nLab
enveloping algebra

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

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Definition

Given a monoid $A$ with multiplication $\mu$ in a symmetric monoidal category $C$ with symmetry $\tau$ its opposite $A^{\mathrm{op}}$ 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 $A\otimes A^{\mathrm{op}}$ and for which the multiplication is given by

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