Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). This construction passes from the non-associative structure L to a (more familiar, and possibly easier to handle) unital associative algebra which captures the important properties of L.
Any associative algebra A over the field K becomes a Lie algebra over K with the Lie bracket:
That is, from an associative product, one can construct a Lie bracket by taking the commutator with respect to that associative product. Denote this Lie algebra by AL.
Construction of the universal enveloping algebra attempts to reverse this process: to a given Lie algebra L over K, find the "most general" unital associative K-algebra A such that the Lie algebra AL contains L; this algebra A is U(L). The important constraint is to preserve the representation theory: the representations of L correspond in a one-to-one 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. Next to Lie algebras, the construction of the universal enveloping algebra has been generalized for Malcev algebras, Bol algebras  and left alternative algebras.
An important topic in Lie algebras studies and probably the main source of their appearance in applications is representation of the Lie algebra. A representation ρ assigns to any element x of a Lie algebra a linear operator ρ(x). The space of linear operators is not only a Lie algebra, but also an associative algebra and so one can consider products ρ(x)ρ(y). The main point to introduce the universal enveloping algebra is to study such products in various representations of a Lie algebra. One obstacle can be immediately seen in a naive attempt to do this: properties of products drastically depend on the representation, not only on the Lie algebra itself. For example for one representation we might have ρ(x)ρ(y) = 0, while in another representation this product may not be zero. Nevertheless it appears to be true that certain properties are universal for all representations, i.e. they hold true for all representations simultaneously. The universal enveloping algebra is a way to grasp all such properties and only them.
Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → UL, we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → AL there exists a unique unital algebra homomorphism g: U → A such that: f(-) = gL (h(-)).
From this universal property, one can prove 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.
for all a and b in (the image in T(L) of) L, where the bracket on the RHS means the given Lie algebra product, in L.
Formally, we define
where I is the two-sided ideal of T(L) generated by elements of the form
The natural map L → T(L) descends 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; and 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 left-invariant differential operators (of all orders) on G; with L lying inside it as the left-invariant vector fields as first-order differential operators.
To relate the above two cases: if L is a vector space V as abelian Lie algebra, the left-invariant 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 often not be generated by first-order operators (see for example Casimir operator of a semi-simple Lie algebra).
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 Poincaré–Birkhoff–Witt theorem gives a precise description of U(L); the most important consequence is that L can be viewed as a linear 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 K-algebras. This functor is left adjoint to the functor which maps an algebra A to the Lie algebra AL. The universal enveloping algebra construction is not exactly inverse to the formation of AL: if we start with an associative algebra A, then U(AL) 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.
The center of the universal enveloping algebra of a simple Lie algebra is described by the Harish-Chandra isomorphism.
- J.M. Perez-Izquierdo, I.P. Shestakov: An envelope for Malcev algebras, Journal of Algebra 272 (2004) 379–393.
- J.M. Perez-Izquierdo: An envelope for Bol algebras, Journal of Algebra 284 (2005) 480–493.
- Rukavicka Josef: An envelope for left alternative algebras, International Journal of Algebra, Vol. 7, 2013, no. 10, 455–462, 
- Dixmier, Jacques (1996) , Enveloping algebras, Graduate Studies in Mathematics 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740
- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics 222 (2nd ed.), Springer
- Musson, Ian M. (2012), Lie Superalgebras and Enveloping Algebras, Graduate Studies in Mathematics 131, Providence, R.I.: American Mathematical Society, ISBN 0-8218-6867-5, Zbl 1255.17001