In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.
Contents
- Informal construction
- Informal interpretation
- Formal definition
- Superalgebras
- Other generalizations
- Universal property
- Other algebras
- PoincarBirkhoffWitt theorem
- Using basis elements
- Coordinate free
- Algebra of symbols
- Representation theory
- Casimir operators
- Rank
- Example Rotation group SO3
- Example Pseudo differential operators
- Examples in particular cases
- Hopf algebras and quantum groups
- References
Universal enveloping algebras play a relatively minor role in the representation theory of Lie groups; the greatest utility is perhaps to give 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 which 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.
Informal construction
An intuitive idea of the algebra can be obtained as follows: imagine the space of all polynomials in one variable x. For some given polynomial p(x), substitute elements
One can arrive at the formal definition by re-interpreting the power series above as an element of the tensor algebra
Informal interpretation
A different informal understanding is obtained by observing that universal enveloping algebras are isomorphic to C*-algebras on the manifold of a Lie group. That is, one can imagine that
More precisely, the isomorphism is to a subspace of the dual vector space
Formal definition
Recall that every Lie algebra
That is, one constructs the space
where
The universal enveloping algebra is obtained by taking the quotient by imposing the relations
for all a and b in the embedding of
can be lifted to define a Lie bracket on the tensor algebra, starting with
This is possible precisely because the tensor product is bilinear, and the Lie bracket is bilinear! This indicates why the construction given here is specific to Lie algebras, and might not work out for other things: the multiplication must be bilinear, in order to be consistent. This also indicates exactly when a universal enveloping algebra can be constructed for some object: if it has a bilinear operator, then the construction can go through.
The lifting is done in such a way as to preserve multiplication as a homomorphism, that is, by definition, one has that
and also that
Observe that, in the above, the relative ordering of a and b was preserved in the first equation, and that the relative ordering of b and c was preserved in the second equation. This is important, since the tensor product is neither commutative nor anti-commutative. Thus, these two are consistency conditions for the Lie bracket on the tensor algebra. These two are sufficient to extend the notion of the Lie bracket to the entire tensor algebra, by appealing to a lemma: since the tensor algebra is a free algebra, any homomorphism on its generating set can be extended to the entire algebra. The only reason for (temporarily) switching to the notation m(a,b) is to make the distributive nature of the above homomorphism more readily apparent, and in keeping with the ordinary notation for multiplication in commutative diagrams of homomorphisms.
In addition to the product rule just discussed, this lifted bracket can be shown to obey the Jacobi identity, and thus, is properly called the Poisson bracket. The result of the lifting is that the tensor algebra of a Lie algebra is a Poisson algebra.
To obtain the universal enveloping algebra, one creates the quotient space
where I is the two-sided ideal over
Note that the above is an element of
and so can be validly used to construct the ideal within
as an element of I, and all elements of I are obtained as linear combinations of elements of the above form. Clearly,
Superalgebras
The analogous construction for Lie superalgebras is straightforward; one need only to keep careful track of the sign, when permuting elements. In this case, the (anti-)commutator of the superalgebra lifts to an (anti-)commuting Poisson bracket.
One can obtain a different result by taking the above construction, and replacing every occurrence of the tensor product by the exterior product. That is, one uses this construction to create the exterior algebra of the Lie group; this construction results in the Gerstenhaber algebra, with the grading naturally coming from the grading on the exterior algebra. (This should not be confused with the Poisson superalgebra).
Other generalizations
The construction has also been generalized for Malcev algebras, Bol algebras and left alternative algebras.
Universal property
The universal enveloping algebra, or rather the universal enveloping algebra together with the canonical map
to a unital associative algebra A (with Lie bracket in A given by the commutator), there exists a unique unital algebra homomorphism
such that
where
This universal property follows from the tensor algebra as a natural transformation. That is, there is a functor T from the category of Lie algebras over K to the category of unital associative K-algebras, taking a Lie algebra to the corresponding free algebra. Similarly, there is also a functor U that takes the same category of Lie algebras to the same category of unital associative K-algebras. The two are related by a natural map that takes T into U: that natural map is the action of quotienting. The universal property passes through the natural map.
The functor U is left adjoint to the functor which maps an algebra A to the Lie algebra AL. (Recall that, given an associate algebra A, one can always build a corresponding Lie algebra AL with underlying vector space A and the Lie bracket given by the commutator of two elements of A). The two are adjoint, but certainly are not inverses: if we start with an associative algebra A, then U(AL) is not equal to A; it is much bigger.
Other algebras
Although the canonical construction, given above, can be applied to other algebras, the result, in general, does not have the universal property. Thus, for example, when the construction is applied to Jordan algebras, the resulting enveloping algebra will contain the special Jordan algebras, but not the exceptional ones: that is, it will not envelope the Albert algebras. Likewise, the Poincaré–Birkhoff–Witt theorem, below, will construct a basis for an enveloping algebra; it just won't be universal. Similar remarks hold for the Lie superalgebras.
Poincaré–Birkhoff–Witt theorem
The Poincaré–Birkhoff–Witt theorem gives a precise description of
Using basis elements
One way is to suppose that the Lie algebra can be given a totally ordered basis, that is, it is the free vector space of a totally ordered set. Recall that a free vector space is defined as the space of all functions from a set X to the field K; it can be given a basis
The Poincaré–Birkhoff–Witt theorem then states that one can obtain a basis for
where
This basis should be easily recognized as the basis of a symmetric algebra. That is,
Coordinate-free
One can also state the theorem in a coordinate-free fashion, avoiding the use of total orders and basis elements. This is convenient when there are difficulties in defining the basis vectors, as there can be for infinite-dimensional Lie algebras. It also gives a more natural form that is more easily extended to other kinds of algebras.
The proper setup requires only a little bit more machinery, most of which should already be apparent. One begins by defining a notation for certain subspaces of the tensor algebra. Let
where
is the m-times tensor product of
More precisely, this is a filtered algebra, since the filtration preserves the algebraic properties of the subspaces. Note that the limit of this filtration is the tensor algebra. By naturality (discussed above), one may define a filtration
Define the space
That is, it is the space
The Poincaré–Birkhoff–Witt theorem then states that
The construction here employs a bit of an empty trick: since the filtered algebra is built out of a graded algebra, the resulting associated algebra is "trivially" isomorphic. That is, in this sketch, one may take
Other algebras
The theorem, applied to Jordan algebras, will yield a basis that is given by the exterior algebra, rather than the symmetric algebra. The resulting algebra will be an enveloping algebra; it will not be universal. As mentioned above, it fails to envelope the exceptional Jordan algebras.
Algebra of symbols
The isomorphism of
The algebra is obtained by taking elements of
that replaces each symbol
for polynomials
The primary issue with this construction is that
A closed form expression is given by
where
and
The universal enveloping algebra of the Heisenberg algebra is the Weyl algebra (modulo the relation that the center be the unit); here, the
Representation theory
The universal enveloping algebra preserves the representation theory: the representations of
The representation theory of semisimple Lie algebras rests on the observation that there is an isomorphism, known as the Kronecker product:
for Lie algebras
where
is just the canonical embedding (with subscripts, respectively for algebras one and two). It is straightforward to verify that this embedding lifts, given the prescription above. See, however, the discussion of the bialgebra structure in the article on tensor algebras for a review of some of the finer points of doing so: in particular, the shuffle product employed there corresponds to the Wigner-Racah coefficients, i.e. the 6j and 9j-symbols, etc.
Also important is that the universal enveloping algebra of a free Lie algebra is isomorphic to the free associative algebra.
Construction of representations typically proceeds by building the Verma modules of the highest weights.
In a typical context where
Casimir operators
The center of
One begins by noting that any element
The lifting is performed by defining
for elements
The algebra
or, equivalently,
where, for an element
The center
From the PBW theorem, it is clear that all such central elements will be linear combinations of symmetric homogenous polynomials in the basis elements
where there are
where the structure constants are
As an example, the quadratic Casimir operator is
where
The center of the universal enveloping algebra of a simple Lie algebra is given in detail by the Harish-Chandra isomorphism.
Rank
The number of algebraically independent Casimir operators of a finite-dimensional semisimple Lie algebra is equal to the rank of that algebra, i.e. is equal to the rank of the Cartan-Weyl basis. This may be seen as follows. For a d-dimensional vector space V, recall that the determinant is the completely antisymmetric tensor on
For a d-dimensional Lie algebra, that is, an algebra whose adjoint representation is d-dimensional, the linear operator
implies that
for elements
The
By linearity, if one expands in the basis,
then the polynomial has the form
that is, a
The center
Example: Rotation group SO(3)
The rotation group SO(3) is of rank one, and thus has one Casimir operator. It is three-dimensional, and thus the Casimir operator must have order (3-1)=2 i.e. be quadratic. Of course, this is the Lie algebra of
The quadratic term can be read off as
and explicit computation shows that
after making use of the structure constants
Example: Pseudo-differential operators
A key observation during the construction of
If the Lie algebra
If the Lie algebra acts on a differentiable manifold, then each Casimir operator corresponds to a higher-order differential on the cotangent manifold, the second-order differential being the most common and most important.
If the action of the algebra is isometric, as would be the case for Riemannian or pseudo-Riemannian manifolds endowed with a metric and the symmetry groups SO(N) and SO (P, Q), respectively, one can then contract upper and lower indecies (with the metric tensor) to obtain more interesting structures. For the quadratic Casimir invariant, this is the Laplacian. Quartic Casimir operators allow one to square the stress–energy tensor, giving rise to the Yang-Mills action. The Coleman–Mandula theorem restricts the form that these can take, when one considers ordinary Lie algebras. However, the Lie superalgebras are able to evade the premises of the Coleman–Mandula theorem, and can be used to mix together space and internal symmetries.
Examples in particular cases
If
which satisfy the following identities under the standard bracket:
this shows us that the universal enveloping algebra has the presentation
as a non-commutative ring.
If
If
To relate the above two cases: if
The center
Another characterization in Lie group theory is of
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.
The universal enveloping algebra of a finite-dimensional Lie algebra is a filtered quadratic algebra.
Hopf algebras and quantum groups
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. This is made precise in the article on the tensor algebra: the tensor algebra has a Hopf algebra structure on it, and because the Lie bracket is consistent with (obeys the consistency conditions for) that Hopf structure, it is inherited by the universal enveloping algebra.
Given a Lie group G, one can construct the vector space C(G) of continuous complex-valued functions on G, and turn it into a C*-algebra. This algebra has a natural Hopf algebra structure: given two functions
and comultiplication as
the counit as
and the antipode as
Now, the Gelfand-Naimark theorem essentially states that every commutative Hopf algebra is isomorphic to the Hopf algebra of continuous functions on some compact topological group G -- the theory of compact topological groups and the theory of commutative Hopf algebras are the same. For Lie groups, this implies that C(G) is isomorphically dual to
These ideas can then be extended to the non-commutative case. One starts by defining the quasi-triangular Hopf algebras, and then performing what is called a quantum deformation to obtain the quantum universal enveloping algebra, or quantum group, for short.