# Universal enveloping algebra

For the universal enveloping W* algebra of a C* algebra, see Sherman–Takeda theorem.

In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.

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 ${\displaystyle g\in {\mathfrak {g}}}$ of the Lie algebra ${\displaystyle {\mathfrak {g}}}$, to obtain the formal power series p(g). Next, apply any and all commutation relations appropriate for that Lie algebra, to identify as equal any other polynomials that arise. In this way, one obtains the universal enveloping algebra ${\displaystyle U({\mathfrak {g}})}$ as the space of all such formal power series. To get a better intuitive idea of the meaning of such a formal power series, suppose that one had a matrix representation ${\displaystyle R({\mathfrak {g}})}$ of ${\displaystyle {\mathfrak {g}}}$, which associates an ordinary matrix R(g) with each element ${\displaystyle g\in {\mathfrak {g}}}$. Substituting the matrix into the formal power series, it becomes well-defined, since the (ordinary) multiplication of matricies is well-defined. One gets some value simply by performing the needed matrix multiplications. This informal construction makes clear where the name "universal enveloping algebra" comes from: one has the glimmer that every Lie group G that one might ever obtain from ${\displaystyle {\mathfrak {g}}}$ is contained inside of it. This should be obvious, if one imagines that every Lie group corresponds to some matrix representation of ${\displaystyle {\mathfrak {g}}}$. It should certainly be clear that every representation of ${\displaystyle {\mathfrak {g}}}$ will be contained in the universal enveloping algebra.

One can arrive at the formal definition by re-interpreting the power series above as an element of the tensor algebra ${\displaystyle T({\mathfrak {g}})}$, and defining the universal enveloping algebra ${\displaystyle U({\mathfrak {g}})}$ as the quotient of this free algebra, by all of the commutation relations of ${\displaystyle {\mathfrak {g}}}$. Insofar as ${\displaystyle {\mathfrak {g}}}$ is just an ordinary vector space, the generators of ${\displaystyle {\mathfrak {g}}}$ correspond to basis elements of the vector space. The monomials of these generators then generate the tensor algebra, on which the quotienting may be performed. This definition is expanded upon, below.

## 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 ${\displaystyle U({\mathfrak {g}})}$ looks like the set C(G) of all continuous complex-valued functions on the manifold of the (simply-connected) Lie group G that corresponds to ${\displaystyle {\mathfrak {g}}}$. This follows from an isomorphism of Hopf algebras: first, the C*-algebra C(G) has a natural Hopf algebra structure; next, ${\displaystyle U({\mathfrak {g}})}$ inherits a Hopf algebra structure from the tensor algebra. It can be shown that the two are isomorphic, as Hopf algebras. Such isomorphisms are studied under the umbrella of Tannaka-Krein duality.

More precisely, the isomorphism is to a subspace of the dual vector space ${\displaystyle U^{*}({\mathfrak {g}})}$. In particular, commutativity in C(G) corresponds to co-commutativity in ${\displaystyle U({\mathfrak {g}})}$ (that is, commutativity in ${\displaystyle U^{*}({\mathfrak {g}}):}$ the tensor algebra is co-commutative). The relaxation of this commutativity condition inspires the study of non-commutative geometry.

## Formal definition

Recall that every Lie algebra ${\displaystyle {\mathfrak {g}}}$ is just a vector space. Thus, one is free to construct the tensor algebra ${\displaystyle T({\mathfrak {g}})}$ from it. The tensor algebra is a free algebra: it simply contains all possible tensor products of all possible vectors in ${\displaystyle {\mathfrak {g}}}$, without any restrictions whatsoever on those products.

That is, one constructs the space

${\displaystyle T({\mathfrak {g}})=K\,\oplus \,{\mathfrak {g}}\,\oplus \,({\mathfrak {g}}\otimes {\mathfrak {g}})\,\oplus \,({\mathfrak {g}}\otimes {\mathfrak {g}}\otimes {\mathfrak {g}})\,\oplus \,\cdots }$

where ${\displaystyle \otimes }$ is the tensor product, and ${\displaystyle \oplus }$ is the direct sum of vector spaces. Here, K is the field over which the Lie algebra is defined. From here, through to the remainder of this article, the tensor product will always be 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 universal enveloping algebra is obtained by taking the quotient by imposing the relations

${\displaystyle a\otimes b-b\otimes a=[a,b]}$

for all a and b in the embedding of ${\displaystyle {\mathfrak {g}}}$ in ${\displaystyle T({\mathfrak {g}}).}$ To avoid the tautological feeling of this equation, keep in mind that the bracket on the right hand side of this equation is actually the Lie algebra product. That is, Lie algebras come with a product ${\displaystyle m(a,b)=[a,b]}$ for which the bracket symbol [-,-] is commonly used. Formally, the Lie bracket

{\displaystyle {\begin{aligned}m:{\mathfrak {g}}\times {\mathfrak {g}}&\to {\mathfrak {g}}\\a\times b&\mapsto m(a,b)=[a,b]\end{aligned}}}

can be lifted to define a Lie bracket on the tensor algebra, starting with

{\displaystyle {\begin{aligned}m:{\mathfrak {g}}\otimes {\mathfrak {g}}&\to {\mathfrak {g}}\\a\otimes b&\mapsto m(a,b)=[a,b]\end{aligned}}}

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

${\displaystyle m(a\otimes b,c)=a\otimes m(b,c)+m(a,c)\otimes b}$

and also that

${\displaystyle m(a,b\otimes c)=m(a,b)\otimes c+b\otimes m(a,c)}$

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

${\displaystyle U({\mathfrak {g}})=T({\mathfrak {g}})/I}$

where I is the two-sided ideal over ${\displaystyle T({\mathfrak {g}})}$ generated by elements of the form

${\displaystyle a\otimes b-b\otimes a-[a,b]}$

Note that the above is an element of

${\displaystyle {\mathfrak {g}}\oplus ({\mathfrak {g}}\otimes {\mathfrak {g}})\subset T({\mathfrak {g}})}$

and so can be validly used to construct the ideal within ${\displaystyle T({\mathfrak {g}})}$. Thus, for example, given ${\displaystyle a,b,c,d,f,g\in {\mathfrak {g}}}$, one can write

${\displaystyle c\otimes d\otimes \cdots \otimes (a\otimes b-b\otimes a-[a,b])\otimes f\otimes g\cdots }$

as an element of I, and all elements of I are obtained as linear combinations of elements of the above form. Clearly, ${\displaystyle I\subset T({\mathfrak {g}})}$ is a subspace. In essence, the universal enveloping algebra is what remains of the tensor algebra after modding out the Poisson algebra structure.

### 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,[1] Bol algebras [2] and left alternative algebras.[3]

## Universal property

The universal enveloping algebra, or rather the universal enveloping algebra together with the canonical map ${\displaystyle h:{\mathfrak {g}}\to U({\mathfrak {g}})}$, possesses a universal property: for any Lie algebra map

${\displaystyle \phi :{\mathfrak {g}}\to A}$

to a unital associative algebra A (with Lie bracket in A given by the commutator), there exists a unique unital algebra homomorphism

${\displaystyle {\widehat {\phi }}:U({\mathfrak {g}})\to A}$

such that

${\displaystyle \phi ={\widehat {\phi }}\circ h}$

where ${\displaystyle h:{\mathfrak {g}}\to U({\mathfrak {g}})}$ is the canonical map, an embedding by the Poincare-Birkhoff-Witt theorem, taking elements of ${\displaystyle {\mathfrak {g}}}$ into ${\displaystyle U({\mathfrak {g}}).}$

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 simply by quotienting 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 ${\displaystyle U({\mathfrak {g}})}$. 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 functions from a set X to the field K; it can be given a basis ${\displaystyle e_{a}:X\to K}$ such that ${\displaystyle e_{a}(b)=\delta _{ab}}$ is the indicator function for ${\displaystyle a,b\in X}$. Let ${\displaystyle h:{\mathfrak {g}}\to T({\mathfrak {g}})}$ 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 ${\displaystyle e_{a}}$, one defines the extension of ${\displaystyle h}$ to be

${\displaystyle h(e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c})=h(e_{a})\otimes h(e_{b})\otimes \cdots \otimes h(e_{c})}$

The Poincaré–Birkhoff–Witt theorem then states that one can obtain a basis for ${\displaystyle U({\mathfrak {g}})}$ from the above, by enforcing the total order of X onto the algebra. That is, ${\displaystyle U({\mathfrak {g}})}$ has a basis

${\displaystyle e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c}}$

where ${\displaystyle a\leq b\leq \cdots \leq c}$, the ordering being that of total order on the set X. The proof of the theorem involves noting if one starts with out-of-order basis elements, these can always be swapped by using the commutator (together with the structure constants), and that the final result is 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, ${\displaystyle U({\mathfrak {g}})}$ and the symmetric algebra are isomorphic as vector spaces (but not as algebras!) The isomorphism, as algebras, is restored by the constructing the algebra of symbols, below.

### 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

${\displaystyle T_{m}{\mathfrak {g}}=K\oplus {\mathfrak {g}}\oplus T^{2}{\mathfrak {g}}\oplus \cdots \oplus T^{m}{\mathfrak {g}}}$

where

${\displaystyle T^{m}{\mathfrak {g}}=T^{\otimes m}{\mathfrak {g}}={\mathfrak {g}}\otimes \cdots \otimes {\mathfrak {g}}}$

is the m-times tensor product of ${\displaystyle {\mathfrak {g}}.}$ The ${\displaystyle T_{m}{\mathfrak {g}}}$ form a filtration:

${\displaystyle K\subset {\mathfrak {g}}\subset T_{2}{\mathfrak {g}}\subset \cdots \subset T_{m}{\mathfrak {g}}\subset \cdots }$

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 ${\displaystyle U_{m}{\mathfrak {g}}}$ whose limit is the universal enveloping algebra ${\displaystyle U({\mathfrak {g}});}$ likewise one defines a filtration ${\displaystyle S_{m}{\mathfrak {g}}}$ of symmetric tensor products ${\displaystyle {\mbox{Sym}}^{m}{\mathfrak {g}}}$ whose limit is the symmetric algebra ${\displaystyle S({\mathfrak {g}})}$. Recall how the symmetric algebra is constructed: the process is exactly the same as described up top, except that one uses a different ideal, the ideal that makes all elements commute:

${\displaystyle S({\mathfrak {g}})=T({\mathfrak {g}})/(a\otimes b-b\otimes a)}$

Define the space

${\displaystyle G_{m}{\mathfrak {g}}=U_{m}{\mathfrak {g}}/U_{m-1}{\mathfrak {g}}}$

That is, it is the space ${\displaystyle U_{m}{\mathfrak {g}}}$ modulo all of the subspaces ${\displaystyle U_{n}{\mathfrak {g}}}$ of strictly smaller dimension. Note that ${\displaystyle G_{m}{\mathfrak {g}}}$ is not the same as ${\displaystyle U^{m}{\mathfrak {g}}.}$ Informally, the modding of ${\displaystyle U_{m}{\mathfrak {g}}}$ by ${\displaystyle U_{m-1}{\mathfrak {g}}}$ can be thought of as specifying a set of basis elements, in such a way that they are all ordered to be larger than all the "previous" basis elements, i.e. those occurring leftmost in the tensor products. One can show that the ${\displaystyle G_{m}{\mathfrak {g}}}$ also form a filtered algebra; its limit is ${\displaystyle G({\mathfrak {g}}).}$ This is the associated graded algebra of the filtration.

The Poincaré–Birkhoff–Witt theorem then states that ${\displaystyle G({\mathfrak {g}})}$ is isomorphic to the symmetric algebra ${\displaystyle S({\mathfrak {g}})}$ (as a vector space, not as an algebra).

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 ${\displaystyle G({\mathfrak {g}})}$ to be isomorphic to ${\displaystyle U({\mathfrak {g}}).}$ This variant of the PBW theorem is still useful, though, as in more general settings, one would have that ${\displaystyle U({\mathfrak {g}})\to G({\mathfrak {g}})}$ is a projection; one then gets PBW-type theorems for the associated graded algebra of a filtered algebra. The notation ${\displaystyle {\mbox{gr}}U({\mathfrak {g}})}$ is sometimes used for ${\displaystyle G({\mathfrak {g}}),}$ serving to remind that it's the filtered algebra.

### 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 ${\displaystyle U({\mathfrak {g}})}$ and ${\displaystyle S({\mathfrak {g}})}$, as vector spaces, leads to the concept of the algebra of symbols ${\displaystyle \star ({\mathfrak {g}})}$. This is the space of symmetric polynomials, endowed with a product, the ${\displaystyle \star }$, that restores the algebraic structure of the universal enveloping algebra. That is, what the PBW theorem took away (isomorphism, as algebras) the algebra of symbols restores.

The algebra is obtained by taking elements of ${\displaystyle S({\mathfrak {g}})}$ and replacing each generator ${\displaystyle e_{i}}$ by an indeterminate, commuting variable ${\displaystyle t_{i}}$ to obtain the space of symmetric polynomials ${\displaystyle K[t_{i}]}$ over the field ${\displaystyle K}$. Indeed, the correspondence is trivial: one simply substitutes the symbol ${\displaystyle t_{i}}$ for ${\displaystyle e_{i}}$. The resulting polynomial is called the symbol of the corresponding element of ${\displaystyle S({\mathfrak {g}})}$. The inverse map is

${\displaystyle w:\star ({\mathfrak {g}})\to U({\mathfrak {g}})}$

that replaces each symbol ${\displaystyle t_{i}}$ by ${\displaystyle e_{i}}$. The algebraic structure is obtained by requiring that the product ${\displaystyle \star }$ act as an isomorphism, that is, so that

${\displaystyle w(p\star q)=w(p)\otimes w(q)}$

for polynomials ${\displaystyle p,q\in \star ({\mathfrak {g}}).}$

The primary issue with this construction is that ${\displaystyle w(p)\otimes w(q)}$ is not trivially, inherently a member of ${\displaystyle U({\mathfrak {g}})}$, 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 ${\displaystyle U({\mathfrak {g}})}$ in the properly ordered basis. An explicit expression for this product can be given: this is the Berezin formula.[4] 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[5]

${\displaystyle p(t)\star q(t)=\left.\exp \left(t_{i}m^{i}\left({\frac {\partial }{\partial u}},{\frac {\partial }{\partial v}}\right)\right)p(u)q(v)\right\vert _{u=v=t}}$

where

${\displaystyle m(A,B)=\log \left(e^{A}e^{B}\right)-A-B}$

and ${\displaystyle m^{i}}$ is just ${\displaystyle m}$ 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 ${\displaystyle \star }$ product is called the Moyal product.

## Representation theory

The universal enveloping algebra preserves the representation theory: the representations of ${\displaystyle {\mathfrak {g}}}$ correspond in a one-to-one manner to the modules over ${\displaystyle U({\mathfrak {g}})}$. In more abstract terms, the abelian category of all representations of ${\displaystyle {\mathfrak {g}}}$ is isomorphic to the abelian category of all left modules over ${\displaystyle U({\mathfrak {g}})}$.

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

${\displaystyle U({\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2})\cong U({\mathfrak {g}}_{1})\otimes U({\mathfrak {g}}_{2})}$

for Lie algebras ${\displaystyle {\mathfrak {g}}_{1},{\mathfrak {g}}_{2}}$. The isomorphism follows from a lifting of the embedding

${\displaystyle i({\mathfrak {g}}_{1}\oplus {\mathfrak {g}}_{2})=i_{1}({\mathfrak {g}}_{1})\otimes 1\oplus 1\otimes i_{2}({\mathfrak {g}}_{2})}$

where

${\displaystyle i:{\mathfrak {g}}\to U({\mathfrak {g}})}$

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 ${\displaystyle {\mathfrak {g}}}$ is acting by infinitesimal transformations, the elements of ${\displaystyle U({\mathfrak {g}})}$ act like differential operators, of all orders.

## Casimir operators

The center of ${\displaystyle U({\mathfrak {g}})}$ is ${\displaystyle Z(U({\mathfrak {g}}))}$ and can be identified with the centralizer of ${\displaystyle {\mathfrak {g}}}$ in ${\displaystyle U({\mathfrak {g}})}$. That is, any element of ${\displaystyle Z(U({\mathfrak {g}}))}$ must commute not only with all of ${\displaystyle U({\mathfrak {g}})}$, but in particular, with the canonical embedding of ${\displaystyle {\mathfrak {g}}}$ into ${\displaystyle U({\mathfrak {g}})}$. Thus, the center is directly useful for classifying representations of ${\displaystyle {\mathfrak {g}}}$. For a finite-dimensional semisimple Lie algebra, the Casimir operators for a distinguished basis form the center ${\displaystyle Z(U({\mathfrak {g}}))}$. These may be constructed as follows.

One begins by noting that any element ${\displaystyle \delta \in {\mbox{Der}}({\mathfrak {g}})}$ in the space of derivations on ${\displaystyle {\mathfrak {g}}}$ can be lifted to a derivation on ${\displaystyle T({\mathfrak {g}}).}$ So, by definition, ${\displaystyle \delta :{\mathfrak {g}}\to {\mathfrak {g}}}$ is a derivation on ${\displaystyle {\mathfrak {g}}}$ if it obeys Leibniz's law:

${\displaystyle \delta ([v,w])=[\delta (v),w]+[v,\delta (w)]}$

The lifting is performed by defining

{\displaystyle {\begin{aligned}\delta (v\otimes w\otimes \cdots \otimes u)=&\,\delta (v)\otimes w\otimes \cdots \otimes u\\&+v\otimes \delta (w)\otimes \cdots \otimes u\\&+v\otimes w\otimes \cdots \otimes \delta (u)\end{aligned}}}

for elements ${\displaystyle v,w,u\in {\mathfrak {g}},}$ so that ${\displaystyle v\otimes w\otimes \cdots \otimes u\in T({\mathfrak {g}}).}$ The ideal used in the canonical construction of ${\displaystyle U({\mathfrak {g}})}$ is invariant under the derevation, and so the derivation descends to ${\displaystyle U({\mathfrak {g}})}$ itself. Thus, both ${\displaystyle T({\mathfrak {g}})}$ and ${\displaystyle U({\mathfrak {g}})}$ are differential algebras.

The algebra ${\displaystyle {\mathfrak {g}}}$ acts on itself by means of the adjoint representation, and this can be extended to an action on ${\displaystyle U({\mathfrak {g}})}$ merely by observing that it is a derivation. That is, by the above,

{\displaystyle {\begin{aligned}{\mbox{ad}}_{x}(v\otimes w\otimes \cdots \otimes u)=&\,{\mbox{ad}}_{x}(v)\otimes w\otimes \cdots \otimes u\\&+v\otimes {\mbox{ad}}_{x}(w)\otimes \cdots \otimes u\\&+v\otimes w\otimes \cdots \otimes {\mbox{ad}}_{x}(u)\end{aligned}}}

or, equivalently,

{\displaystyle {\begin{aligned}\left[x,(v\otimes w\otimes \cdots \otimes u)\right]=&\,[x,v]\otimes w\otimes \cdots \otimes u\\&+v\otimes [x,w]\otimes \cdots \otimes u\\&+v\otimes w\otimes \cdots \otimes [x,u]\end{aligned}}}

where, for an element ${\displaystyle x\in {\mathfrak {g}},}$ one uses the adjoint endomorphism ${\displaystyle {\mbox{ad}}_{x}:{\mathfrak {g}}\to {\mathfrak {g}}}$ with ${\displaystyle {\mbox{ad}}_{x}\in {\mbox{ad}}_{\mathfrak {g}},}$ and recall the definition ${\displaystyle {\mbox{ad}}_{x}(y)=[x,y].}$

The center ${\displaystyle Z(U({\mathfrak {g}}))}$ then corresponds to linear combinations of all elements ${\displaystyle z=v\otimes w\otimes \cdots \otimes u\in U({\mathfrak {g}})}$ that are in the kernel of ${\displaystyle {\mbox{ad}}_{\mathfrak {g}};}$ that is, the center is precisely those elements ${\displaystyle z\in U({\mathfrak {g}})}$ that commute with all elements ${\displaystyle x\in {\mathfrak {g}};}$ that is, for which ${\displaystyle {\mbox{ad}}_{x}(z)=[x,z]=0.}$

From the PBW theorem, it is clear that all such central elements will be linear combinations of symmetric homogenous polynomials in the basis elements ${\displaystyle e_{a}}$ of the Lie algebra. The Casimir invariants are the irreducible homogenous polynomials of a given, fixed degree. That is, given a basis ${\displaystyle e_{a}}$, a Casimir operator of order ${\displaystyle m}$ has the form

${\displaystyle C_{(m)}=\kappa ^{ab\cdots c}e_{a}\otimes e_{b}\otimes \cdots \otimes e_{c}}$

where there are ${\displaystyle m}$ terms in the tensor product, and ${\displaystyle \kappa ^{ab\cdots c}}$ is a completely symmetric tensor of order ${\displaystyle m}$ belonging to the adjoint representation. That is, ${\displaystyle \kappa ^{ab\cdots c}}$ can be (should be) thought of as an element of ${\displaystyle \left({\mbox{ad}}_{\mathfrak {g}}\right)^{\otimes m}.}$ 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 ${\displaystyle [x,C_{(m)}]=0}$, it follows that

${\displaystyle f_{ij}^{\;\;k}\kappa ^{jl\cdots m}+f_{ij}^{\;\;l}\kappa ^{kj\cdots m}+\cdots +f_{ij}^{\;\;m}\kappa ^{kl\cdots j}=0}$

where the structure constants are

${\displaystyle [e_{i},e_{j}]=f_{ij}^{\;\;k}e_{k}}$

As an example, the quadratic Casimir operator is

${\displaystyle C_{(2)}=\kappa ^{ij}e_{i}\otimes e_{j}}$

where ${\displaystyle \kappa ^{ij}}$ is the inverse matrix of the Killing form ${\displaystyle \kappa _{ij}.}$ That the Casimir operator ${\displaystyle C_{(2)}}$ belongs to the center ${\displaystyle Z(U({\mathfrak {g}}))}$ 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 d-dimensional vector space V, recall that the determinant is the completely antisymmetric tensor on ${\displaystyle V^{\otimes d}}$. Given a matrix M, one may write the characteristic polynomial of M as

${\displaystyle \det(tI-M)=\sum _{n=0}^{d}p_{n}t^{n}}$

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

${\displaystyle {\mbox{ad}}:{\mathfrak {g}}\to {\mbox{End}}({\mathfrak {g}})}$

implies that ${\displaystyle {\mbox{ad}}_{x}}$ is a d-dimensional endomorphism, and so one has the characteristic equation

${\displaystyle \det(tI-{\mbox{ad}}_{x})=\sum _{n=0}^{d}p_{n}(x)t^{n}}$

for elements ${\displaystyle x\in {\mathfrak {g}}.}$ The non-zero roots of this characteristic polynomial (that are roots for all x) form the root system of the algebra. In general, there are only r such roots; this is the rank of the algebra. This implies that the highest value of n for which the ${\displaystyle p_{n}(x)}$ is non-vanishing is r.

The ${\displaystyle p_{n}(x)}$ are homogeneous polynomials of degree d-n. This can be seen in several ways: Given a constant ${\displaystyle k\in K}$, ad is linear, so that ${\displaystyle {\mbox{ad}}_{kx}=k\,{\mbox{ad}}_{x}.}$ By plugging and chugging in the above, one obtains that

${\displaystyle p_{n}(kx)=k^{d-n}p_{n}(x).}$

By linearity, if one expands in the basis,

${\displaystyle x=\sum _{i=1}^{d}x_{i}e_{i}}$

then the polynomial has the form

${\displaystyle p_{n}(x)=x_{a}x_{b}\cdots x_{c}\kappa ^{ab\cdots c}}$

that is, a ${\displaystyle \kappa }$ is a tensor of rank ${\displaystyle m=d-n}$. By linearity and the commutativity of addition, i.e. that ${\displaystyle {\mbox{ad}}_{x+y}={\mbox{ad}}_{y+x},}$, one concludes that this tensor must be completely symmetric. This tensor is exactly the Casimir invariant of order m.

The center ${\displaystyle Z({\mathfrak {g}})}$ corresponded to those elements ${\displaystyle z\in Z({\mathfrak {g}})}$ for which ${\displaystyle {\mbox{ad}}_{x}(z)=0}$ for all x; by the above, these clearly corresponds to the roots of the characteristic equation. One concludes that the roots form a space of rank r and that the Casimir invariants span this space. That is, the Casimir invariants generate the center ${\displaystyle Z(U({\mathfrak {g}})).}$

### 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 ${\displaystyle A_{1}.}$ As an elementary exercise, one can compute this directly. Changing notation to ${\displaystyle e_{i}=L_{i},}$ with ${\displaystyle L_{i}}$ belonging to the adjoint rep, a general algebra element is ${\displaystyle xL_{1}+yL_{2}+zL_{3}}$ and direct computation gives

${\displaystyle \det \left(xL_{1}+yL_{2}+zL_{3}-tI\right)=-t^{3}-(x^{2}+y^{2}+z^{2})t+2xyz}$

The quadratic term can be read off as ${\displaystyle \kappa ^{ij}=\delta ^{ij}}$, and so the squared angular momentum operator for the rotation group is that Casimir operator. That is,

${\displaystyle C_{(2)}=L^{2}=e_{1}\otimes e_{1}+e_{2}\otimes e_{2}+e_{3}\otimes e_{3}}$

and explicit computation shows that

${\displaystyle [L^{2},e_{k}]=0}$

after making use of the structure constants

${\displaystyle [e_{i},e_{j}]=\epsilon _{ij}^{\;\;k}e_{k}}$

### Example: Pseudo-differential operators

A key observation during the construction of ${\displaystyle U({\mathfrak {g}})}$ above was that it was a differential algebra, by dint of the fact that any derivation on the Lie algebra can be lifted to ${\displaystyle U({\mathfrak {g}})}$. Thus, one is led to a ring of pseudo-differential operators, from which one can construct Casimir invariants.

If the Lie algebra ${\displaystyle {\mathfrak {g}}}$ 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 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 ${\displaystyle {\mathfrak {g}}}$ is abelian (that is, the bracket is always 0), then ${\displaystyle U({\mathfrak {g}})}$ is commutative; and if a basis of the vector space ${\displaystyle {\mathfrak {g}}}$ has been chosen, then ${\displaystyle U({\mathfrak {g}})}$ can be identified with the polynomial algebra over K, with one variable per basis element.

If ${\displaystyle {\mathfrak {g}}}$ is the Lie algebra corresponding to the Lie group G, then ${\displaystyle U({\mathfrak {g}})}$ can be identified with the algebra of left-invariant differential operators (of all orders) on G; with ${\displaystyle {\mathfrak {g}}}$ lying inside it as the left-invariant vector fields as first-order differential operators.

To relate the above two cases: if ${\displaystyle {\mathfrak {g}}}$ is a vector space V 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 ${\displaystyle Z({\mathfrak {g}})}$ consists of the left- and right- invariant differential operators; this, in the case of G not commutative, will often not be generated by first-order operators (see for example Casimir operator of a semi-simple Lie algebra).

Another characterization in Lie group theory is of ${\displaystyle U({\mathfrak {g}})}$ as the convolution algebra of distributions supported only at the identity element e of G.

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 ${\displaystyle \phi ,\psi \in C(G)}$, one defines multiplication as

${\displaystyle (\nabla (\phi ,\psi ))(x)=\phi (x)\psi (x)}$

and comultiplication as

${\displaystyle (\Delta (\phi ))(x\otimes y)=\phi (xy),}$

the counit as

${\displaystyle \epsilon (\phi )=\phi (e)}$

and the antipode as

${\displaystyle (S(\phi ))(x)=\phi (x^{-1}).}$

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 ${\displaystyle U({\mathfrak {g}})}$; more precisely, it is isomorphic to a subspace of the dual space ${\displaystyle U^{*}({\mathfrak {g}}).}$

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.