Universal enveloping algebra

In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a 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.
The idea of the universal enveloping algebra is to embed a Lie algebra formula_1 into an associative algebra formula_2 with identity in such a way that the abstract bracket operation in formula_1 corresponds to the commutator formula_4 in formula_2 and the algebra formula_2 is generated by the elements of formula_1. There may be many ways to make such an embedding, but there is a unique "largest" such formula_2, called the universal enveloping algebra of formula_1.
Generators and relations.
Let formula_1 be a Lie algebra, assumed finite-dimensional for simplicity, with basis formula_11. Let formula_12 be the structure constants for this basis, so that
Then the universal enveloping algebra is the associative algebra (with identity) generated by elements formula_14 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 formula_16.
Consider, for example, the Lie algebra sl(2,C), spanned by the matrices
which satisfy the commutation relations formula_18, formula_19, and formula_20. The universal enveloping algebra of sl(2,C) is then the algebra generated by three elements formula_21 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 formula_23 matrices. For example, the formula_23 matrix formula_25 satisfies formula_26, as is easily verified. But in the universal enveloping algebra, the element formula_27 does not satisfy formula_28—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 formula_29 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 formula_30 first, then factors of formula_31, etc. For example, whenever we have a term that contains formula_32 (in the "wrong" order), we can use the relations to rewrite this as formula_33 plus a linear combination of the formula_34'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 formula_36's being non-negative integers, span the enveloping algebra. (We allow formula_37, meaning that we allow terms in which no factors of formula_34 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 formula_39 themselves are linearly independent. It is therefore common—if potentially confusing—to identify the formula_34's with the generators formula_41 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 formula_1 may be an algebra of formula_43 matrices, the universal enveloping of formula_1 is does not consist of (finite-dimensional) matrices. In particular, there is no finite-dimensional algebra that contains the universal enveloping of formula_1; 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 formula_25, formula_47 and formula_48 as formula_23 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 formula_50. The universal enveloping algebra is the "largest" unital associative algebra generated by elements of formula_16 with a Lie bracket compatible with the original Lie algebra.
Formal definition.
Recall that every Lie algebra formula_1 is in particular a vector space. Thus, one is free to construct the tensor algebra formula_53 from it. The tensor algebra is a free algebra: it simply contains all possible tensor products of all possible vectors in formula_1, without any restrictions whatsoever on those products.
That is, one constructs the space
where formula_56 is the tensor product, and formula_57 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 formula_58 that is bilinear, skew-symmetric and satisfies the Jacobi identity. We wish to define a Lie bracket [-,-] that is a map formula_59 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 formula_60 as
This is a consistent, coherent definition, because both sides are bilinear, and both sides are skew symmetric (the Jacobi identity will follow shortly). The above defines the bracket on formula_62; it must now be lifted to formula_63 for arbitrary formula_64 This is done recursively, by "defining"
and likewise
It is straightforward to verify that the above definition is bilinear, and is 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 formula_67 one says that it has been "lifted" to all of formula_53 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 explicitly a Poisson algebra. It is a unital associative algebra with a Lie bracket that is compatible with the Lie algebra bracket; it is compatible 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 formula_69 of formula_1 is defined as the quotient space
where the equivalence relation formula_72 is given by
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 formula_69 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 ultimately equivalent) way. Forget, for a moment, the above lifting, and instead consider the two-sided ideal generated by elements of the form
This generator is an element of
A general member of the ideal will have the form
for some formula_78 All elements of are obtained as linear combinations of elements of this form. Clearly, formula_79 is a subspace. It is an ideal, in that if formula_80 and formula_81 then formula_82 and formula_83 Establishing that this is an ideal is important, because ideals are precisely those things that one can quotient with; ideals lie in the kernel of the quotienting map. That is, one has the short exact sequence
where each arrow is a linear map, and the kernel of that map is given by the image of the previous map. The universal enveloping algebra can then be defined as 
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 formula_86 endowed with multiplication formula_87 that takes elements formula_88 "If" the multiplication is bilinear, then the same construction and definitions can go through. One starts by lifting formula_89 up to formula_90 so that the lifted formula_89 obeys all of the same properties that the base formula_89 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.
The universal enveloping algebra, or rather the universal enveloping algebra together with the canonical map formula_96, 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
for all formula_99. Then there exists a "unique" unital algebra homomorphism
such that
where formula_96 is the canonical map. (The map formula_103 is obtained by embedding formula_1 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.)
To put it differently, if formula_105 is a linear map into a unital algebra formula_106 satisfying formula_98, then formula_108 extends to an algebra homomorphism of formula_100. Since formula_110 is generated by elements of formula_1, the map formula_112 must be uniquely determined by the requirement that
The point is that because there are no other relations in the universal enveloping algebra besides those coming from the commutation relations of formula_1, the map formula_112 is well defined, independent of how one writes a given element formula_116 as a linear combination of products of Lie algebra elements.
The universal property of the enveloping algebra immediately implies that every representation of formula_1 acting on a vector space formula_86 extends uniquely to a representation of formula_69. (Take formula_120.) This observation is important because it allows (as discussed below) the Casimir elements to act on formula_86. These operators (from the center of formula_69) 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.
The Poincaré–Birkhoff–Witt theorem gives a precise description of formula_69. 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 finite supported functions from a set to the field (finitely supported means that only finitely many values are non-zero); it can be given a basis formula_124 such that formula_125 is the indicator function for formula_126. Let formula_127 be the injection into the tensor algebra; this is used to give the tensor algebra a basis as well. This is done by lifting: given some arbitrary sequence of formula_128, one defines the extension of formula_103 to be
The Poincaré–Birkhoff–Witt theorem then states that one can obtain a basis for formula_69 from the above, by enforcing the total order of onto the algebra. That is, formula_69 has a basis
where formula_134, 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. That is, the underlying vector spaces of formula_69 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: this is what one gets, if one mods out by all commutators, without specifying what the values of the commutators are. The second step is to apply the specific commutation relations from formula_136 The first step is universal, and does not depend on the specific formula_136 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 formula_138 whose limit is the universal enveloping algebra formula_139
First, a notation is needed for an ascending sequence of subspaces of the tensor algebra. Let
where
is the -times tensor product of formula_136 The formula_143 form a filtration:
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 formula_145
It was already established, above, that quotienting by the ideal is a natural transformation that takes one from formula_53 to formula_139 This also works naturally on the subspaces, and so one obtains a filtration formula_138 whose limit is the universal enveloping algebra formula_139
Next, define the space
This is the space formula_138 modulo all of the subspaces formula_152 of strictly smaller filtration degree. Note that formula_153 is "not at all" the same as the leading term formula_154 of the filtration, as one might naively surmise. It is not constructed through a set subtraction mechanism associated with the filtration.
Quotienting formula_138 by formula_156 has the effect of setting all Lie commutators defined in formula_138 to zero. One can see this by observing that the commutator of a pair of elements whose products lie in formula_158 actually gives an element in formula_156. 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.
Since commutators of elements whose products are defined in formula_158 lie in formula_156, the quotienting that defines formula_153 has the effect of setting all commutators to zero. What PBW states is that the commutator of elements in formula_153 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. This is the algebra where all commutators vanish. It can be defined as a filtration formula_164 of symmetric tensor products formula_165. Its limit is the symmetric algebra formula_166. 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:
Thus, one can view the Poincaré–Birkhoff–Witt theorem as stating that formula_168 is isomorphic to the symmetric algebra formula_166, both as a vector space "and" as a commutative algebra.
The formula_153 also form a filtered algebra; its limit is formula_171 This is the associated graded algebra of the filtration.
The construction above, due to its use of quotienting, implies that the limit of formula_168 is isomorphic to formula_139 In more general settings, with loosened conditions, one finds that formula_174 is a projection, and one then gets PBW-type theorems for the associated graded algebra of a filtered algebra. To emphasize this, the notation formula_175 is sometimes used for formula_176 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.
Suppose formula_177 is a real Lie group with Lie algebra formula_1. Following the modern approach, we may identify formula_1 with the space of left-invariant vector fields (i.e., first-order left-invariant differential operators). Specifically, if we initially think of formula_1 as the tangent space to formula_177 at the identity, then each vector in formula_1 has a unique left-invariant extension. We then identify the vector in the tangent space with the associated left-invariant vector field. Now, the commutator (as differential operators) of two left-invariant vector fields is again a vector field and again left-invariant. We can then define the bracket operation on formula_1 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.
We may then consider left-invariant differential operators of arbitrary order. Every such operator formula_106 can be expressed (non-uniquely) as a linear combination of products of left-invariant vector fields. The collection of all left-invariant differential operators on formula_177 forms an algebra, denoted formula_186. It can be shown that formula_186 is isomorphic to the universal enveloping algebra formula_69.
In the case that formula_1 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. Specifically, the algebra formula_186 of left-invariant differential operators is generated by elements (the left-invariant vector fields) that satisfy the commutation relations of formula_1. Thus, by the universal property of the enveloping algebra, formula_186 is a quotient of formula_69. Thus, if the PBW basis elements are linearly independent in formula_186—which one can establish analytically—they must certainly be linearly independent in formula_69. (And, at this point, the isomorphism of formula_186 with formula_69 is apparent.)
Algebra of symbols.
The underlying vector space of formula_166 may be given a new algebra structure so that formula_69 and formula_166 are isomorphic "as associative algebras". This leads to the concept of the algebra of symbols formula_201: the space of symmetric polynomials, endowed with a product, the formula_202, 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 algebra is obtained by taking elements of formula_166 and replacing each generator formula_204 by an indeterminate, commuting variable formula_205 to obtain the space of symmetric polynomials formula_206 over the field formula_207. Indeed, the correspondence is trivial: one simply substitutes the symbol formula_205 for formula_204. The resulting polynomial is called the symbol of the corresponding element of formula_166. The inverse map is
that replaces each symbol formula_205 by formula_204. The algebraic structure is obtained by requiring that the product formula_202 act as an isomorphism, that is, so that
for polynomials formula_216
The primary issue with this construction is that formula_217 is not trivially, inherently a member of formula_69, as written, and that one must first perform a tedious reshuffling of the basis elements (applying the structure constants as needed) to obtain an element of formula_69 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
and formula_222 is just formula_89 in the chosen basis.
The universal enveloping algebra of the Heisenberg algebra is the Weyl algebra (modulo the relation that the center be the unit); here, the formula_202 product is called the Moyal product.
Representation theory.
The universal enveloping algebra preserves the representation theory: the representations of formula_1 correspond in a one-to-one manner to the modules over formula_69. In more abstract terms, the abelian category of all representations of formula_1 is isomorphic to the abelian category of all left modules over formula_69.
The representation theory of semisimple Lie algebras rests on the observation that there is an isomorphism, known as the Kronecker product:
for Lie algebras formula_230. 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.
In a typical context where formula_1 is acting by "infinitesimal transformations", the elements of formula_69 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 of formula_69 is formula_236 and can be identified with the centralizer of formula_1 in formula_139 Any element of formula_236 must commute with all of formula_240 and in particular with the canonical embedding of formula_1 into formula_242 Because of this, the center is directly useful for classifying representations of formula_1. For a finite-dimensional semisimple Lie algebra, the Casimir operators form a distinguished basis from the center formula_236. These may be constructed as follows.
The center formula_236 corresponds to linear combinations of all elements formula_246 that that commute with all elements formula_247 that is, for which formula_248 That is, they are in the kernel of formula_249 Thus, a technique is needed for computing that kernel. What we have is the action of the adjoint representation on formula_250 we need it on formula_139 The easiest route is to note that formula_252 is a derivation, and that the space of derivations can be lifted to formula_53 and thus to formula_139 This implies that both of these are differential algebras.
By definition, formula_255 is a derivation on formula_1 if it obeys Leibniz's law:
(It would not be facetious to note that the Lie bracket becomes the Lie derivative when acting on a manifold; the above is a hint for how this is plays out.) The lifting is performed by "defining"
Since formula_259 is a derivation for any formula_260 the above defines formula_259 acting on formula_53 and formula_139 
From the PBW theorem, it is clear that all central elements are linear combinations of symmetric homogenous polynomials in the basis elements formula_128 of the Lie algebra. The Casimir invariants are the irreducible homogenous polynomials of a given, fixed degree. That is, given a basis formula_128, a Casimir operator of order formula_89 has the form
where there are formula_89 terms in the tensor product, and formula_269 is a completely symmetric tensor of order formula_89 belonging to the adjoint representation. That is, formula_269 can be (should be) thought of as an element of formula_272 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. That is, from formula_273, it follows that
where the structure constants are
As an example, the quadratic Casimir operator is
where formula_277 is the inverse matrix of the Killing form formula_278 That the Casimir operator formula_279 belongs to the center formula_236 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 on formula_281. 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
implies that formula_284 is a -dimensional endomorphism, and so one has the characteristic equation
for elements formula_286 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 for which the formula_287 is non-vanishing is 
The formula_287 are homogeneous polynomials of degree This can be seen in several ways: Given a constant formula_289, ad is linear, so that formula_290 By plugging and chugging in the above, one obtains that
By linearity, if one expands in the basis,
then the polynomial has the form
that is, a formula_294 is a tensor of rank formula_295. By linearity and the commutativity of addition, i.e. that formula_296, one concludes that this tensor must be completely symmetric. This tensor is exactly the Casimir invariant of order 
The center formula_297 corresponded to those elements formula_298 for which formula_299 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 and that the Casimir invariants span this space. That is, the Casimir invariants generate the center formula_300
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 formula_301 As an elementary exercise, one can compute this directly. Changing notation to formula_302 with formula_303 belonging to the adjoint rep, a general algebra element is formula_304 and direct computation gives
The quadratic term can be read off as formula_306, 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.
A key observation during the construction of formula_69 above was that it was a differential algebra, by dint of the fact that any derivation on the Lie algebra can be lifted to formula_69. Thus, one is led to a ring of pseudo-differential operators, from which one can construct Casimir invariants.
If the Lie algebra formula_1 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.
If formula_313, then it has a basis of matricesformula_314which satisfy the following identities under the standard bracket:formula_315, formula_316, and formula_317this shows us that the universal enveloping algebra has the presentationformula_318as a non-commutative ring.
If formula_1 is "abelian" (that is, the bracket is always ), then formula_69 is commutative; and if a basis of the vector space formula_1 has been chosen, then formula_69 can be identified with the polynomial algebra over , with one variable per basis element.
If formula_1 is the Lie algebra corresponding to the Lie group , then formula_69 can be identified with the algebra of left-invariant differential operators (of all orders) on ; with formula_1 lying inside it as the left-invariant vector fields as first-order differential operators.
To relate the above two cases: if formula_1 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.
The center formula_297 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).
Another characterization in Lie group theory is of formula_69 as the convolution algebra of distributions supported 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. This algebra has a natural Hopf algebra structure: given two functions
formula_329, 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 is isomorphically dual to formula_69; more precisely, it is isomorphic to a subspace of the dual space formula_335
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.