In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the universal enveloping algebra of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
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. 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 group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebr ...
s. This dual can be shown, by the
Gelfand–Naimark theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra ''A'' is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 ...
, to contain the
C* algebra of the corresponding Lie group. This relationship generalizes to the idea of
Tannaka–Krein duality between
compact topological group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s 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
.
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 consists 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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
. 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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. Thus, one is free to construct the
tensor algebra from it. The tensor algebra is a
free algebra: it simply contains all possible
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
s 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 as follows. First, recall that the bracket operation on a Lie algebra is a bilinear map
that is
bilinear,
skew-symmetric and satisfies the
Jacobi identity. We wish to define a Lie bracket
,-that is a map
that is also bilinear, skew symmetric and obeys the Jacobi identity.
The lifting can be done grade by grade. Begin by ''defining'' the bracket on
as
: