Universal enveloping algebra
From Academic Kids

In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). This construction passes from the nonassociative structure L to a (more familiar, and possibly easier to handle) unital associative algebra which captures the important properties of L.
To understand the basic idea of this construction, first note that any associative algebra A over the field K becomes a Lie algebra over K with the bracket
 [a,b] = ab − ba.
That is, from an associative product, one can construct a Lie bracket by simply taking the commutator with respect to that associative product. We denote this Lie algebra by A_{L}.
Construction of the universal enveloping algebra attempts to reverse this process: to a given Lie algebra L over K we find the "most general" unital associative Kalgebra A such that the Lie algebra A_{L} contains L; this algebra A is U(L). The important constraint is to preserve the representation theory: the representations of L correspond in a onetoone manner to the modules over U(L). In a typical context where L is acting by infinitesimal transformations, the elements of U(L) act like differential operators, of all orders.
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Universal property
Let L be any Lie algebra over K. Given a unital associative Kalgebra U and a Lie algebra homomorphism
 h: L → U_{L},
(notation as above) we say that U is the universal enveloping algebra of L if it satisfies the following universal property: for any unital associative Kalgebra A and Lie algebra homomorphism
 f: L → A_{L}
there exists a unique unital algebra homomorphism
 g: U → A
such that
 f = gh.
Direct construction
For general reasons having to do with universal properties, we can say that if a Lie algebra has a universal enveloping algebra, then this enveloping algebra is uniquely determined by L (up to a unique algebra isomorphism). By the following construction, which suggests itself on general grounds (for instance, as part of a pair of adjoint functors), we establish that indeed every Lie algebra does have a universal enveloping algebra.
Starting with the tensor algebra T(L) on the vector space underlying L, we take U(L) to be the quotient of T(L) made by imposing the relations
 a.b − b.a = [a,b]
for all a and b in (the image in T(L) of) L, where the "." on the LHS denotes the associative multiplication in T(L), and the bracket on the RHS now means the given Lie algebra product, in L.
Formally, we define
 U(L) = T(L)/I
where
 I = ([a,b] − a.b + b.a  a, b in L)
is the (twosided) ideal in T(L) generated by all elements of the form [a,b] − a.b + b.a for a,b in L.
The natural map L → T(L) gives rise to a map h : L → U(L), and this is the Lie algebra homomorphism used in the universal property given above.
The analogous construction for Lie superalgebras is straightforward.
Examples in particular cases
If L is abelian (that is, the bracket is always 0), then U(L) is commutative; if a basis of the vector space L has been chosen, then U(L) can be identified with the polynomial algebra over K, with one variable per basis element.
If L is the Lie algebra corresponding to the Lie group G, U(L) can be identified with the algebra of leftinvariant differential operators (of all orders) on G; with L lying inside it as the leftinvariant vector fields as firstorder differential operators.
To relate the above two cases: if L is a vector space V as abelian Lie algebra, the leftinvariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the partial derivatives of first order.
The center of U(L) is called Z(L) and consists of the left and right invariant differential operators; this in the case of G not commutative will not be generated by firstorder operators (see for example Casimir operator).
Another characterisation in Lie group theory is of U(L) as the convolution algebra of distributions supported only at the identity element e of G.
The algebra of differential operators in n variables with polynomial coefficients may be obtained starting with the Lie algebra of the Heisenberg group. See Weyl algebra for this; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.
Further description of structure
The fundamental PoincaréBirkhoffWitt theorem gives a precise description of U(L); the most important consequence is that L can be viewed as a subspace of U(L). More precisely: the canonical map h : L → U(L) is always injective. Furthermore, U(L) is generated as a unital associative algebra by L.
L acts on itself by the Lie algebra adjoint representation, and this action can be extended to a representation of L on U(L): L acts as an algebra of derivations on T(L), and this action respects the imposed relations, so it actually acts on U(L). (This is the purely infinitesimal way of looking at the invariant differential operators mentioned above.)
Under this representation, the elements of U(L) invariant under the action of L (i.e. such that any element of L acting on them gives zero) are called invariant elements. They are generated by the Casimir invariants.
As mentioned above, the construction of universal enveloping algebras is part of a pair of adjoint functors. U is a functor from the category of Lie algebras over K to the category of unital associative Kalgebras. This functor is left adjoint to the functor which maps an algebra A to the Lie algebra A_{L}. It should be noted that the universal enveloping algebra construction is not exactly inverse to the formation of A_{L}: if we start with an associative algebra A, then U(A_{L}) is not equal to A; it is much bigger.
The facts about representation theory mentioned earlier can be made precise as follows: the abelian category of all representations of L is isomorphic to the abelian category of all left modules over U(L).
The construction of the group algebra for a given group is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural comultiplications which turn them into Hopf algebras.de:Universelle einhüllende Algebra