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. 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

.

Function composition is associative

The associativity of the universal enveloping algebra can be understood as arising from the associativity of function composition, one has that

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 (usually the reals or the complex numbers

Generators and relations

be the structure constants for this basis, so that

subject to the relations

and no other relations.

Consider, for example, the Lie algebra sl(2,C), spanned by the matrices

subject to the relations

and no other relations. We emphasize that the universal enveloping algebra is not 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 are all linearly independent in the universal enveloping algebra.

Finding a basis

plus a linear combination'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

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.

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. The universal enveloping algebra is the "largest" unital associative algebra with a Lie bracket compatible with the original Lie algebra.

Formal definition

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, without any restrictions whatsoever on those products.

That is, one constructs the space

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.

(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 in the conventional sense of a "lift" from a base space (here, the Lie algebra) to a covering space (here, the tensor algebra).

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 is defined as the quotient space

where the equivalence relation is given by

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

.) A general member of the ideal will be linear combinations of elements of the form

. Since is an ideal, we can quotient by it. The universal enveloping algebra can then be defined as

Superalgebras and other generalizations

If 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.

Another possibility is to use something other than the tensor algebra as the covering algebra. One such possibility is to use the exterior algebra; that is, to replace every occurrence of the tensor product by the exterior product. If the base algebra is a Lie algebra, then the result is the Gerstenhaber algebra; it is the exterior algebra of the corresponding Lie group. As before, it has a grading naturally coming from the grading on the exterior algebra. (The Gerstenhaber algebra should not be confused with the Poisson superalgebra; both invoke anticommutation, but in different ways.)

The construction has also been generalized for Malcev algebras, Bol algebras and left alternative algebras.

Universal property

, possesses a universal property. Suppose we have any Lie algebra map

to a unital associative algebra (with Lie bracket in given by the commutator). More explicitly, this means that we assume

. Then there exists a unique unital algebra homomorphism

such that

into its tensor algebra and then composing with the quotient map to the universal enveloping algebra. This map is an embedding, by the Poincaré–Birkhoff–Witt theorem.)

is a linear map must be uniquely determined by the requirement that

1. .

as a linear combination of products of Lie algebra elements.

) act as scalars and provide important information about the representations. The quadratic Casimir element is of particular importance in this regard.

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 contains the special Jordan algebras, but not the exceptional ones: that is, it does not envelope the Albert algebras. Likewise, the Poincaré–Birkhoff–Witt theorem, below, constructs a basis for an enveloping algebra; it just won't be universal. Similar remarks hold for the Lie superalgebras.

Poincaré–Birkhoff–Witt theorem

. This can be done in either one of two different ways: either by reference to an explicit vector basis on the Lie algebra, or in a coordinate-free fashion.

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 finitely supported functions from a set to the field is the indicator function to be

from the above, by enforcing the total order of has a basis

, the ordering being that of total order on the set . The proof of the theorem involves noting that, if one starts with out-of-order basis elements, these can always be swapped by using the commutator (together with the structure constants). The hard part of the proof is establishing that the final result is unique and independent of the order in which the swaps were performed.

This basis should be easily recognized as the basis of a symmetric algebra and the symmetric algebra are isomorphic, and it is the PBW theorem that shows that this is so. See, however, the section on the algebra of symbols, below, for a more precise statement of the nature of the isomorphism.

It is useful, perhaps, to split the process into two steps. In the first step, one constructs the free Lie algebra It can also be precisely defined: the basis elements are given by Hall words, a special case of which are the Lyndon words; these are explicitly constructed to behave appropriately as commutators.

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. This is accomplished by constructing a filtration

First, a notation is needed for an ascending sequence of subspaces of the tensor algebra. Let

where

is the form a filtration:

More precisely, this is a filtered algebra, since the filtration preserves the algebraic properties of the subspaces. Note that the limit

It was already established, above, that quotienting by the ideal is a natural transformation

Next, define the space

is not at all of the filtration, as one might naively surmise. It is not constructed through a set subtraction mechanism associated with the filtration.

. This is perhaps not immediately obvious: to get this result, one must repeatedly apply the commutation relations, and turn the crank. The essence of the Poincaré–Birkhoff–Witt theorem is that it is always possible to do this, and that the result is unique.

is necessarily zero. What is left are the elements that are not expressible as commutators.

In this way, one is lead immediately to the symmetric algebra. It is constructed by appeal to the same notion of naturality as before. One starts with the same tensor algebra, and just uses a different ideal, the ideal that makes all elements commute:

, both as a vector space and as a commutative algebra.

This is the associated graded algebra of the filtration.

is a projection, and one then gets PBW-type theorems for the associated graded algebra of a filtered algebra serving to remind that it is the filtered algebra.

Other algebras

The theorem, applied to Jordan algebras, yields the exterior algebra, rather than the symmetric algebra. In essence, the construction zeros out the anti-commutators. The resulting algebra is an enveloping algebra, but is not universal. As mentioned above, it fails to envelop the exceptional Jordan algebras.

Left-invariant differential operators

as the tangent space as the commutator on the associated left-invariant vector fields. This definition agrees with any other standard definition of the bracket structure on the Lie algebra of a Lie group.

.

arises as the Lie algebra of a real Lie group, one can use left-invariant differential operators to give an analytic proof of the Poincaré–Birkhoff–Witt theorem is apparent.)

Algebra of symbols

are isomorphic as associative algebras. This leads to the concept of the algebra of symbols: the space of symmetric polynomials, that places the algebraic structure of the Lie algebra onto what is otherwise a standard associative algebra. That is, what the PBW theorem obscures (the commutation relations) the algebra of symbols restores into the spotlight.

. The resulting polynomial is called the symbol. The inverse map is

act as an isomorphism, that is, so that

, as written, and that one must first perform a tedious reshuffling of the basis elements (applying the structure constants in the properly ordered basis. An explicit expression for this product can be given: this is the Berezin formula. It follows essentially from the Baker–Campbell–Hausdorff formula for the product of two elements of a Lie group.

A closed form expression is given by

where

in the chosen basis.

The universal enveloping algebra of the Heisenberg algebra is the Weyl algebra product is called the Moyal product.

Representation theory

The universal enveloping algebra preserves the representation theory: the representations correspond in a one-to-one manner to the modules. In more abstract terms, the abelian category of all representations is isomorphic.

The representation theory of semisimple Lie algebras rests on the observation that there is an isomorphism, known as the Kronecker product:

. The isomorphism follows from a lifting of the embedding

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.

is acting by infinitesimal transformations act like differential operators, of all orders. (See, for example, the realization of the universal enveloping algebra as left-invariant differential operators on the associated group, as discussed above.)

Casimir operators

The center. For a finite-dimensional semisimple Lie algebra, the Casimir operators. These may be constructed as follows.

Thus, a technique is needed for computing that kernel. What we have is the action of the adjoint representation is a derivation This implies that both of these are differential algebras.

if it obeys Leibniz's law:

, the Lie bracket is that of vector fields.) The lifting is performed by defining

From the PBW theorem, it is clear that all central elements are linear combinations of symmetric homogeneous polynomials of the Lie algebra. The Casimir invariants has the form

Recall that the adjoint representation is given directly by the structure constants, and so an explicit indexed form of the above equations can be given, in terms of the Lie algebra basis; this is originally a theorem of Israel Gel'fand, it follows that

where the structure constants are

As an example, the quadratic Casimir operator is

is the inverse matrix of the Killing form follows from the fact that the Killing form is invariant under the adjoint action.

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 -dimensional vector space , recall that the determinant is the completely antisymmetric tensor. Given a matrix , one may write the characteristic polynomial of as

For a -dimensional Lie algebra, that is, an algebra whose adjoint representation is -dimensional, the linear operator

is a -dimensional endomorphism, and so one has the characteristic equation

The non-zero roots of this characteristic polynomial (that are roots for all ) form the root system of the algebra. In general, there are only such roots; this is the rank of the algebra. This implies that the highest value of is non-vanishing is

are homogeneous polynomials of degree By plugging and chugging in the above, one obtains that

By linearity, if one expands in the basis,

then the polynomial has the form

, one concludes that this tensor must be completely symmetric. This tensor is exactly the Casimir invariant of order

for all by the above, these clearly corresponds to the roots of the characteristic equation. One concludes that the roots form a space of rank

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)= and direct computation gives

, and so the squared angular momentum operator for the rotation group is that Casimir operator. That is,

and explicit computation shows that

after making use of the structure constants

Example: Pseudo-differential operators

. Thus, one is led to a ring of pseudo-differential operators, from which one can construct Casimir invariants.

acts on a space of linear operators, such as in Fredholm theory, then one can construct Casimir invariants on the corresponding space of operators. The quadratic Casimir operator corresponds to an elliptic operator.

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 indices (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

, then it has a basis of matriceswhich satisfy the following identities under the standard bracket:this shows us that the universal enveloping algebra has the presentationas a non-commutative ring.

is abelian (that is, the bracket is always is commutative; and if a basis of the vector space can be identified with the polynomial algebra over , with one variable per basis element.

is the Lie algebra corresponding to the Lie group can be identified with the algebra of left-invariant differential operators (of all orders) on lying inside it as the left-invariant vector fields as first-order differential operators.

is a vector space 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.

consists of the left- and right- invariant differential operators; this, in the case of not commutative, is often not generated by first-order operators (see for example Casimir operator of a semi-simple Lie algebra).

as the convolution algebra of distributionss supporteded only at the identity element of .

The algebra of differential operators in 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 that 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 , one can construct the vector space of continuous complex-valued functions on , and turn it into a C*-algebra, one defines multiplication as

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 —the theory of compact topological groups and the theory of commutative Hopf algebras are the same. For Lie groups, this implies that

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.

See also

• Milnor–Moore theorem

• Harish-Chandra homomorphism

References

• Shlomo Sternberg (2004), Lie algebras, Harvard University.