Universal Enveloping Algebra

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

The idea of the universal enveloping algebra is to embed a Lie algebra $\mathfrak$ into an associative algebra $\mathcal$ with identity in such a way that the abstract bracket operation in $\mathfrak$ corresponds to the commutator $xy-yx$ in $\mathcal$ and the algebra $\mathcal$ is generated by the elements of $\mathfrak$. There may be many ways to make such an embedding, but there is a unique "largest" such $\mathcal$, called the universal enveloping algebra of $\mathfrak$.

Generators and relations

Let $\mathfrak$ be a Lie algebra, assumed finite-dimensional for simplicity, with basis $X_1,\ldots X_n$. Let $c_$ be the structure constants for this basis, so that
: _i,X_j\sum_^n c_X_k.
Then the universal enveloping algebra is the associative algebra (with identity) generated by elements $x_1,\ldots x_n$ subject to the relations
:$x_i x_j - x_j x_i=\sum_^n c_x_k$
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 $\mathfrak g$.

Consider, for example, the Lie algebra sl(2,C), spanned by the matrices
:$X = \begin 0 & 1\\ 0 & 0 \end \qquad Y = \begin 0 & 0\\ 1 & 0 \end \qquad H = \begin 1 & 0\\ 0 & -1 \end ~,$
which satisfy the commutation relations ,X2X, ,Y-2Y, and ,YH. The universal enveloping algebra of sl(2,C) is then the algebra generated by three elements $x,y,h$ subject to the relations
:$hx-xh=2x,\quad hy-yh=-2y,\quad xy-yx=h,$
and no other relations. We emphasize that the universal enveloping algebra ''is not'' the same as (or contained in) the algebra of $2\times 2$ matrices. For example, the $2\times 2$ matrix $X$ satisfies $X^2=0$, as is easily verified. But in the universal enveloping algebra, the element $x$ does not satisfy $x^2=0$—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 $1,x,x^2,x^3,\ldots$ 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 $x_1$ first, then factors of $x_2$, etc. For example, whenever we have a term that contains $x_2 x_1$ (in the "wrong" order), we can use the relations to rewrite this as $x_1 x_2$ plus a linear combination of the $x_j$'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
:$x_1^x_2^\cdots x_n^$
with the $k_j$'s being non-negative integers, span the enveloping algebra. (We allow $k_j=0$, meaning that we allow terms in which no factors of $x_j$ 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 $x_1,\ldots, x_n$ themselves are linearly independent. It is therefore common—if potentially confusing—to identify the $x_j$'s with the generators $X_j$ 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 $\mathfrak$ may be an algebra of $n\times n$ matrices, the universal enveloping of $\mathfrak$ does not consists of (finite-dimensional) matrices. In particular, there is no finite-dimensional algebra that contains the universal enveloping of $\mathfrak$; 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 $X$, $Y$ and $H$ as $2\times 2$ 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 $x_i x_j -x_j x_i-\Sigma c_x_k$. The universal enveloping algebra is the "largest" unital associative algebra generated by elements of $\mathfrak g$ with a Lie bracket compatible with the original Lie algebra.

Formal definition

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

That is, one constructs the space
:$T(\mathfrak) = K \,\oplus\, \mathfrak \,\oplus\, (\mathfrak \otimes \mathfrak) \,\oplus\, (\mathfrak \otimes \mathfrak \otimes \mathfrak) \,\oplus\, \cdots$

where $\otimes$ is the tensor product, and $\oplus$ 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 $\mathfrak\times\mathfrak\to\mathfrak$ that is bilinear, skew-symmetric and satisfies the Jacobi identity. We wish to define a Lie bracket ,-that is a map $T(\mathfrak)\otimes T(\mathfrak)\to T(\mathfrak)$ 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 $\mathfrak \otimes \mathfrak \to \mathfrak$ as

:a \otimes b - b \otimes a = ,b/math>

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 $T^2(\mathfrak)=\mathfrak \otimes \mathfrak$; it must now be lifted to $T^n(\mathfrak)$ for arbitrary $n.$ This is done recursively, by ''defining''

: \otimes b, c= a \otimes ,c+ ,cotimes b

and likewise

: , b\otimes c= ,botimes c + b\otimes ,c/math>

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 $T(\mathfrak);$ one says that it has been "lifted" to all of $T(\mathfrak)$ 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 $U(\mathfrak)$ of $\mathfrak$ is defined as the quotient space

:$U(\mathfrak) = T(\mathfrak)/\sim$

where the equivalence relation $\sim$ is given by

:a\otimes b - b \otimes a = ,b/math>

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 $U(\mathfrak)$ 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

:a\otimes b - b \otimes a - ,b/math>

This generator is an element of
:$\mathfrak \oplus (\mathfrak\otimes\mathfrak) \subset T(\mathfrak)$

A general member of the ideal will have the form

:c\otimes d \otimes \cdots \otimes (a\otimes b - b \otimes a - ,b \otimes f \otimes g \cdots

for some $a,b,c,d,f,g\in\mathfrak.$ All elements of are obtained as linear combinations of elements of this form. Clearly, $I\subset T(\mathfrak)$ is a subspace. It is an ideal, in that if $j\in I$ and $x\in T(\mathfrak),$ then $j\otimes x\in I$ and $x\otimes j\in I.$ 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

:$0\to I \to T(\mathfrak) \to T(\mathfrak)/I \to 0$

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

:$U(\mathfrak) = T(\mathfrak)/I$

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 $V$ endowed with multiplication $m:V\times V\to V$ that takes elements $a\times b\mapsto m(a,b).$ ''If'' the multiplication is bilinear, then the same construction and definitions can go through. One starts by lifting $m$ up to $T(V)$ so that the lifted $m$ obeys all of the same properties that the base $m$ does – symmetry or antisymmetry or whatever. The lifting is done ''exactly'' as before, starting with
:$\begin m: V \otimes V &\to V \\ a \otimes b &\mapsto m(a,b) \end$

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

:$m(a \otimes b,c)= a \otimes m(b,c) + m(a,c) \otimes b$
and also that
:$m(a,b\otimes c)= m(a,b) \otimes c + b \otimes m(a,c)$
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 $h:\mathfrak\to U(\mathfrak)$, possesses a universal property. Suppose we have any Lie algebra map
:$\varphi: \mathfrak \to A$
to a unital associative algebra (with Lie bracket in given by the commutator). More explicitly, this means that we assume
:\varphi( ,Y=\varphi(X)\varphi(Y)-\varphi(Y)\varphi(X)
for all $X,Y\in\mathfrak$. Then there exists a ''unique'' unital algebra homomorphism

:$\widehat\varphi: U(\mathfrak) \to A$
such that
:$\varphi = \widehat \varphi \circ h$

where $h:\mathfrak\to U(\mathfrak)$ is the canonical map. (The map $h$ is obtained by embedding $\mathfrak$ 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 $\varphi:\mathfrak\rightarrow A$ is a linear map into a unital algebra $A$ satisfying \varphi( ,Y=\varphi(X)\varphi(Y)-\varphi(Y)\varphi(X), then $\varphi$ extends to an algebra homomorphism of $\widehat\varphi: U(\mathfrak) \to A$. Since $U(\mathfrak)$ is generated by elements of $\mathfrak$, the map $\widehat$ must be uniquely determined by the requirement that
:$\widehat(X_\cdots X_)=\varphi(X_)\cdots \varphi(X_),\quad X_\in\mathfrak$.
The point is that because there are no other relations in the universal enveloping algebra besides those coming from the commutation relations of $\mathfrak$, the map $\widehat$ is well defined, independent of how one writes a given element $x\in U(\mathfrak)$ as a linear combination of products of Lie algebra elements.

The universal property of the enveloping algebra immediately implies that every representation of $\mathfrak$ acting on a vector space $V$ extends uniquely to a representation of $U(\mathfrak)$. (Take $A=\mathrm(V)$.) This observation is important because it allows (as discussed below) the Casimir elements to act on $V$. These operators (from the center of $U(\mathfrak)$) 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 $U(\mathfrak)$. 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 $e_a:X\to K$ such that $e_a(b) = \delta_$ is the indicator function for $a,b\in X$. Let $h:\mathfrak\to T(\mathfrak)$ 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 $e_a$, one defines the extension of $h$ to be

:$h(e_a\otimes e_b \otimes\cdots \otimes e_c) = h(e_a) \otimes h(e_b) \otimes\cdots \otimes h(e_c)$

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

:$e_a\otimes e_b \otimes\cdots \otimes e_c$

where $a\le b \le \cdots \le c$, 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 $U(\mathfrak)$ 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 $\mathfrak.$ The first step is universal, and does not depend on the specific $\mathfrak.$ 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 $U_m \mathfrak$ whose limit is the universal enveloping algebra $U(\mathfrak).$

First, a notation is needed for an ascending sequence of subspaces of the tensor algebra. Let
:$T_m\mathfrak = K\oplus \mathfrak\oplus T^2\mathfrak \oplus \cdots \oplus T^m\mathfrak$
where
:$T^m\mathfrak = T^ \mathfrak = \mathfrak\otimes \cdots \otimes \mathfrak$

is the -times tensor product of $\mathfrak.$ The $T_m\mathfrak$ form a filtration:
:$K\subset \mathfrak\subset T_2\mathfrak \subset \cdots \subset T_m\mathfrak \subset\cdots$

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 $T(\mathfrak).$

It was already established, above, that quotienting by the ideal is a natural transformation that takes one from $T(\mathfrak)$ to $U(\mathfrak).$ This also works naturally on the subspaces, and so one obtains a filtration $U_m \mathfrak$ whose limit is the universal enveloping algebra $U(\mathfrak).$

Next, define the space
:$G_m\mathfrak = U_m \mathfrak/U_ \mathfrak$
This is the space $U_m \mathfrak$ modulo all of the subspaces $U_n \mathfrak$ of strictly smaller filtration degree. Note that $G_m\mathfrak$ is ''not at all'' the same as the leading term $U^m\mathfrak$ of the filtration, as one might naively surmise. It is not constructed through a set subtraction mechanism associated with the filtration.

Quotienting $U_m \mathfrak$ by $U_ \mathfrak$ has the effect of setting all Lie commutators defined in $U_m \mathfrak$ to zero. One can see this by observing that the commutator of a pair of elements whose products lie in $U_ \mathfrak$ actually gives an element in $U_ \mathfrak$. This is perhaps not immediately obvious: to get this result, one must repeatedly apply the com