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. 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 and evaluations of Grothendieck polynomials. See my recent preprint arXiv:1704.00839 for more on it.
On the other hand, recall that a Rota-Baxter algebra of weight $\beta$ 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, 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
(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
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"?
