem 1c were stated by P. Kolesnikov in [29] in the context of Gröbner—Shirshov bases for preassociative algebras. J.-L. Loday asked V. Dotsenko about the solution of Problem 1b around 2009 [13]. The discussion of Problem 1 in the case of restricted pre-Lie algebras can be found in [12]. The analogues of Problem 1 for Koszul-dual objects, di- and trialgebras, were solved in [34, 26].

Recently, in [24] and [23], the author solved Problem 1a in pre- and postalgebra cases with the help of embedding of pre-Lie (postLie) algebras into Lie RB-algebras [25] and the Gröbner—Shirshov bases technique developed for associative RB-algebras [28]. Actually, the solution of Problem 1a for pre-Lie algebras can be derived from the results concerned Hopf preassociative algebras and so called brace algebras stated in 2002 independently by F. Chapaton [10] and M. Ronco [39]. In 2018, V. Dotsenko and P. Tamaroff by means of the general approach arising from the category theory solved Problem 1b for pre-Lie algebras stating that the pair of varieties $(\mathrm{pre}\mathrm{Lie},\mathrm{pre}\mathrm{As})$ is a PBW-pair [13].

The current work is devoted to the complete solution of Problem 1 in both pre- and postalgebra cases. Let us briefly describe the idea of the solution. For this, we need one more embedding problem.

Let $A$ be an associative algebra with an RB-operator $R$. Then the algebra $A^{(-)}$ is a Lie RB-algebra under the product $[x,y]=xy-yx$ and the same action of $R$. Thus, we can state the analogue of Problem 1 for the varieties of Lie and associative RB-algebras.

Problem 2. a) Prove that every Lie RB-algebra injectively embeds into its universal enveloping associative RB-algebra.

b) Construct the universal enveloping associative RB-algebra for a given Lie RB-algebra.

We are not asking whether the pair $(\textrm{RB}\mathrm{As},\textrm{RB}\mathrm{Lie})$ of the varieties of associative and Lie RB-algebras forms a PBW-pair, since it is easy to disprove it just in 2-dimensional case. To the moment, we are far from the solution Problem 2, and the current work as well as [23, 24] can be also considered like a step in such direction.

The sketch of the solution of Problem 1c is following. At first, we embed a pre- or post-Lie algebra $C$ into its universal enveloping Lie RB-algebra $L$ [22]. At second, with the help of Gröbner—Shirshov bases we embed the Lie RB-algebra $L$ into its universal enveloping associative RB-algebra $A$ with an RB-operator $R$. At third, we show that a subalgebra $U(C)$ generated by $C$ in the induced pre- or postassociative algebra on the space $A$ is the universal enveloping pre- or postassociative algebra for $C$.

Actually, the same solution algorithm was used earlier by author in [23] and [24] but with the following change: in the first step it was considered injective enveloping Lie RB-algebra from [25] instead of $L$, the universal enveloping one constructed in [22]. Surprisingly, a posteriori we may say that at least in the pre-Lie algebra case the injective enveloping preassociative algebra of a given pre-Lie algebra obtained earlier in [24] is an universal one.

At the end of Introduction, let us collect all stated in the work connections between universal enveloping algebras of different kind in the following commutative diagram,

\textstyle{\mathrm{pre}/\mathrm{post}\mathrm{Lie}}$$\textstyle{\textrm{RB}\mathrm{Lie}}$$\textstyle{\mathrm{pre}/\mathrm{post}\mathrm{As}}$$\textstyle{\textrm{RB}\mathrm{As}}

where every arrow maps the algebra from corresponding variety into its universal enveloping one. An associative RB-algebra $A$ with an RB-operator $P$ of weight $\lambda=0$ ($\lambda\neq 0$) is an enveloping for a pre-Lie (postLie) algebra $C$ in the sense that the following equalities

[TABLE]

hold for any $a,b\in C$.

2 Preliminaries

2.1 Some required formulas

Given a Lie algebra $L$, denote the product $[[\ldots[[y,x],x]\ldots],x]\in L^{p+1}$ by $[y,x^{(p)}]$.

Lemma [24]. Given a Lie algebra $L$, the equality

[TABLE]

holds in the universal enveloping algebra $U(L)$ for any $x,y\in L$ and $l\geq 0$.

It follows immediately from (1) that

[TABLE]

where the sign $\hat{}$ shows the omitting action of $R$. From (3) and (4), we derive the formula

[TABLE]

where $b_{l}=b_{l+1}=\ldots=b_{l+k}$ and $R(b_{l})^{s}$ means $(R(b_{l}))^{s}$.

In advance we will use the formula (5) with maybe absent $R(b_{1})$ and $R(b_{l+k+1})$, it means that we omit all $b_{1}$ and $R(b_{1})$ in the summands and all indexes $i_{j}$ start with two when $R(b_{1})$ is absent. We do the same if $R(b_{l+k+1})$ is absent.

2.2 Embedding of pre- and postalgebras into RB-algebras

Theorem 1 [1, 2, 3, 14, 19, 33]. Let $A$ be an RB-algebra of a variety $\mathrm{Var}$ and weight $\lambda=0$ ($\lambda\neq 0$). With respect to the operations

[TABLE]

$A$ is a pre-$\mathrm{Var}$-algebra (post-$\mathrm{Var}$-algebra).

Denote the pre- and post-$\mathrm{Var}$-algebra obtained in Theorem 1 as $A^{(R)}_{\lambda}$.

Given a pre-$\mathrm{Var}$-algebra $\langle C,\succ,\prec\rangle$, universal enveloping RB-$\mathrm{Var}$-algebra $U$ of $C$ is the universal algebra in the class of all RB-$\mathrm{Var}$-algebras of weight zero such that there exists homomorphism from $C$ to $U^{(R)}_{0}$. Analogously universal enveloping RB-$\mathrm{Var}$-algebra of a post-$\mathrm{Var}$-algebra is defined.

Theorem 2 [25]. Every pre-$\mathrm{Var}$-algebra (post-$\mathrm{Var}$-algebra) could be embedded into its universal enveloping RB-algebra of the variety $\mathrm{Var}$ and weight $\lambda=0$ ($\lambda\neq 0$).

Let us briefly describe the idea of the proof of Theorem 2. Given a pre-$\mathrm{Var}$-algebra (post-$\mathrm{Var}$-algebra) $C$, we define the product on the space $\hat{C}=C\oplus C^{\prime}$, where $C^{\prime}$ is a copy of $C$, in such a way that $\hat{C}$ is an algebra of the variety $\mathrm{Var}$. Then we define on $\hat{C}$ the linear operator $P$ which occurs to be a Rota—Baxter operator of weight $\lambda=0$ ($\lambda\neq 0$). Finally, Theorem 2 was stated by embedding $C$ into $\hat{C}^{(P)}_{\lambda}$ by the map $c\to c^{\prime}$.

2.3 Gröbner—Shirshov bases for associative RB-algebras

Let $R\mathrm{As}\langle X\rangle$ denote the free associative algebra generated by a set $X$ with a linear map $R$ in the signature. One can construct a linear basis of $R\mathrm{As}\langle X\rangle$ (see, e.g., [15]) by induction. At first, all elements from $S(X)$, the free semigroup generated by $X$, lie in the basis. At second, if we have basic elements $a_{1},a_{2},\ldots,a_{k}$, $k\geq 1$, then the word $w_{1}R(a_{1})w_{2}\ldots w_{k}R(a_{k})w_{k+1}$ lies in the basis of $R\mathrm{As}\langle X\rangle$. Here $w_{2},\ldots,w_{k}\in S(X)$ and $w_{1},w_{k+1}\in S(X)\cup\emptyset$, where $\emptyset$ denotes the empty word. Let us denote the basis obtained as $RS(X)$. Given a word $u$ from $RS(X)$, the number of appearances of the symbol $R$ in $u$ is denoted by $\deg_{R}(u)$, the $R$-degree of $u$. We call an element from $RS(X)$ of the form $R(w)$ as $R$-letter. By $X_{\infty}$ we denote the union of $X$ and the set of all $R$-letters. Given $u\in RS(X)$, define $\deg u$ (degree of $u$) as the length of $u$ in the alphabet $X_{\infty}$. In [15], $\deg u$ was called the breadth of $u$.

Suppose that $X$ is a well-ordered set with respect to $<$. Let us introduce by induction the deg-lex order on $S(X)$. At first, we compare two words $u$ and $v$ by the length: $u<v$ if $|u|<|v|$. At second, when $|u|=|v|$, $u=x_{i}u^{\prime}$, $v=x_{j}v^{\prime}$, $x_{i},x_{j}\in X$, we have $u<v$ if either $x_{i}<x_{j}$ or $x_{i}=x_{j}$, $u^{\prime}<v^{\prime}$. We compare two words $u$ and $v$ from $RS(X)$ by $R$-degree: $u<v$ if $\deg_{R}(u)<\deg_{R}(v)$. If $\deg_{R}(u)=\deg_{R}(v)$, we compare $u$ and $v$ in deg-lex order as words in the alphabet $X_{\infty}$. Here we define each $x$ from $X$ to be less than all $R$-letters and $R(a)<R(b)$ if and only if $a<b$.

Let $*$ be a symbol not containing in $X$. By a $*$-bracketed word on $X$, we mean a basic word from $R\mathrm{As}\langle X\cup\{*\}\rangle$ with exactly one occurrence of $*$. The set of all $*$-bracketed words on $X$ is denoted by $RS^{*}(X)$. For $q\in RS^{*}(X)$ and $u\in R\mathrm{As}\langle X\rangle$, we define $q|_{u}$ as the bracketed word obtained by replacing the letter $*$ in $q$ by $u$.

The order defined above is monomial, i.e., from $u<v$ it follows that $q|_{u}<q|_{v}$ for all $u,v\in RS(X)$ and $q\in RS^{*}(X)$.

Given $f\in R\mathrm{As}\langle X\rangle$, by $\bar{f}$ we mean the leading word in $f$. We call $f$ monic if the coefficient of $\bar{f}$ in $f$ is 1.

Definition 1 [28]. Let $f,g\in R\mathrm{As}\langle X\rangle$. If there exist $\mu,\nu,w\in RS(X)$ such that $w=\bar{f}\mu=\nu\bar{g}$ with $\deg w<\deg(\bar{f})+\deg(\bar{g})$, then we define $(f,g)_{w}$ as $f\mu-\nu g$ and call it the composition of intersection of $f$ and $g$ with respect to $(\mu,\nu)$. If there exist $q\in RS^{*}(X)$ and $w\in RS(X)$ such that $w=\bar{f}=q|_{\bar{g}}$, then we define $(f,g)^{q}_{w}$ as $f-q|_{g}$ and call it the composition of inclusion of $f$ and $g$ with respect to $q$.

Definition 2 [28]. Let $S$ be a subset of monic elements from $R\mathrm{As}\langle X\rangle$ and $w\in RS(X)$.

(1) For $u,v\in R\mathrm{As}\langle X\rangle$, we call $u$ and $v$ congruent modulo $(S,w)$ and denote this by $u\equiv v\mod(S,w)$ if $u-v=\sum c_{i}q_{i}|_{s_{i}}$ with $c_{i}\in\Bbbk$, $q_{i}\in RS^{*}(X)$, $s_{i}\in S$ and $q_{i}|_{\overline{s_{i}}}<w$.

(2) For $f,g\in R\mathrm{As}\langle X\rangle$ and suitable $w,\mu,\nu$ or $q$ that give a composition of intersection $(f,g)_{w}$ or a composition of inclusion $(f,g)^{q}_{w}$, the composition is called trivial modulo $(S,w)$ if $(f,g)_{w}$ or $(f,g)^{q}_{w}\equiv 0\mod(S,w)$.

(3) The set $S\subset R\mathrm{As}\langle X\rangle$ is called a Gröbner—Shirshov basis if, for all $f,g\in S$, all compositions of intersection $(f,g)_{w}$ and all compositions of inclusion $(f,g)^{q}_{w}$ are trivial modulo $(S,w)$.

Theorem 3 [28]. Let $S$ be a set of monic elements in $R\mathrm{As}\langle X\rangle$, let $<$ be a monomial ordering on $RS(X)$ and let $Id(S)$ be the $R$-ideal of $R\mathrm{As}\langle X\rangle$ generated by $S$. If $S$ is a Gröbner—Shirshov basis in $R\mathrm{As}\langle X\rangle$, then $R\mathrm{As}\langle X\rangle=\Bbbk Irr(S)\oplus Id(S)$ where $Irr(S)=RS(X)\setminus\{q|_{\bar{s}}\mid q\in RS^{*}(X),s\in S\}$ and $Irr(S)$ is a linear basis of $R\mathrm{As}\langle X\rangle/Id(S)$.

3 PBW-theorem for pre-Lie and postLie algebras

Let $A$ be an associative algebra with an RB-operator $R$. Then the algebra $A^{(-)}$ is a Lie RB-algebra under the product $[x,y]=xy-yx$ and the same action of $R$.

Let $L$ be a Lie RB-algebra with an RB-operator $P$ of weight $\lambda$. Suppose that there exists a subset $X_{0}=\{x_{\alpha}\mid\alpha\in\Omega\}$ in $L$ such that $X=\{x_{\alpha,k}:=P^{k}(x_{\alpha})\mid k\in\mathbb{N},\,\alpha\in\Omega\}$ is a linear basis of $L$. Our goal is to construct the universal enveloping associative RB-algebra of $L$ (via Gröbner—Shirshov bases). This will lead us to the proof of the Poincaré—Birkhoff—Witt (PBW) theorem for the pairs (pre-Lie, preAs) and (postLie, postAs).

We may assume that the set $\Omega$ is well-ordered, so we define an order $<$ on the set $X$: $x_{\alpha,k}<x_{\beta,l}$ if $\alpha<\beta$ or $\alpha=\beta$ and $k<l$.

Consider the set $S$ of the following elements in $R\mathrm{As}\langle X\rangle$:

[TABLE]

where $\Delta$ is defined as the RHS of (5) for $l=l_{1}+\ldots+l_{s}+s+1$ and

[TABLE]

Here $\vec{x}_{q_{i}}=x_{q_{i}}^{1}\ldots x_{q_{i}}^{l_{i}}$ for $x_{q_{i}}^{j}\in X$ and $\tilde{x}$ denotes $x_{\alpha,t-1}$ for $x=x_{\alpha,t}\in X$, i.e., $R(\tilde{x})=x$.

In (8), (9), we have

[TABLE]

and $x_{\beta,r}$ is greater than any letter from $\vec{x}_{q_{s}}$.

By $R(z_{1})$ we denote either that $z_{1}\in RS(X)\setminus S(X)$ or that $R(z_{1})$ is absent, i.e., $R(z_{1})=\emptyset$. The same holds for $R(z_{s+1})$. In particular, the values $s=1$, $k=0$, $R(z_{1})=R(z_{2})=\vec{x}_{q_{1}}=\emptyset$, transform (9) to the relation $R(x_{\beta,r})-x_{\beta,r+1}$.

Remark 1. In (9), we use associative words $\vec{x}_{\alpha}\in S(X)$ instead of ordered polynomials from $\Bbbk[X]$, otherwise we will have to reduce the products of such polynomials from $\Bbbk[X]$ to the ordered ones in all possible compositions from $S$.

Theorem 4. The set $S$ is a a Gröbner—Shirshov basis in $R\mathrm{As}\langle X\rangle$.

Proof. All compositions between two elements from (7) are trivial, as it is the method to construct the universal enveloping associative algebra for a given Lie algebra. Also, compositions of intersection between (8) and (8) are trivial, it is a way to get the free associative RB-algebra. Thus, all compositions of intersection which are not at the same time compositions of inclusion are trivial.

Let us compute a composition of inclusion between (7) and (9). Let

[TABLE]

satisfy all conditions described above. We apply the relation (7): $xy-yx-[x,y]$ to the subword

[TABLE]

Suppose that $j<s$. Since the image of an RB-operator is a subalgebra, $[x,y]$ lies in the linear combination of elements from $S(X\setminus X_{0})$. Define $w^{\prime}=w|_{xy\to yx}$ and $w^{\prime\prime}=w|_{xy\to[x,y]}$, here $w|_{\alpha\to\beta}$ means the word $w$ with the subword $\alpha$ replaced by $\beta$.

On the one hand, we have modulo $(S,w)$

[TABLE]

where $\Sigma_{w}$ is the expression in the RHS in the brackets under the action of $R$ from (5).

On the other hand, modulo $(S,w)$

[TABLE]

So, the composition equals $\dfrac{1}{k+1}(R(\Sigma_{w})-R(\Sigma_{w^{\prime}})-R(\Sigma_{w^{\prime\prime}}))$ and we may rewrite it briefly as

[TABLE]

for corresponding index set $S$ and $\alpha_{s},\beta_{s}\in RS(X)$. We get zero, since

[TABLE]

and $R$ has zero kernel on $L$.

Consider the case $j=s$. The triviality of the corresponding composition of inclusion one can derive from the following fact. Denote as $A(z_{1},\vec{x}_{q_{1}},z_{2},\ldots,z_{s},\vec{x}_{q_{s}},z_{s+1})$ the expression (4) for $t=l_{1}+\ldots+l_{s}+s+1$,

[TABLE]

where $z_{1},\ldots,z_{s+1}$ and the words $\vec{x}_{q_{i}}=x_{q_{i}}^{1}\ldots x_{q_{i}}^{l_{i}}$ satisfy the conditions for (9) except the one concerned $x_{\beta,r}$. We also have that the word $\vec{x}_{q_{s}}=x_{q_{1}}^{1}\ldots x_{q_{1}}^{l_{s}}$ contains the biggest letter $x_{\beta,r}$, $r\geq 1$, on the positions

[TABLE]

So, the composition of inclusion is trivial if

[TABLE]

for $w$ greater than all terns involved in $A(\ldots)$.

To prove (11), we will proceed on by induction on $l_{s}=|\vec{x}_{q_{s}}|$. For $l_{s}=1$, we are done by (9).

Consider the equality

[TABLE]

where $\vec{x}_{q_{s},0}$ is obtained from $\vec{x}_{q_{s}}$ by arising all letters $x_{\beta,r}$ with preserving order of all remaining letters, $w_{i}$ is obtained from the word $x_{q_{s}}^{k_{i}+1}\ldots x_{q_{s}}^{l_{s}}$ by arising all $p-i$ letters $x_{\beta,r}$. For $w_{i}=w_{i}^{1}w_{i}^{2}\ldots w_{i}^{l_{s}-k_{i}}$, $w_{i}^{j}\in X$, the bracket $[x_{q_{s}}^{k_{i}},w_{i}]$ in (12) means

[TABLE]

By (9) and Lemma, we deduce that

[TABLE]

where $\vec{x}_{q_{s}}(i,j)=x_{q_{s}}^{1}\ldots x_{q_{s}}^{k_{i}-1}\ldots w_{i}^{1}\ldots w_{i}^{j-1}[x_{q_{s}}^{k_{i}},w_{i}^{j}]w_{i}^{j+1}\ldots w_{i}^{l_{s}-k_{i}}x_{\beta,r}^{p-i}$. The equality $A(z_{1},\ldots,z_{s},\vec{x}_{q_{s}}(q,i),z_{s+1})\equiv 0$ modulo $(S,w)$ follows from the inductive hypothesis.

Consider a composition of inclusion between (8) and (9). Let $w=R(w_{0})$ be defined by (10) and $b\in RS(X)$. At first, we have modulo $(S,w)$

[TABLE]

where $\Sigma_{w}$ is the expression in the RHS in the brackets under the action of $R$ from (5). At second, modulo $(S,w)$

[TABLE]

So, the composition of inclusion multiplied by $(k+1)$ equals

[TABLE]

Depending on the last factor, $\Sigma_{w}$ splits into the sum $\Sigma_{w}^{\prime}R(z_{s+1})+\Sigma_{w}^{\prime\prime}z_{s+1}$. Applying the formulas

[TABLE]

we deduce

[TABLE]

Writing down the sum $R(\Sigma_{w}^{\prime\prime}R(z_{s+1})b)$, $u$ equals to zero by the definition of $\Sigma^{\prime}_{w}$, $\Sigma^{\prime\prime}_{w}$ and Lemma.

Compute a composition of inclusion between (9) and (9). Suppose that we have $w$ defined by (10) and

[TABLE]

satisfying all conditions written above for (7).

Consider the case $m\leq s$. By $\Sigma_{w}$, as earlier, we mean the expression in the RHS in the brackets under the action of $R$ from (5) for $w$. By $\Delta$, denote the expression in the RHS in the brackets under the action of $R$ from (5) for $z_{m}$. We also define

[TABLE]

$\Sigma_{w}=\Sigma_{w}^{\prime}+\Sigma_{w}^{\prime\prime}$, where we collect all summands from $\Sigma_{w}$ with the factor $R(z_{m})$ in the sum $\Sigma_{w}^{\prime}$ and all others in $\Sigma_{w}^{\prime\prime}$.

On the one hand, modulo $(S,w)$ we get

[TABLE]

On the other hand,

[TABLE]

Subtracting (14) from (13), we get $\frac{1}{(k+1)(m+1)}R(\Sigma_{w}^{\prime\prime}|_{z_{m}\to C})$, where

[TABLE]

for $D=H-(m+1)x_{\gamma,o}x_{\gamma,o+1}^{m}$. Equality of $D$ to zero follows from Lemma.

The proof in the case $m=s+1$ is slightly different if only $R(a_{1})=\emptyset$ and $n=1$. Then we need to apply the equality A\big{(}z_{1},\vec{x}_{q_{1}},z_{2},\ldots,z_{s},\vec{x}_{q_{s}}x_{\beta,r}x_{\beta,r+1}^{k}\vec{x}_{t_{1}}x_{\gamma,o+1}^{m+1},a_{2}\big{)}\equiv 0 modulo $(S,w)$. Such application is correct, since all terms involved in it have less $R$-degree than $w$.

It is easy to verify that the remaining compositions of inclusion between (7) and (8) are trivial. $\Square$

Corollary 1. The quotient $A$ of $R\mathrm{As}\langle X\rangle$ by $Id(S)$ is the universal enveloping associative RB-algebra for the Lie algebra $L$ with the RB-operator $R$. Moreover, $L$ injectively embeds into $A^{(-)}$.

Proof. By (8), $A$ is an associative RB-algebra. By (7)–(9), we have that $A$ is enveloping of $L$ for both: the Lie bracket $[,]$ and the action of $R$. Thus, $A$ is an associative enveloping of $L$.

Let us prove that $A$ is the universal enveloping one. At first, $A$ is generated by $L$. At second, all elements from $S$ are identities in the universal enveloping associative RB-algebra $U_{RB}(L)$. Indeed, (7) are enveloping conditions for the product, (8) is the RB-identity, the relations (9) as the application of (5) are direct consequences of the RB-identity.

By Theorems 3 and 4 we get the injectivity of embedding $L$ into $A^{(-)}$. $\Square$

Let $C$ be a pre- or post-Lie algebra with a linear basis $Y$. By Theorem 2, $C$ can be injectively embedded into the Lie algebra $L$ with the RB-operator $P$ of weight $\lambda$ and such subset $X_{0}=\{x_{\alpha}\mid\alpha\in\Omega\}\subset L$ that $X=\{x_{\alpha,k}:=P^{k}(x_{\alpha})\mid k\in\mathbb{N},\,\alpha\in\Omega\}$ is a linear basis of $L$. Here $\Omega$ is a well-ordered set. Then, by Corollary 1, we embed the Lie RB-algebra $L$ into its universal enveloping associative algebra $A$ with the RB-operator $R$. Thus, the subalgebra (in pre- or postalgebra sense) $U(C)$ in $A^{(R)}_{\lambda}$ generated by the set $Y$ is an enveloping pre- or postassociative algebra of $C$.

Now, we state the main result of the work, the analogue of the Poincaré—Birkhoff—Witt theorem for pre-Lie and postLie algebras.

Theorem 5. a) Let $C$ be a pre- or post-Lie algebra, then $U(C)$ is the universal enveloping pre- or postassociative algebra of $C$.

b) [13] The pairs $(\mathrm{pre}\mathrm{Lie},\mathrm{pre}\mathrm{As})$ and $(\mathrm{post}\mathrm{Lie},\mathrm{post}\mathrm{As})$ are PBW-pairs.

Proof. a) Consider the postLie algebra case, the proof when $C$ is a pre-Lie algebra is the same. Let $Y$ be a basis of $C$. It is easy to show that $A$ is the universal enveloping associative RB-algebra for $C$ in the sense of the equalities (2). Indeed, given an enveloping associative RB-algebra $D$ of $C$, define $M$ as the Lie RB-subalgebra of $D^{(-)}$ generated by the image of $C$. By the universality of $L$, we have that $M$ is the homomorphic image of the Lie RB-algebra $L$. Thus, $D$ as the enveloping of $M$ is the homomorphic image of the universal enveloping associative RB-algebra of $M$. The last one is the homomorphic image of $A$.

Consider the universal enveloping postassociative algebra $V(C)$ of $C$. Due to Theorem 2, we may embed $V(C)$ in its universal enveloping associative RB-algebra $Z$ with RB-operator $Q$. Since $Z$ is the homomorphic image of $A$, $V(C)$ as the subalgebra in $Z^{(Q)}_{\lambda}$ generated by $Y$ is the homomorphic image of $U(C)$.

b) We get it by the construction. $\Square$

Corollary 2. The pairs $(\mathrm{pre}\mathrm{Lie},\textrm{RB}\mathrm{As})$ and $(\mathrm{post}\mathrm{Lie},\textrm{RB}\mathrm{As})$ are PBW-pairs.

Corollary 3. The universal enveloping associative RB-algebra of $U(C)$ is isomorphic to $A$.

As another corollary, we obtain the commutative diagram from Introduction.

4 Universal enveloping pre/post-associative algebra

4.1 Post-Lie case

Given a postLie algebra $C$ with a linear basis $Y$, we want to construct a linear basis of the universal enveloping postassociative algebra $U(C)$. Define $Y^{+}=Y\cup Y^{\prime}\subset X\subset L$ for $Y^{\prime}=\{P(y)\mid y\in Y\}$.

Given a well-ordered set $Z$, define $\mathrm{Com}(Z):=\{w=w_{1}\ldots w_{k}\in S(Z)\mid w_{j}\in Z,\,w_{1}\leq w_{2}\leq\ldots\leq w_{k}\}$.

Let us define a set $E\subset RS(Y)$ by induction. At first, $Y\subset E$. At second, the word

[TABLE]

lies in $E$ for $s\geq 1$, $w_{1},\ldots,w_{s}\in\mathrm{Com}(Y^{+})$, $z_{2},\ldots,z_{s}\in E\cap RS(Y)\setminus Y$, $z_{1}\in E\cap RS(Y)\setminus Y$ or $R(z_{1})=\emptyset$ (the same holds for $z_{s+1}$), if

1. at least one of $w_{i}$ contains a letter from $Y$,

2. every $z_{i}$ is not of the following form

[TABLE]

with the same conditions written for $a$. Moreover, $u_{1},\ldots,u_{t}\in\mathrm{Com}(Y^{\prime})$ and $y\in Y$ is greater than all letters from $u_{t}$.

Theorem 6. The set $\{e+Id(S)\mid e\in E\}$ forms a linear basis of $U(C)$.

Proof. At first, Theorems 3 and 4 imply that the set $\{e+Id(S)\mid e\in E\}$ is a linearly independent set of elements in $R\mathrm{As}\langle Y\rangle$.

At second, we show that $U(C)\subset\Bbbk E+Id(S)$. Let us prove it by induction of the summary $R$-degree $r$ of factors $a,b$ involved in the process of generating the algebra $U(C)$. By the definition, $Y\subset E$. If $r=0$, we get only linear combinations of elements of the form $w\in S(Y^{+})$ with at least one letter from $Y$, where we identify $R(y)$ with $P(y)$ for every $y\in Y$. Because of (7), we have $w\in E+Id(S)$.

Let us prove the inductive step for $r>0$. Suppose that $a,b\in E$, more precisely,

[TABLE]

where

[TABLE]

by $R(z_{1})$, as above, we mean that either $z_{1}\in E\cap RS(Y)\setminus Y$ or $R(z_{1})=\emptyset$. The same holds for $R(z_{s+1}),R(q_{1}),R(q_{t+1})$.

To prove Theorem, it is enough to state that $a\succ b,a\prec b,a\cdot b\in\Bbbk E+Id(S)$. We have modulo $Id(S)$

[TABLE]

where $z_{s+1}\circ q_{1}=z_{s+1}\succ q_{1}+z_{s+1}\prec q_{1}+z_{s+1}\cdot q_{1}$. In the third case, we have an element from $E$. In the first one, we need to express $w_{s}u_{1}\in U(L)$ via basic elements from $\mathrm{Com}(Y^{+})$ and so $a\cdot b\in\Bbbk E$. Finally, in the second case, after rewriting $z_{s+1}\circ q_{1}$ as a linear combination of elements from $E+Id(S)$, we only need to check the condition 2 of the definition of $E$. If it’s required, we apply the relation (9) and we are done by induction on $r$.

We have modulo $Id(S)$

[TABLE]

By the inductive hypothesis, $a\circ q_{1}\in\Bbbk E+Id(S)$. We need to check the condition 2 of the definition of $E$ for the first $R$-letter of the product. As above, we apply (9).

The case of $a\prec b$ can be considered analogously. $\Square$

Remark 2. We may reformulate Theorem 6 in terms avoiding $Id(S)$ and any factorization. For this, we need to define by induction the products $\succ,\prec,\cdot$ on the space $\Bbbk E$.

4.2 Pre-Lie case

Given a pre-Lie algebra $C$ with a linear basis $Y$, we construct a linear basis of the universal enveloping preassociative algebra $U(C)$.

Let us define a set $E\subset RS(Y)$ by induction. At first, $Y\subset E$. At second, the word

[TABLE]

lies in $E$ for $s\geq 1$, $w_{1},\ldots,w_{s}\in\mathrm{Com}(Y^{+})$, $z_{2},\ldots,z_{s}\in E\cap RS(Y)\setminus Y$, $z_{1}\in E\cap RS(Y)\setminus Y$ or $R(z_{1})=\emptyset$ (the same holds for $z_{s+1}$), if

1. the only $w_{i}$ contains a letter from $Y$,

2. every $z_{i}$ is not of the following form

[TABLE]

with the same conditions written for $a$; $u_{1},\ldots,u_{t}\in\mathrm{Com}(Y^{\prime})$ and $y\in Y$ is greater than all letters from $u_{t}$.

Theorem 7. The set $\{e+Id(S)\mid e\in E\}$ forms a linear basis of $U(C)$.

Proof. Analogously to the proof of Theorem 6. $\Square$

Example. If $C$ is a one-dimensional pre-Lie algebra with linear basis $\{y\}$, then $\{y(P(y))^{k}\}$ is a linear basis of $U(C)$. Indeed, because of the condition 2, all basic elements have no $R$-letters, so we have unique $y$ and several $P(y)$ to form the elements from $E$. Moreover, in the case $C=\mathrm{Span}\{y\}$ we have the isomorphism $U_{gr}(C)\cong\mathrm{pre}\mathrm{Com}\langle y\rangle$, where $U_{gr}(C)$ means the associated graded preassociative algebra obtained from $U(C)$ by the filtration by the degree and $\mathrm{pre}\mathrm{Com}\langle Z\rangle$ means a free precommutative algebra generated by the set $Z$. If $C=\mathrm{Span}\{y\}$ has the trivial product, then $U(C)\cong\mathrm{pre}\mathrm{Com}\langle y\rangle$.

Remark 3. Note that given a pre-Lie algebra $C$, the injective enveloping preassociative algebra $T$ constructed in [24] is isomorphic to $U(C)$, so it is the universal enveloping one. In the current paper, we have embedded $C$ into its universal enveloping Lie RB-algebra, but in [24] $C$ was embedded into the enveloping Lie RB-algebra $\hat{C}$ arisen from the proof of Theorem 2. A posteriori, we conclude that it is enough to embed $C$ into the doubling (as a vector space) Lie RB-algebra to preserve all required connections for the construction of the universal enveloping preassociative algebra $U(C)$.

Acknowledgements

This work was supported by the Austrian Science Foundation FWF grant P28079.