In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. Section 9.5 In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka–Krein duality between compact topological groups and their representations.
From an analytic viewpoint, the universal enveloping algebra of the Lie algebra of a Lie group may be identified with the algebra of left-invariant differential operators on the group.
Informal construction The idea of the universal enveloping algebra is to embed a Lie algebra into an associative algebra with identity in such a way that the abstract bracket operation in corresponds to the commutator in and the algebra is generated by the elements of . There may be many ways to make such an embedding, but there is a unique "largest" such , called the universal enveloping algebra of .
Function composition is associative The associativity of the universal enveloping algebra can be understood as arising from the associativity of function composition. Consider the case where is just "some non-associative algebra" (without any appeal to it being a Lie algebra). For any element , left-multiplication by defines a map, that is, a function The collection of all such maps, using function composition as a product, forms an associative algebra. Specifically, for any , one has that
An enveloping algebra is an algebra of all such functions with function composition as the product. It is manifestly associative, because function composition is associative.
That this algebra is indeed an algebra over a field, i.e. a vector space, i.e. linear, is also manifest: the can be added, and also multiplied by the scalars of (usually the reals or the complex numbers, or more generally any ring or field). As a vector space, a primary task is to find a vector space basis for this enveloping algebra, and then define function composition in terms of this basis. That is, given one wishes to find such that , and more generally, to find the smallest such set of that span the space. By linearity, it is enough to work with the basis elements of
Generators and relations Let be a Lie algebra, assumed finite-dimensional for simplicity, with basis . Let be the structure constants for this basis, so that
Then the universal enveloping algebra is the associative algebra (with identity) generated by elements subject to the relations
and no other relations. Below we will make this "generators and relations" construction more precise by constructing the universal enveloping algebra as a quotient of the tensor algebra over .
Consider, for example, the Lie algebra sl(2,C), spanned by the matrices
which satisfy the commutation relations , , and . The universal enveloping algebra of sl(2,C) is then the algebra generated by three elements subject to the relations
and no other relations. We emphasize that the universal enveloping algebra is not the same as (or contained in) the algebra of matrices. For example, the matrix satisfies , as is easily verified. But in the universal enveloping algebra, the element does not satisfy because we do not impose this relation in the construction of the enveloping algebra. Indeed, it follows from the Poincaré–Birkhoff–Witt theorem (discussed § below) that the elements are all linearly independent in the universal enveloping algebra.
Finding a basis In general, elements of the universal enveloping algebra are linear combinations of products of the generators in all possible orders. Using the defining relations of the universal enveloping algebra, we can always re-order those products in a particular order, say with all the factors of first, then factors of , etc. For example, whenever we have a term that contains (in the "wrong" order), we can use the relations to rewrite this as plus a linear combination of the 's. Doing this sort of thing repeatedly eventually converts any element into a linear combination of terms in ascending order. Thus, elements of the form
with the 's being non-negative integers, span the enveloping algebra. (We allow , meaning that we allow terms in which no factors of occur.) The Poincaré–Birkhoff–Witt theorem, discussed below, asserts that these elements are linearly independent and thus form a basis for the universal enveloping algebra. In particular, the universal enveloping algebra is always infinite dimensional.
The Poincaré–Birkhoff–Witt theorem implies, in particular, that the elements themselves are linearly independent. It is therefore common—if potentially confusing—to identify the 's with the generators of the original Lie algebra. That is to say, we identify the original Lie algebra as the subspace of its universal enveloping algebra spanned by the generators. Although may be an algebra of matrices, the universal enveloping of does not consist of (finite-dimensional) matrices. In particular, there is no finite-dimensional algebra that contains the universal enveloping of ; the universal enveloping algebra is always infinite dimensional. Thus, in the case of sl(2,C), if we identify our Lie algebra as a subspace of its universal enveloping algebra, we must not interpret , and as matrices, but rather as symbols with no further properties (other than the commutation relations).
Formalities The formal construction of the universal enveloping algebra takes the above ideas, and wraps them in notation and terminology that makes it more convenient to work with. The most important difference is that the free associative algebra used in the above is narrowed to the tensor algebra, so that the product of symbols is understood to be the tensor product. The commutation relations are imposed by constructing a quotient space of the tensor algebra quotiented by the smallest two-sided ideal containing elements of the form . The universal enveloping algebra is the "largest" unital associative algebra generated by elements of with a Lie bracket compatible with the original Lie algebra.
Formal definition Recall that every Lie algebra is in particular a vector space. Thus, one is free to construct the tensor algebra from it. The tensor algebra is a free algebra: it simply contains all possible tensor products of all possible vectors in , without any restrictions whatsoever on those products.
That is, one constructs the space
where is the tensor product, and is the direct sum of vector spaces. Here, 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 (where it is defined) to the tensor algebra (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
and
It is straightforward to verify that the above definition is bilinear and 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 one says that it has been "lifted" to all of 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 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 of is defined as the quotient space
where the equivalence relation is given by
for all . 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 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 generated by elements of the form
for . (Note that these generators are elements of .) A general member of the ideal will be linear combinations of elements of the form
where all lower-case letters are elements of . Since is an ideal, we can quotient by it. The universal enveloping algebra can then be defined as Section 9.3
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 endowed with multiplication that takes elements If the multiplication is bilinear, then the same construction and definitions can go through. One starts by lifting up to so that the lifted obeys all of the same properties that the base does – symmetry or antisymmetry or whatever. The lifting is done exactly as before, starting with
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
and also that
This extension is consistent by appeal to a lemma on free objects: 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 superalgebras 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.