Let $\mathbf{k}$ be a commutative ring, and $\beta$ an element of $\mathbf{k}$. Fix a positive integer $n$, and set $\left[n\right] = \left\{1,2,\ldots,n\right\}$.

The *$n$-th type-A subdivision algebra over
$\mathbf{k}$ for parameter $\beta$* is the commutative $\mathbf{k}$-algebra
$B$ with

* generators $x_{i,j}$ indexed by all the $n\left(  n-1\right)  /2$ pairs
$\left(  i,j\right)  $ of integers satisfying $1\leq i<j\leq n$;

* relations $x_{i,j}x_{j,k}=x_{i,k}\left(  x_{i,j}+x_{j,k}+\beta\right)  $ for all $\left(  i,j,k\right)  \in\left[  n\right]  ^{3}$ satisfying $i<j<k$.

Alternatively, we can define $B$ in a more symmetric fashion: Namely, $B$ is
the commutative $\mathbf{k}$-algebra with

* generators $x_{i,j}$ indexed by all the $n\left(  n-1\right)  $ pairs
$\left(  i,j\right)  $ of distinct integers in $\left[n\right]$;

* relations $x_{i,j}+x_{j,i}=-\beta$ whenever $i\neq j$, as well as
$x_{i,j}x_{j,k}+x_{j,k}x_{k,i}+x_{k,i}x_{i,j}+\beta\left(  x_{i,j}
+x_{j,k}+x_{k,i}\right)  +\beta^{2}=0$ whenever $i,j,k$ are distinct elements
of $\left[n\right]$.

The $\mathbf{k}$-algebra $B$ has appeared in various contexts. It was
originally [introduced by Karola Mészáros as the abelianization of
Anatol Kirillov's quasi-classical Yang-Baxter
algebra](http://www.ams.org/journals/tran/2011-363-08/S0002-9947-2011-05265-7/).
It is a deformation of the Orlik-Terao algebra of the braid arrangement of
type $A_{n-1}$ (with the case $\beta=0$ corresponding to the Orlik-Terao
algebra). It is probably isomorphic to a $\mathbf{k}$-subalgebra of the
localization of the polynomial ring $\mathbf{k}\left[  q_{1},q_{2}
,\ldots,q_{n}\right]  $ at the multiplicative subset generated by the
differences $q_{i}-q_{j}$ for $i<j$ (here I say "probably" because I can only
show this for $\beta=0$, in which case it is isomorphic to the $\mathbf{k}
$-subalgebra generated by all $\dfrac{1}{q_{i}-q_{j}}$). It comes up [in the
computation of volumes of flow polytopes](https://arxiv.org/abs/1502.03997v3)
and [evaluations of Grothendieck
polynomials](https://arxiv.org/abs/1705.02418v3). See [my recent preprint
arXiv:1704.00839](http://www.cip.ifi.lmu.de/~grinberg/algebra/subdiv-v7.pdf)
for more on it.

On the other hand, recall that a [*Rota-Baxter algebra of weight $\beta$*](http://www.ams.org/notices/200911/rtx091101436p.pdf) means a $\mathbf{k}$-algebra $R$ equipped with a $\mathbf{k}$-linear map $P:R\rightarrow R$
(called its *Rota-Baxter operator*) that satisfies
\begin{equation}
P\left(  a\right)  P\left(  b\right)  =P\left(  P\left(  a\right)  b\right)
+P\left(  aP\left(  b\right)  \right)  +\beta P\left(  ab\right)
\label{eq.rota-baxter.def} \tag{1}
\end{equation}
for all $a,b\in R$. (Some authors, like those of the [Wikipedia
page](https://en.wikipedia.org/wiki/Rota-Baxter_algebra), prefer to put the
$\beta P\left(  ab\right)  $ addend on the left instead of the right hand
side, but this just boils down to replacing $\beta$ by $-\beta$.)

The axiom \eqref{eq.rota-baxter.def} of the Rota-Baxter algebra is uncannily
similar to the relations
\begin{equation}
x_{i,j}x_{j,k}=x_{i,k}\left(  x_{i,j}+x_{j,k}+\beta\right)  \label{eq.relB} \tag{2}
\end{equation}
of the algebra $B$. Indeed, represent each monomial in the indeterminates
$x_{i,j}$ as a multigraph on the vertex set $\left\{  1,2,\ldots,n\right\}  $,
where each indeterminate $x_{i,j}$ appearing in the monomial contributes an
edge $ij$ to the multigraph. Then, \eqref{eq.relB} can be visually rewritten
as
[![enter image description here][1]][1]
(where all vertices other than $i,j,k$ are omitted). Now, imagine writing an
"$a$" between the $i$ and the $j$, and writing a "$b$" between the $j$ and the
$k$, and interpreting each edge as a signal to apply $P$ to whatever stands
under the edge. The above equality thus becomes
[![enter image description here][2]][2]
which is precisely \eqref{eq.rota-baxter.def}.

> **Question.** Can this resemblance be turned into anything concrete (e.g., an
action of $B$ on Rota-Baxter algebras?). Barring that, can we define "Rota-Baxter algebras of other types"?

  [1]: https://i.sstatic.net/6W3Gp.png
  [2]: https://i.sstatic.net/ogarQ.png