Typed Brauer and partition diagram categories

Let $R$ be a commutative unital ring and let $\delta \in R$. A set partition of a finite set is a collection of disjoint nonempty subsets, called blocks, whose union is the whole set; a perfect matching is a set partition whose blocks each have size two. For $a,b\in \mathbb{N}=\{0,1,2,\dots \}$, place $a$ ordered input vertices on a top edge and $b$ ordered output vertices on a bottom edge. The Brauer module ${\mathcal{B}}_{\delta }(a,b)$ is the free $R$-module on perfect matchings of these $a+b$ vertices. It is zero if $a+b$ is odd. The partition module ${\mathcal{P}}_{\delta }(a,b)$ is the free $R$-module on set partitions of the same vertices.

For $D\in {\mathcal{B}}_{\delta }(a,b)$ and $E\in {\mathcal{B}}_{\delta }(b,c)$, glue the output vertices of $D$ to the corresponding input vertices of $E$. Take the equivalence relation generated by the two matchings and these identifications. Each equivalence class meeting an outer vertex consists of exactly two outer vertices and gives an edge of a new matching. Each class meeting no outer vertex is a closed loop. Define $E\circ D={\delta }^{\mathrm{\ell }}H$, where $H$ is the resulting outer matching and $\mathrm{\ell }$ is the number of removed loops. For partition diagrams use the equivalence relation generated by the two partitions and the same gluing; each outer equivalence class gives a block and each class with no outer vertices contributes one factor $\delta$. Extend composition $R$-bilinearly.

The identity at $a$ is the matching joining input $i$ to output $i$, respectively the partition into those $a$ through-blocks. Juxtaposition defines tensor product on objects by addition and on diagrams by disjoint union. The Brauer algebra $B_a(\delta )$ and partition algebra $P_a(\delta )$ are the endomorphism algebras at object $a$.

These are abstract diagram modules over arbitrary $R$. This definition asserts no faithfulness of any tensor representation.