# Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras

^{1}

^{2}

## Abstract

**:**

_{λ}Com, postCom) is proved to be a Poincaré–Birkhoff–Witt-pair (PBW)-pair and the pair (RBCom, preCom) is proven not to be.

## 1. Introduction

## 2. Preliminaries

#### 2.1. RB-Operator

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 2.2. Yang–Baxter Equation

**Example**

**4.**

**Proposition**

**1**

**.**Let $r=\sum {a}_{i}\otimes {b}_{i}$ be a solution of the AYBE on an associative algebra A. A linear map $R:A\to A$ defined as $R\left(x\right)=\sum {a}_{i}x{b}_{i}$ is a RB-operator on A of zero weight.

**Example 5**

**.**Up to conjugation, transpose and a scalar multiple all nonzero solutions of the AYBE on ${M}_{2}\left(\mathbb{C}\right)$ are $({e}_{11}+{e}_{22})\otimes {e}_{12}$, ${e}_{12}\otimes {e}_{12}$, ${e}_{22}\otimes {e}_{12}$, and ${e}_{11}\otimes {e}_{12}-{e}_{12}\otimes {e}_{11}$.

#### 2.3. PBW-Pair of Varieties

#### 2.4. Free Commutative RB-Algebra

**Example**

**6.**

- —
- all monomials from $F\left[X\right]$ lie in T;
- —
- if $u\in T$, then $R\left(u\right),R\left(u\right)w\in T$ for a monomial w from $F\left[X\right]$.

#### 2.5. Pre-Commutative Algebra

#### 2.6. Post-Commutative Algebra

#### 2.7. Embedding of Loday Algebras into RB-Algebras

**Theorem 1**

**.**

- (a)
- Any pre-$\mathrm{Var}$-algebra can be embedded into its universal enveloping RB-algebra of the variety $\mathrm{Var}$ and zero weight.
- (b)
- Any post-$\mathrm{Var}$-algebra can be embedded into its universal enveloping RB-algebra of the variety $\mathrm{Var}$ and nonzero weight.

## 3. Universal Enveloping Rota–Baxter Algebra of Pre-Commutative Algebra

- (1)
- elements of ${E}_{0}$ are E-words of type 1;
- (2)
- given an E-word u, we define $R\left(u\right)$ as an E-word of type 2;
- (3)
- given an E-word x, we define $R\left(x\right)w$ for $w\in {E}_{0}$ as an E-word of type 3 if $R\left(x\right)$ is good.

**Example**

**7.**

- (a)
- ${w}_{1},{w}_{2}\in {E}_{0}$;
- (b)
- $R\left(x\right)$ is good (otherwise $R\left(x\right){w}_{1}$ is not an E-word);
- (c)
- the length of ${w}_{1}$ is greater than 1 (otherwise $R\left(R\left(x\right){w}_{1}\right)$ is not a good E-word).

**Theorem**

**2.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

- (a)
- Let $x=R\left(b\right)$, $b\in B$. Recall that $A=F\left[B\right]/I$ for $I=\langle (d\succ a)c-(d\succ c)a,\phantom{\rule{0.166667em}{0ex}}a,d,c\in B\rangle $. Define a map $\u22a2:F\left[B\right]\to A$ on monomials as follows:$$\u22a2\left(aw\right)=(b\succ a)\xb7w$$We check that $\u22a2\left(I\right)=0$. By Equation (6) and the definition of I,$$\begin{array}{ll}b\ast \left(\right(d\succ a)cu-(d\succ c\left)au\right)& =(b\succ (d\succ a\left)\right)\xb7cu-(b\succ (d\succ c\left)\right)\xb7au\\ & =u\xb7\left(\right(b\succ (d\succ a))\xb7c-(b\succ (d\succ c))\xb7a)\\ & =u\xb7\left(\right((b\succ d+d\succ b)\succ a)\xb7c-((b\succ d+d\succ b)\succ c)\xb7a)=0\end{array}$$Thus, the map ⊢ can be considered as a map $\u22a2:A\to A$, and it coincides on ${E}_{0}$ with the map $z\to R\left(b\right)\ast z$. Finally,$$(x,y,z)=\left(R\right(b),a,c)=(b\succ a)\xb7c-R\left(b\right)\ast (a\xb7c)=(b\succ a)\xb7c-b\u22a2ac=0$$
- (b)
- The case $x=R\left(R\right(y\left)b\right)$, $b\in B$, can be derived from (a).
- (c)
- If x is good, then associativity follows from Lemma 1.

- (a)
- Let $y=R\left(c\right)$, $c\in B$.
- (a1)
- If $x=R\left(b\right)$, $b\in B$, then by Equation (6),$$\begin{array}{ll}(x,y,z)& =R\left(R\right(b)\ast c+b\ast R(c\left)\right)\ast a-R\left(b\right)\ast (c\succ a)\\ & =R(b\succ c+c\succ b)\ast a-(b\succ (c\succ a\left)\right)\\ & =\left(\right(b\succ c+c\succ b)\succ a)-(b\succ (c\succ a\left)\right)=0\end{array}$$
- (a2)
- If $x=R\left(R\right(p\left)b\right)$, $b\in B$, then compute$$\begin{array}{ll}(x,y,z)& =\left(R\right(R\left(p\right)b)\ast R(c\left)\right)\ast a-R\left(R\right(p\left)b\right)\ast (c\succ a)\\ & =R\left(R\right(R\left(p\right)b)\ast c)\ast a+R\left(R\right(p)b\ast R(c\left)\right)\ast a+R(p\ast R(b\left)\right)\ast (c\succ a)-R\left(p\right)(b\succ (c\succ a\left)\right)\\ & =R\left(R\right(p\left)\right(b\succ c\left)\right)\ast a-R\left(R\right(p\ast R\left(b\right)\left)c\right)\ast a+R\left(R\right(R\left(p\right)c)\ast b)\ast a\\ & \phantom{\rule{10pt}{0ex}}+R\left(R\right(p\ast R\left(c\right)\left)b\right)\ast a+R(p\ast R(b\left)\right)\ast (c\succ a)-R\left(p\right)(b\succ (c\succ a\left)\right)\\ & =R\left(p\right)\left(\right(b\succ c)\succ a)-R(p\ast R((b\succ c)\left)\right)a+R\left(\right(p\ast R\left(b\right))\ast R(c\left)\right)a-R(p\ast R(b\left)\right)(c\succ a)\\ & \phantom{\rule{10pt}{0ex}}+R\left(R\right(p\left)\right(c\succ b\left)\right)\ast a-R\left(R\right(p\ast R\left(c\right)\left)b\right)\ast a+R(p\ast R(c\left)\right)(b\succ a)-R\left(\right(p\ast R\left(c\right))\ast R(b\left)\right)a\\ & \phantom{\rule{10pt}{0ex}}+R(p\ast R(b\left)\right)\ast (c\succ a)-R\left(p\right)(b\succ (c\succ a\left)\right)\end{array}$$Substituting the equalities$$\begin{array}{c}R\left(R\right(p\left)\right(c\succ b\left)\right)\ast a=R\left(p\right)\left(\right(c\succ b)\succ a)-R(p\ast R(c\succ b\left)\right)a\\ R\left(R\right(p\ast R\left(c\right)\left)b\right)\ast a=R(p\ast R(c\left)\right)(b\succ a)-R\left(\right(p\ast R\left(c\right))\ast R(b\left)\right)a\end{array}$$
- (a3)
- Let x be good; then$$(x,y,z)=R\left(R\right(u)c+u\ast R(c\left)\right)\ast a-R\left(u\right)(c\succ a)$$

- (b)
- Let $y=R\left(R\right(t\left)c\right)$, $c\in B$.$$\begin{array}{ll}(x,y,z)& =\left(R\right(u)\ast R(R\left(t\right)c\left)\right)\ast a-R\left(u\right)\ast \left(R\right(R\left(t\right)c)\ast a)\\ & =R\left(R\right(u)\ast R(t\left)c\right)\ast a+R(u\ast R(R\left(t\right)c\left)\right)\ast a-R\left(u\right)\ast \left(R\right(t\left)\right(c\succ a)-R(t\ast R\left(c\right)\left)a\right)\\ & =R\left(R\right(R\left(u\right)\ast t\left)c\right)\ast a+R\left(R\right(u\ast R\left(t\right))\ast c)\ast a+R(u\ast R(R\left(t\right)c\left)\right)\ast a+R\left(R\right(u)\ast (t\ast R\left(c\right)\left)\right)\ast a\\ & \phantom{\rule{10pt}{0ex}}+R(u\ast R(t\ast R\left(c\right)\left)\right)\ast a-R\left(R\right(u)\ast t)(c\succ a)-R(u\ast R(t\left)\right)\ast (c\succ a)\end{array}$$The expression $R\left(R\right(u)\ast t)e)$, $e\in B$, is well-defined; this easily follows from the fact that $R\left(t\right)$ is good. We give the first summand of the right-hand side (RHS) of Equation (18):$$R\left(R\right(R\left(u\right)\ast t\left)c\right)\ast a=R\left(R\right(u)\ast t)(c\succ a)-R\left(\right(R\left(u\right)\ast t)\ast R(c\left)\right)\ast a$$$$\begin{array}{ll}(x,y,z)& =R\left(R\right(u\ast R\left(t\right))\ast c)\ast a+R(u\ast R(t)\ast R(c\left)\right)\ast a-R(u\ast R(t\left)\right)\ast (c\succ a)\\ & =\left(R\right(u\ast R\left(t\right)),R(c),a)=0\end{array}$$

**Proof of**

**Theorem 2.**

**Example**

**8.**

**Example**

**9.**

**Corollary**

**1.**

**Proof.**

## 4. Universal Enveloping Rota–Baxter Algebra of Post-Commutative Algebra

- (1)
- elements of B are $pE$-words of type 1;
- (2)
- given a $pE$-word u, we define $R\left(u\right)$ as a $pE$-word of type 2;
- (3)
- given a $pE$-word u, we define ${R}^{2}\left(u\right)a$ for $a\in B$ as a $pE$-word of type 3.

**Example**

**10.**

**Example**

**11.**

**Theorem**

**3.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

- (a)
- Let $x=R\left({R}^{2}\left(p\right)b\right)$ be a $pE$-word of type 2, $y=R\left(c\right)$, $z=a$; $a,b,c\in B$.$$\begin{array}{ll}x\ast (y\ast z)& =R\left({R}^{2}\left(s\right)b\right)\ast (c\succ a)\\ & ={R}^{2}\left(s\right)(b\succ (c\succ a))-R(R\left(p\right)\ast R\left(b\right))\ast (c\succ a)-R(R\left(s\right)\ast b)\ast (c\succ a)\end{array}$$$$(x\ast y)\ast z=R(R\left({R}^{2}\left(s\right)b\right)\ast c)\ast a+R({R}^{2}\left(s\right)b\ast R\left(c\right))\ast a+R\left({R}^{2}\left(s\right)(b\perp c)\right)\ast a$$Applying induction on r, we give the more detailed RHS of Equation (29):$$\begin{array}{c}R(R\left({R}^{2}\left(s\right)b\right)\ast c)\ast a=R({R}^{2}\left(s\right)(b\succ c)-R(R\left(s\right)\ast R\left(b\right))\ast c-R(R\left(s\right)\ast b)\ast c)\ast a\hfill \\ ={R}^{2}\left(s\right)((b\succ c)\succ a)-R(R\left(s\right)\ast R(b\succ c))\ast a-R(R\left(s\right)\ast (b\succ c))\ast a\\ -R\left(R\right(s)\ast R(b\left)\right)\ast (c\succ a)+R\left(R\right(s)\ast R(b)\ast R(c\left)\right)\ast a+R\left(R\right(s)\ast (b\succ c\left)\right)\ast a\\ -R\left(R\right(s)\ast b)\ast (c\succ a)+R\left(R\right(s)\ast (c\succ b\left)\right)\ast a+R\left(R\right(s)\ast (b\perp c\left)\right)\ast a\\ =R\left(R\right(s)\ast R(b)\ast R(c\left)\right)\ast a-R\left(R\right(s)\ast R(b\succ c\left)\right)\ast a\\ +R\left(R\right(s)\ast (c\succ b\left)\right)\ast a+R\left(R\right(s)\ast (b\perp c\left)\right)\ast a\\ \hfill -R(R\left(s\right)\ast R\left(b\right))\ast (c\succ a)-R(R\left(s\right)\ast b)\ast (c\succ a)+{R}^{2}\left(s\right)((b\succ c)\succ a)\end{array}$$$$\begin{array}{c}R({R}^{2}\left(s\right)b\ast R\left(c\right))\ast a=R(R\left({R}^{2}\left(s\right)c\right)\ast b)\ast a\hfill \\ +R\left(R\right(R\left(s\right)\ast R\left(c\right))\ast b)\ast a+R\left(R\right(R\left(s\right)\ast c)\ast b)\ast a\\ =R\left({R}^{2}\left(s\right)(c\succ b)\right)\ast a-R(R(R\left(s\right)\ast R\left(c\right))\ast b)\ast a-R(R(R\left(s\right)\ast c)\ast b)\ast a\\ +R\left(R\right(s)\ast R(c\left)\right)\ast (b\succ a)-R\left(R\right(s)\ast R(c)\ast R(b\left)\right)\ast a-R\left(R\right(s)\ast (c\succ b\left)\right)\ast a\\ \hfill +R\left(R\right(s)\ast c)\ast (b\succ a)-R\left(R\right(s)\ast (b\succ c\left)\right)\ast a-R\left(R\right(s)\ast (b\perp c\left)\right)\ast a\end{array}$$$$R\left({R}^{2}\left(s\right)(b\perp c)\right)\ast a={R}^{2}\left(s\right)\ast ((b\perp c)\succ a)-R(R\left(s\right)\ast R(b\perp c))\ast a-R(R\left(s\right)\ast (b\perp c))\ast a$$We give the second row of Equation (31) as$$\begin{array}{c}R\left({R}^{2}\left(s\right)(c\succ b)\right)\ast a-R(R(R\left(s\right)\ast R\left(c\right))\ast b)\ast a-R(R(R\left(s\right)\ast c)\ast b)\ast a\hfill \\ ={R}^{2}\left(s\right)((c\succ b)\succ a)-R(R\left(s\right)\ast R(c\succ b))\ast a-R(R\left(s\right)\ast (c\succ b))\ast a\\ -R\left(R\right(s)\ast R(c\left)\right)\ast (b\succ a)+R\left(R\right(s)\ast R(c)\ast R(b\left)\right)\ast a+R\left(R\right(s)\ast (c\succ b\left)\right)\ast a\\ \hfill -R\left(R\right(s)\ast c)\ast (b\succ a)+R\left(R\right(s)\ast (b\succ c\left)\right)\ast a+R\left(R\right(s)\ast (b\perp c\left)\right)\ast a\end{array}$$
- (b)
- $x=R\left(u\right)$, $y=R\left({R}^{2}\left(t\right)b\right)$, $z=a$; $a,b\in B$. Applying induction on r, we have$$\begin{array}{c}(x\ast y)\ast z=R(R\left(u\right)\ast {R}^{2}\left(t\right)b)\ast a+R(u\ast R\left({R}^{2}\left(t\right)b\right))\ast a+R(u\ast {R}^{2}\left(t\right)b)\ast a\hfill \\ =R(R\left(u\right)\ast R\left(t\right)+u\ast {R}^{2}\left(t\right)+u\ast R\left(t\right))\ast (b\succ a)\\ -R((R\left(u\right)\ast R\left(t\right)+u\ast {R}^{2}\left(t\right)+u\ast R\left(t\right))\ast R\left(b\right))\ast a\\ -R((R\left(u\right)\ast R\left(t\right)+u\ast {R}^{2}\left(t\right)+u\ast R\left(t\right))\ast b)\ast a\\ \hfill +R(u\ast R\left({R}^{2}\left(t\right)b\right))\ast a+R(u\ast {R}^{2}\left(t\right)b)\ast a\end{array}$$$$\begin{array}{c}x\ast (y\ast z)=R\left(u\right)\ast ({R}^{2}\left(t\right)\ast (b\succ a)-R(R\left(t\right)\ast R\left(b\right))a-R(R\left(t\right)\ast b)\ast a)\hfill \\ =R(R\left(u\right)\ast R\left(t\right)+u\ast {R}^{2}\left(t\right)+u\ast R\left(t\right))\ast (b\succ a)\\ -R\left(R\right(u)\ast R(t)\ast R(b\left)\right)\ast a-R(u\ast R(R\left(t\right)\ast R\left(b\right)\left)\right)\ast a-R(u\ast R(t)\ast R(b\left)\right)\ast a\\ \hfill -R\left(R\right(u)\ast R(t)\ast b)\ast a-R(u\ast R(R\left(t\right)\ast b\left)\right)\ast a-R(u\ast R(t)\ast b)\ast a\end{array}$$

**Proof of**

**Theorem 3.**

**Corollary**

**2.**

**Problem**

**1.**

## Acknowledgments

## Conflicts of Interest

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Gubarev, V.
Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras. *Axioms* **2017**, *6*, 33.
https://doi.org/10.3390/axioms6040033

**AMA Style**

Gubarev V.
Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras. *Axioms*. 2017; 6(4):33.
https://doi.org/10.3390/axioms6040033

**Chicago/Turabian Style**

Gubarev, Vsevolod.
2017. "Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras" *Axioms* 6, no. 4: 33.
https://doi.org/10.3390/axioms6040033