As we mentioned before, the Lie algebra cannot detect other componentsof the Lie group, or discrete normal subgroups. It is often useful to note thatdiscrete normal subgroups of a connected group are always in the center (proof:the image of an element under conjugation is connected and discrete, so is justone point). So for example, On, SOnand P Onall have the same Lie algebra.Subalgebras of Lie algebras B are defined in the obvious way and are ana-logues of subgroups. The analogues of normal subgroups are subalgebras A suchthat [A, B] ⊆ A (corresponding to the fact that a subgroup is normal if and only,if aba−1b−1is in A for all a ∈ A and b ∈ B) so that B/A is a Lie algebra inthe obvious way. Much of the terminology for groups is extended to Lie alge-bras in the most obvious way. For example a Lie algebra is called abelian if thebracket is always 0, and is called solvable if there is a chain of ideals with abelianquotients. Unfortunately the definition of simple Lie algebras and simple Liegroups is not completely standardized and is not consistent with the definitionof a simple abstract group. The definition of a simple lie algebra has a trap:a Lie algebra is called simple if it has no ideals other than 0 and itself, and ifthe Lie algebra is NON-ABELIAN. In particular the 1-dimensional Lie algebrais NOT usually considered to be simple. The definition of simple is sometimesmodified slightly for Lie groups: a connected Lie group is called simple if its Liealgebra is simple. This corresponds to the Lie group being non-abelian havingno normal subgroups other than itself and discrete subgroups of its center, sofor example SL2(R) is considered to be a simple Lie group even though it is notsimple as an abstract group.Subgroups of Lie groups are closely related to subgroups of the Lie algebra.Informally, we can get a subalgebra by taking the tangent space of the subgroup,and can get a subgroup by taking the elements “generated” by the infinitesimalgroup elements of the Lie algebra. However it is rather tricky to make thisrigorous, as the following examples show.Example 28 The rational numbers and the integers are subgroup of the reals,but these do not correspond to subgroups of the Lie algebra of the reals. It isclear why not: the integers are not connected so cannot be detected by lookingnear the identity, and the rationals are not closed.This suggests that subalgebras should correspond to closed connected sub-groups, but this fails for more subtle reasons:Example 29 Consider the compact abelian group G = R2/Z2, a 2-dimensionaltorus. For any element (a, b) of its Lie algebra R2we get a homomorphism ofR to G, whose image is a subgroup. If the ratio a/b is rational we get a closedsubgroup isomorphic to S1as the image. However if the ratio is irrational theimage is a copy of R that is dense in G, and in particular is not a closed subgroup.You might think this problem has something to do with the fact that G is notsomply connected, because it disappears if we replace G by its universal cover.However G is a subgroup of the simple connected group SU(3) so we run intoexactly the same problem even for simply connected compact groups.In infinite dimensions the correspondence between subgroups and subalge-bras is even more subtle, as the following examples show.19Example 30 Let us try to find a Lie algebra of the unitary group in infinitedimensions. One possible choice is the Lie algebra of bounded skew Hermitianoperators. This is a perfectly good Lie algebra, but its elements do not corre-spond to all the 1-parameter subgroups of the unitary group. We recall fromHilbert space theory that 1-parameter subgroups of the unitary group corespondto UNBOUNDED skew Hermitian operators (typical example: translation onL2(R) corresponds to d/dx, which is not defined everywhere.) So if we definethe Lie algebra like this, then 1-dimensional subgroups need not correspond to1-dimensional subalgebras. So instead we might try to define the Lie algebra tobe unbounded skew hermitian operators. But these are not even closed underaddition, never mind the Lie bracket, because two such operators might haveno non-zero vectors in their domains of definition.Example 31 It is reasonable to regard the Lie algebra of smooth vector fieldsas something like the Lie algebra of diffeomorphisms of the manifold. Howeverelements of this Lie algebra need not correspond to 1-parameter subgroups. Tosee what can go wrong, consider the vector field x2d/dx on the real line. If we tryto find the flow corresponding to this we have to solve dx/dt = x2with solutionx = x0/(1 − x0t). However this blows up at finite time t. so we do not geta 1-parameter group of diffeomorphisms. A similar example is the vector fieldd/dx on the positive real line: it corresponds to translations, but translationsare not diffeomorphisms of the positive line because you fall off the edge.So in infinite dimensions 1-parameter subgroups need not correspond to 1-dimensional subalgebras of the Lie algebra, and 1-dimensional subalgebras neednot correspond to 1-parameter subgroups.3 The Poincar´e-Birkhoff-Witt theoremWe defined the Lie algebra of a Lie group as the left-invariant normalized dif-ferential operators of order at most 1, and simply threw away the higher orderoperators. This turns out to lose no information, because we can reconstructthese higher order differential operators from the Lie algebra by taking the “uni-versal enveloping algebra”, which partly justifies the claim that the Lie algebracaptures the Lie group locally.The universal enveloping algebra Ug of a Lie algebra g is the associativealgebra generated by the module g, with the relations [A, B] = AB − BA. Amodule over a Lie algebra g is a vector space together with a linear map f fromg to operators on the space such that f([a, b]) = f (a)f(b) − f(b)f(a).Exercise 32 Show that modules over the algebra U g are the same as modulesover the Lie algebra g, and show that U g is universal in the sense that any mapfrom the Lie algebra to an associative algebra such that [A, B] = AB − BAfactors through it. (Category theorists would say that the universal envelopingalgebra is a functor that is left adjoint to the functor taking an associativealgebra to its underlying Lie algebra.)The universal enveloping algebra of a Lie algebra can be thought of as thering of all left invariant differential operators on the group (while the Lie algebraconsists of the normalized ones of order at