nLab enveloping algebra

There is also a distinct notion of an enveloping algebra of a Lie algebra.

---

Definition

Given a monoid $A$ with multiplication $\mu$ in a symmetric monoidal category $C$ with symmetry $\tau$ its opposite $A^{op}$ is the same underlying object $A$ with multiplication $\mu\circ\tau$. The enveloping monoid $A^e$ of $A$ is the monoid whose underlying object is $A\otimes A^{op}$ and for which the multiplication is given by

$$(\mu\otimes (\mu\circ\tau))\circ(id\otimes\tau\otimes id): (A\otimes A^{op})\otimes (A\otimes A^{op})\to A\otimes A^{op}$$

where $\tau=\tau_{A,A}$ is the symmetry $A\otimes A\to A\otimes A$.

The enveloping monoid is sometimes called the enveloping algebra, especially if the monoidal category is a category of vector spaces. The left $A^e$-modules in $C$ are in 1-1 correspondence with the $A$-$A$-bimodules in $C$.