11institutetext: Tomasz Brzeziński 22institutetext: Department of Mathematics,
Swansea University, Swansea University Bay Campus, Fabian Way, Swansea,
Swansea SA1 8EN, U.K., 22email<EMAIL_ADDRESS>33institutetext:
Faculty of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245
Białystok, Poland, 33email<EMAIL_ADDRESS>
# Homothetic Rota-Baxter systems and Dyckm-algebras

Tomasz Brzeziński

###### Abstract

It is shown that generalized Rota-Baxter operators introduced in [W.A.
Martinez, E.G. Reyes & M. Ronco, Int. J. Geom. Meth. Mod. Phys. 18 (2021)
2150176] are a special case of Rota-Baxter systems [T. Brzeziński, J. Algebra
460 (2016), 1–25]. The latter are enriched by homothetisms and then shown to
give examples of Dyckm-algebras.

##  1 Introduction

Rota-Baxter operators first appeared in Bax:ana  in analysis of differential
operators on commutative Banach algebras, then brought to combinatorics
Rot:Bax , and are now intensively studied e.g. in probability, renormalization
of quantum field theories, the theory of operads, dialgebras, trialgebras,
dendriform algebras, pre-Lie algebras etc.; see Guo:int  for progress up to
the early 2010s.

A Rota-Baxter operator of weight $\lambda$ on an associative algebra $A$ over
a field ${\mathbb{K}}$ is a linear operator $R:A\to A$, such that, for all
$a,b\in A$,

$R(a)R(b)=R\left(R(a)b+aR(b)+\lambda\ ab\right),$ (1)

where $\lambda$ is a scalar. The pair $(A,R)$ is often referred to as a Rota-
Baxter algebra.

In a recently published article MarRey:gen  Martinez, Reyes & Ronco introduced
a generalization of Rota-Baxter operators that involves a pair of scalars
$(\alpha,\beta)$. A generalized Rota-Baxter operator of weights
$(\alpha,\beta)$ on an associative algebra $A$ is a linear map $\bar{R}:A\to
A$, such that, for all $a,b\in A$,

$\bar{R}(a)\bar{R}(b)=\bar{R}\left(\bar{R}(a)b+a\bar{R}(b)+\alpha\
ab\right)+\beta ab.$ (2)

The authors of MarRey:gen  then proceed to show that such operators of weights
$(3,2)$ give rise to a class of Dyckm-algebras introduced and studied in
LopPre:alg , LopPre:sim  in context of dendriform structures.

This note has two aims. First, we show that generalized Rota-Baxter operators
are examples of Rota-Baxter systems Brz:Rot . Second, we enrich Rota-Baxter
systems with homothetisms Red:ver  or self-permutable bimultiplications
Mac:ext  that were introduced in studies of ring extensions and constrain them
in such a way as to produce examples of Dyckm-algebras that extend those in
MarRey:gen .

##  2 Rota-Baxter systems, homothetisms and Dyckm-algebras

###  2.1 Rota-Baxter systems and generalized Rota-Baxter operators

The following notion was introduced in Brz:Rot . An associative algebra $A$
(over a field ${\mathbb{K}}$) together with a pair of linear operators
$R,S:A\longrightarrow A$ is called a Rota-Baxter system if, for all $a,b\in
A$,

$\displaystyle R(a)R(b)=R\left(R(a)b+aS(b)\right),$ (3a) $\displaystyle
S(a)S(b)=S\left(R(a)b+aS(b)\right).$ (3b)

As explained in Das:gen , conditions (3) can be recast in the form of a single
Nijenhuis operator. Recall from CarGra:qua  that a Nijenhuis tensor or
operator on an associative algebra $B$ is a linear function
$N:B\longrightarrow B$ such that, for all $a,b\in B$,

$N(a)N(b)=N(N(a)b+aN(b)-N(ab)).$ (4)

Starting with an algebra $A$ we can form an algebra $B$ on the vector space
$A\oplus A\oplus A$ with the product

$\begin{pmatrix}a\cr b\cr c\end{pmatrix}\begin{pmatrix}a^{\prime}\cr
b^{\prime}\cr c^{\prime}\end{pmatrix}=\begin{pmatrix}ab\cr
a^{\prime}b^{\prime}\cr ac^{\prime}+cb^{\prime}\end{pmatrix}$

Then $(A,R,S)$ is a Rota-Baxter system if and only if

$N=\begin{pmatrix}0&0&R\\\ 0&0&S\\\ 0&0&0\end{pmatrix}$

is a Nijenhuis operator on $B$.

If $R$ is a Rota-Baxter operator of weight $\lambda$, then setting
$S=R+\lambda\ \mathrm{id}$ one obtains a Rota-Baxter system. In a similar way,
one can interpret a generalized Rota-Baxter operator of weights
$(\alpha,\beta)$ as a Rota-Baxter system. More precisely,

###### Lemma 1

Let $\bar{R}$ be a generalized Rota-Baxter operator of weights
$(\alpha,\beta)$ and let $\lambda,\mu\in{\mathbb{K}}$ be such that

$\alpha=\lambda+\mu,\qquad\beta=\lambda\mu.$ (5)

Set

$R=\bar{R}+\lambda\ \mathrm{id},\qquad S=\bar{R}+\mu\ \mathrm{id}.$ (6)

Then $(A,R,S)$ is a Rota-Baxter system.

This is checked by a straightforward calculation (left to the reader). As a
consequence and in view of (Brz:Rot, , Proposition 2.5), one concludes, for
example that there is a dendriform algebra associated to a generalized Rota-
Baxter operator $\bar{R}$ (cf. (MarRey:gen, , Proposition 18)).

Of course, if ${\mathbb{K}}$ is not an algebraically closed field, equations
(5) for $\lambda$ and $\mu$ might not have solutions. However, the most
interesting case of a generalized Rota-Baxter operator studied in MarRey:gen
corresponds to the weights $(3,2)$, hence one can take $\lambda=1$ and $\mu=2$
then.

###  2.2 Homothetisms and homothetic Rota-Baxter systems

The following notion originates from studies of homology of rings and ring
extensions in Red:ver  and Mac:ext . Let $A$ be an associative algebra. By a
double operator $\sigma$ on $A$ we mean a pair of linear operators
$\sigma=(\stackrel{{\scriptstyle\rightarrow}}{{\sigma}},\stackrel{{\scriptstyle\leftarrow}}{{\sigma}})$
on $A$,

$\stackrel{{\scriptstyle\rightarrow}}{{\sigma}}:A\longrightarrow A,\quad
a\longmapsto\sigma
a,\qquad\stackrel{{\scriptstyle\leftarrow}}{{\sigma}}:A\longrightarrow A,\quad
a\longmapsto a\sigma.$

The somewhat unusual way of writing the argument to the left of the operator
(in the definition of $\stackrel{{\scriptstyle\leftarrow}}{{\sigma}}$) proves
very practical and economical in expressing the action of double operators
with additional properties, in particular those that we are introducing
presently.

A double operator $\sigma$ on $A$ is called a bimultiplication Mac:ext  or a
bitranslation Pet:ide  if, for all $a,b\in A$,

$\sigma(ab)=(\sigma a)b,\qquad(ab)\sigma=a(b\sigma)\quad\&\quad a(\sigma
b)=(a\sigma)b.$ (7)

A bimultiplication $\sigma$ is called a double homothetism Red:ver  or is said
to be self-permutable Mac:ext  provided that, for all $a\in A$,

$(\sigma a)\sigma=\sigma(a\sigma).$ (8)

The first two conditions in (7) mean that
$\stackrel{{\scriptstyle\rightarrow}}{{\sigma}}$ is a right and
$\stackrel{{\scriptstyle\leftarrow}}{{\sigma}}$ is a left $A$-module
homomorphism. A bimultiplication is called simply a multiplication in Hoc:coh
. In functional analysis, in particular in the context of $C^{*}$-algebras,
bimultiplications are known as multipliers Hel:mul , Bus:dou . The set of all
bimultiplications is a unital algebra, known as a multiplier algebra. The
relations (8) mean that endomorphisms
$\stackrel{{\scriptstyle\rightarrow}}{{\sigma}}$ and
$\stackrel{{\scriptstyle\leftarrow}}{{\sigma}}$ mutually commute in the
endomorphism algebra of the vector space $A$. Put together, conditions (7) and
(8) mean that one needs not put any brackets in strings of letters that
involve elements of $A$ and $\sigma$.

Any element of $A$, say $s\in A$, induces a double homothetism
$\bar{s}=(\stackrel{{\scriptstyle\rightarrow}}{{s}},\stackrel{{\scriptstyle\leftarrow}}{{s}})$
on $A$,

$\stackrel{{\scriptstyle\rightarrow}}{{s}}:a\longmapsto
sa,\qquad\stackrel{{\scriptstyle\leftarrow}}{{s}}:a\longmapsto as.$

Such double homothetisms are said to be inner and they form an ideal in the
multiplier algebra. In applications of bimultiplications to ring extensions,
most recently in connecting extensions of integers to trusses (sets with an
associative binary operation distributing over a ternary abelian heap
operation) AndBrz:ide  the key role is played by the quotient of the
multiplier algebra by the ideal of inner homothetisms; in particular in the
theory of operator algebras this is known as the corona algebra.

The rescaling by $\lambda\in{\mathbb{K}}$ understood as a pair
$(\lambda\mathrm{id},\lambda\mathrm{id})$ is a double homothetism, which we
will also denote by $\lambda$. If $A$ has the identity $1$, then, of course
$\lambda$ is an inner homothetism, $\overline{\lambda 1}$.

We are now ready to define the main notion of this note.

###### Definition 1

Let $(A,R,S)$ be a Rota-Baxter system and let $\sigma$ be a double homothetism
on $A$. We say that $(A,R,S,\sigma)$ is a homothetic Rota-Baxter system if,
for all $a\in A$,

$S(a)\sigma-\sigma R(a)=\sigma a\sigma.$ (9)

If $\bar{R}$ is a generalized Rota-Baxter operator of weights $(\alpha,\beta)$
such that

$\gamma:=\pm\sqrt{\alpha^{2}-4\beta}\in{\mathbb{K}},$

then $(A,R,S,\gamma)$ is a homothetic Rota-Baxter system, where $(A,R,S)$
corresponds to $\bar{R}$ through Lemma 1 (the sign depends on the choice of
$\lambda$ and $\mu$, i.e. whether $\lambda<\mu$ or $\lambda>\mu$).

###  2.3 Dyckm-algebras

The combinatorial, algebraic and operadic aspects of a certain class of
lattice paths counted by Fuss-Catalan numbers led López, Préville-Ratelle &
Ronco in LopPre:alg  to introduce the following notion.

Let $m$ be a natural number. A ${\mathbb{K}}$-vector space $A$ together with
$m+1$ linear operations $\ast_{i}:A\otimes A\longrightarrow A$, $i=0,\ldots,m$
such that, for all $a,b,c\in A$

$a\ast_{i}(b\ast_{j}c)=(a\ast_{i}b)\ast_{j}c,\qquad 0\leq i<j\leq m,$ (10a)
$a\ast_{0}(b\ast_{0}c)=\left(\sum_{i=0}^{m}a\ast_{i}b\right)\ast_{0}c,$ (10b)
$a\ast_{m}\left(\sum_{i=0}^{m}b\ast_{i}c\right)=(a\ast_{m}b)\ast_{m}c,$ (10c)
$a\ast_{i}\left(\sum_{k=0}^{i}b\ast_{k}c\right)=\left(\sum_{k=i}^{m}a\ast_{k}b\right)\ast_{i}c,\qquad
1\leq i\leq m-1,$ (10d)

is called a Dyckm-algebra.

Dyckm-algebras generalize associative algebras (the $m=0$ case) and Loday’s
dendriform algebras (the $m=1$ case) (Lod:dia, , Section 5). In MarRey:gen  it
is shown that one can associate Dyckm-algebras to any generalized Rota-Baxter
operator of weights $(3,2)$ (see Theorem 20 and Proposition 21 in MarRey:gen
for explicit formulae). Aided by this observation we will associate Dyckm-
algebras to any homothetic Rota-Baxter system.

##  3 Dyckm-algebras from homothetic Rota-Baxter systems

The main result of this note is contained in the following theorem.

###### Theorem 3.1

Let $(A,R,S,\sigma)$ be a homothetic Rota-Baxter system and let $m$ be any
natural number. Define $m+1$ linear operations $\ast_{i}:A\otimes
A\longrightarrow A$, $i=0,\ldots,m$ as follows:

$a\ast_{i}b=\begin{cases}R(a)b,&i=0,\\\ (-1)^{i+1}a\sigma
b,&i=1,\ldots,m-1,\\\ aS(b)-\frac{1+(-1)^{m}}{2}a\sigma b,&i=m,\end{cases}$
(11)

for all $a,b\in A$. Then $\left(A,\\{\ast_{i}\\}_{i=0}^{m}\right)$ is a Dyckm-
algebra.

###### Proof

We will carefully check that all the relations between different operations
listed in (10) hold. Starting with (10a), if $i\neq 0$ and $j\neq m$, these
equations reduce to the equality $a\sigma(b\sigma c)=(a\sigma b)\sigma c$
which follows by (7) and (8). For $i=0$ and $j\neq m$, (10a) amounts to the
equality $R(a)(b\sigma c)=(R(a)b)\sigma c$, which holds by (7). The proofs of
(10a) for $j=m$, and all the remaining equalities in (10) depend on the parity
of $m$. So, we will consider two separate cases in turn.

Assume that $m$ is odd. Then (10a) with $i=0$ and $j=m$ follows immediately by
the associativity of $A$, while $i\neq 0$ and $j=m$ amounts to the equality
$a(b\sigma S(c))=(ab)\sigma S(c)$, which holds by (7).

Since $m$ is odd (10b) reduces to

$a\ast_{0}(b\ast_{0}c)=(a\ast_{0}b+a\ast_{m}b)\ast_{0}c,$

that is

$R(a)R(b)c=R(R(a)b+aS(b))c,$

and this immediately follows by (3a). In a similar way (10c) follows by (3b).

We split checking (10d) into two cases. If $i$ is even, then (10d) reduces to

$a\ast_{i}(b\ast_{0}c)=(a\ast_{m}b+a\ast_{m-1}b)\ast_{i}c,$

which amounts to the equality

$a\sigma R(b)c=(aS(b)-a\sigma b)\sigma c,$ (12)

that follows by the constraint (9). If $i$ is odd, then (10d) is equivalent to

$a\ast_{i}(b\ast_{0}c+b\ast_{1}c)=(a\ast_{m}b)\ast_{i}c,$

that is,

$R(a)(R(b)c+b\sigma c)=(aS(b))\sigma c$ (13)

and thus again follows by the definition of a double homothetism and (9). This
completes the proof of the theorem for $m$ odd.

Assume now that $m$ is even. We look back at two remaining cases in (10a). If
$i=0$ and $j=m$, then

$\displaystyle a\ast_{0}(b\ast_{m}c)$ $\displaystyle=R(a)bS(c)-R(a)(b\sigma
c)$ $\displaystyle=R(a)bS(c)-(R(a)b)\sigma c=(a\ast_{0}b)\ast_{m}c,$

by the associativity of $A$ and (7). In a similar way, if $i\neq 0$ and $j=m$,
(10a) is equivalent to the equality

$a\sigma(bS(c))-a\sigma(b\sigma c)=(a\sigma b)S(c)-(a\sigma b)\sigma c,$

which follows by (7) and (8).

Since $m$ is even (10b) reduces to

$a\ast_{0}(b\ast_{0}c)=(a\ast_{0}b+a\ast_{1}b+a\ast_{m}b)\ast_{0}c,$

and, in view of the definition of $\ast_{m}$, this immediately follows by
(3a). In a similar way (10c) follows by (3b).

As for the $m$-odd case, we split checking (10d) into two cases. If $i$ is
even, then (10d) reduces to

$a\ast_{i}(b\ast_{0}c)=(a\ast_{m}b)\ast_{i}c,$

which is the same as (12), while, for $i$ odd, (10d) is equivalent to

$a\ast_{i}(b\ast_{0}c+b\ast_{1}c)=(a\ast_{m}b+a\ast_{m-1}b)\ast_{i}c,$

that is already proven equality (12). This completes the proof of the theorem.

If $\bar{R}$ is a generalized Rota-Baxter operator of weights $(3,2)$, then
the corresponding Rota-Baxter system $R=\bar{R}+\mathrm{id}$,
$S=\bar{R}+2\mathrm{id}$ is constrained by the homothetism $1$, induced by the
rescaling by the identity in ${\mathbb{K}}$, and thus Theorem 3.1 implies
(MarRey:gen, , Theorem 20 & Proposition 21).

###### Acknowledgements.

The research is partially supported by the National Science Centre, Poland,
grant no. 2019/35/B/ST1/01115.

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