There is also a distinct notion of an enveloping algebra of a Lie algebra.
Given a monoid with multiplication in a symmetric monoidal category with symmetry its opposite is the same underlying object with multiplication . The enveloping monoid of is the monoid whose underlying object is and for which the multiplication is given by
where is the symmetry .
The enveloping monoid is sometimes called the enveloping algebra, especially if the monoidal category is a category of vector spaces. The left -modules in are in 1-1 correspondence with the --bimodules in .