Editing Universal enveloping algebra (section)

===Superalgebras and other generalizations===
The above construction focuses on Lie algebras and on the Lie bracket, and its skewness and antisymmetry. To some degree, these properties are incidental to the construction. Consider instead some (arbitrary) algebra (not a Lie algebra) over a vector space, that is, a vector space <math>V</math> endowed with multiplication <math>m:V\times V\to V</math> that takes elements <math>a\times b\mapsto m(a,b).</math>  ''If'' the multiplication is bilinear, then the same construction and definitions can go through. One starts by lifting <math>m</math> up to <math>T(V)</math> so that the lifted <math>m</math> obeys all of the same properties that the base <math>m</math> does – symmetry or antisymmetry or whatever. The lifting is done ''exactly'' as before, starting with
:<math>\begin{align}
m: V \otimes V &\to V \\
a \otimes b &\mapsto m(a,b)
\end{align}</math>

This is consistent precisely because the tensor product is bilinear, and the multiplication is bilinear. The rest of the lift is performed so as to preserve multiplication as a [[homomorphism]]. ''By definition'', one writes

:<math>m(a \otimes b,c)= a \otimes m(b,c) + m(a,c) \otimes b</math>
and also that
:<math>m(a,b\otimes c)= m(a,b) \otimes c + b \otimes m(a,c)</math>
This extension is consistent by appeal to a lemma on [[free object]]s: since the tensor algebra is a [[free algebra]], any homomorphism on its generating set can be extended to the entire algebra. Everything else proceeds as described above: upon completion, one has a unital associative algebra; one can take a quotient in either of the two ways described above.

The above is exactly how the universal enveloping algebra for [[Lie superalgebra]]s is constructed. One need only to carefully keep track of the sign, when permuting elements. In this case, the (anti-)commutator of the superalgebra lifts to an (anti-)commuting Poisson bracket.

Another possibility is to use something other than the tensor algebra as the covering algebra. One such possibility is to use the [[exterior algebra]]; that is, to replace every occurrence of the tensor product by the [[exterior product]].  If the base algebra is a Lie algebra, then the result is the [[Gerstenhaber algebra]]; it is the [[exterior algebra]] of the corresponding Lie group. As before, it has a grading [[natural transformation|naturally]] coming from the grading on the exterior algebra. (The Gerstenhaber algebra should not be confused with the [[Poisson superalgebra]]; both invoke anticommutation, but in different ways.)

The construction has also been generalized for [[Malcev algebra]]s,<ref>{{cite journal | last1 = Perez-Izquierdo | first1 = J.M. | last2 = Shestakov | first2 = I.P. | year = 2004 | title = An envelope for Malcev algebras | journal = Journal of Algebra | volume = 272 | pages = 379–393 | doi=10.1016/s0021-8693(03)00389-2| hdl = 10338.dmlcz/140108 | hdl-access = free }}</ref> [[Bol loop|Bol algebras]]<ref>{{cite journal | last1 = Perez-Izquierdo | first1 = J.M. | year = 2005 | title = An envelope for Bol algebras | journal = Journal of Algebra | volume = 284 | issue = 2| pages = 480–493 | doi=10.1016/j.jalgebra.2004.09.038| doi-access = free }}</ref> and [[alternative algebra|left alternative algebras]].{{citation needed|date=November 2019}}