Editing Universal enveloping algebra (section)

==Formal definition==
Recall that every Lie algebra <math>\mathfrak{g}</math> is in particular a [[vector space]]. Thus, one is free to construct the [[tensor algebra]] <math>T(\mathfrak{g})</math> from it. The tensor algebra is a [[free algebra]]: it simply contains all possible [[tensor product]]s of all possible vectors in <math>\mathfrak{g}</math>, without any restrictions whatsoever on those products.

That is, one constructs the space
:<math>T(\mathfrak{g}) = K \,\oplus\, \mathfrak{g} \,\oplus\, (\mathfrak{g} \otimes \mathfrak{g}) 
\,\oplus\, (\mathfrak{g} \otimes \mathfrak{g} \otimes \mathfrak{g}) \,\oplus\, \cdots </math>

where <math>\otimes</math> is the tensor product, and <math>\oplus</math> is the [[direct sum]] of vector spaces. Here, {{math|''K''}} is the field over which the Lie algebra is defined. From here, through to the remainder of this article, the tensor product is always explicitly shown. Many authors omit it, since, with practice, its location can usually be inferred from context. Here, a very explicit approach is adopted, to minimize any possible confusion about the meanings of expressions.

The first step in the construction is to "lift" the Lie bracket from the Lie algebra <math>\mathfrak{g}</math> (where it is defined) to the tensor algebra <math>T(\mathfrak{g})</math> (where it is not), so that one can coherently work with the Lie bracket of two tensors. The lifting is done recursively. Let us define

:<math>[a\otimes b, c] = a \otimes [b,c] + [a,c]\otimes b</math>

and

:<math>[a, b\otimes c] = [a,b]\otimes c + b\otimes [a,c]</math>

It is straightforward to verify that the above definition is [[Bilinear form|bilinear]] and [[Skew-symmetric bilinear form|skew-symmetric]]; one can also show that it obeys the [[Jacobi identity]]. The final result is that one has a Lie bracket that is consistently defined on all of <math>T(\mathfrak{g});</math> one says that it has been "lifted" to all of <math>T(\mathfrak{g})</math> in the conventional sense of a "lift" from a base space (here, the Lie algebra) to a [[covering space]] (here, the tensor algebra).

The result of this lifting is that <math>T(\mathfrak{g})</math> becomes a [[Poisson algebra]]: a [[unital associative algebra]] with a Lie bracket that is compatible with the original Lie algebra bracket (by construction). It is not the ''smallest'' such algebra, however; it contains far more elements than needed. One can get something smaller by projecting back down. The universal enveloping algebra <math>U(\mathfrak{g})</math> of <math>\mathfrak{g}</math> is defined as the [[Quotient space (linear algebra)|quotient space]]

:<math>U(\mathfrak{g}) = T(\mathfrak{g})/\sim</math>

where the [[equivalence relation]] <math>\sim</math> is given by

:<math>a\otimes b - b \otimes a = [a,b]</math>

for all <math>a,b\in T(\mathfrak{g})</math>. That is, the Lie bracket defines the equivalence relation used to perform the quotienting. The result is still a unital associative algebra, and one can still take the Lie bracket of any two members. Computing the result is straight-forward, if one keeps in mind that each element of <math>U(\mathfrak{g})</math> can be understood as a [[coset]]: one just takes the bracket as usual, and searches for the coset that contains the result. It is the ''smallest'' such algebra; one cannot find anything smaller that still obeys the axioms of an associative algebra.

The universal enveloping algebra is what remains of the tensor algebra after modding out the [[Poisson algebra]] structure. (This is a non-trivial statement; the tensor algebra has a rather complicated structure: it is, among other things, a [[Hopf algebra]]; the Poisson algebra is likewise rather complicated, with many peculiar properties. It is compatible with the tensor algebra, and so the modding can be performed. The Hopf algebra structure is conserved; this is what leads to its many novel applications, e.g. in [[string theory]]. However, for the purposes of the formal definition, none of this particularly matters.)

The construction can be performed in a slightly different (but equivalent) way. Consider the [[two-sided ideal]] {{math|''I''}} generated by elements of the form

:<math>a\otimes b - b \otimes a - [a,b]</math>

for <math>a,b\in\mathfrak{g}</math>. (Note that these generators are elements of <math>\mathfrak{g} \oplus (\mathfrak{g}\otimes\mathfrak{g}) \subset T(\mathfrak{g})</math>.) A general member of the ideal {{math|''I''}} will be linear combinations of elements of the form

:<math>\cdots \otimes c \otimes d \otimes (a\otimes b - b \otimes a - [a,b]) \otimes f \otimes g \otimes \cdots</math>

where all lower-case letters are elements of <math>\mathfrak{g}</math>. Since {{math|''I''}} is an ideal, we can quotient by it. The universal enveloping algebra can then be defined as<ref>{{harvnb|Hall|2015}} Section 9.3</ref>

:<math>U(\mathfrak{g}) = T(\mathfrak{g})/I</math>

===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}}