The universal enveloping algebra is given by “generators and relations”, andwhen objects are given in this way it is usually easy to find upp er bounds forthe size of such objects by algebraic manipulation, but harder to show they donot collapse further. A good way to find a lower bound on the size is to findan explicit representation on something. Some cases of the PBW theorem areeasy: for example, if our Lie algebra is known to be the Lie algebra of a Liegroup, then the UEA should be differential operators, which come with a naturalrepresentation on smooth functions, or we can even restrict to formal powerseries expansions of smooth functions at the identity. There are enough suchfunctions to see that the elements gn11gn22... of the UEA are linearly independentas they have linearly independent actions on smooth functions.If we want to prove the PBW theorem more generally we have to workharder: even over the reals it is not at all obvious that any Lie algebra is theLie algebra of a Lie group. (At first sight this seems easy: all one has todo is find a faithful represe ntation on a vector space, and exponentiate thisto a group action. For abelian Lie algebras this is trivial, and at the oppositeextreme when the algebra has no center it is also trivial as we can use the adjointrepresentation. Every Lie algebra can be obtained by starting with these twotype and taking extensions, but it the trouble is that it is hard to show that onecan find a faithful representation of an extension of two algebras with faithfulrepresentations.)In general we have to build a suitable representation from the Lie algebra.We will do this by building something that “ought” to be U g, and then definingan action of g on this by left multiplication and checking that it works.Step 1: Construction of V . We choose a well-ordered basis of g and defineV to have a basis of monomials that are formal products abc... of elements ofthe basis with a ≤ b ≤ c ≤ · · · . (There is an obvious map from V onto Ug; ouraim is to prove that it is an isomorphism.)Step 2: Construction of the action of g on V . Suppose a is a basis element ofg and bc · · · is in V . We define a(bc · · · ) to be abc · · · if a ≤ b, and [ab](c · · · ) +b(a(c · · · ) if a > b. This is well defined by induction on the length of an elementof V and by using the fact that the basis is well ordered.Step 3. Check that this action of g on V satisfies [a, b] = ab − ba. Thisholds on c · · · whenever b ≤ c by definition of the action, and similarly it holdswhenever a ≤ c, so we can assume that both a and b are greater than c. Wecan also assume by induction that [x, y](z) = x(y(z)) − y(x(z) for any elementsx, y of the Lie algebra and any element z of V of length less than that of c · · · .We calculate both sides of the identity we have to prove, pushing c to the left,as follows:a(b(c · · · )) = a(c(b(· · · )) + a([b, c](· · · )) (1)= c(a(b(· · · ))) + [a, c](b(· · · ) + [a, [b, c]](· · · ) + [b, c](a(· · · )) (2)and similarlyb(a(c · · · )) = c(b(a(· · · ))) + [b, c](a(· · · ) + [b, [a, c]](· · · ) + [a, c](b(· · · ))[a, b](c · · · ) = c([a, b](· · · )) + [[a, b], c](· · · )Comparing everything we see thata(b(c · · · )) − b(a(c · · · )) − [a, b](c · · · )22is equal to[[a, b], c](· · · ) + [[b, c], a](· · · ) + [[c, a], b](· · · )We finally use the Jacobi identity for g to see that this vanishes, thus showingthat we indeed have an action of g on V .This completes the proof of the PBW theorem.The PBW theorem shows that we have found “all” identities satisfied by theLie bracket, at least over fields, because any Lie algebra is a subalgebra of the Liealgebra of some associative algebra. For Jordan algebras the analogous result isnot true. Jordan algebras are analogous to Lie algebras except that the Jordanproduct a ◦ b is ab + ba rather than ab − ba. This satisfies identities a ◦ b = b ◦ aand the Jordan identity. However these identities are not enough to force theJordan algebra to be a subalgebra of the Jordan algebra of an associative ring:the Jordan algebras with this property are called special, and satisfy furtherindependent identities (the smallest of which has degree 8). There is a 27-dimensional Jordan algebra (Hermitian matrices over the Cayley numbers) thatis not special. Another example is Lie superalgebras: here we need the extraidentity [a, [a, a]] = 0 for a odd in order to get a faithful representation in a ring.The PBW theorem underlies the later calculations of characters of irreduciblerepresentations of Lie algebras. These representations can be written in terms of“Verma modules”, an Verma modules in turn can be identified with the universalenveloping algebras of Lie algebras. The PBW theorem gives complete controlover the “size” of the Verma module, in other words its character, which inturn leads to character formulas for the irreducible representations. A relatedapplication is the construction of e xceptional simple Lie algebras (and Kac–Moody Lie algebras): the hard part of the construction is to show that thesealgebras are non-zero, which ultimately reduces to showing that certain universalenveloping algebras are non-zero.The UEA of a Lie algebra is not just an associative algebra; it is also a Hopfalgebra.Definition 36 A Hopf algebra is a group.In order to understand this, we need to explain what a group is. It is a set Gwith an associative product, identity, and inverse. However it also has furtherstructure as follows. Given a group action on sets X, Y , there is an action onX × Y given by g(x × y) = g( x) × g(y). This action is given by the diagonalmap g 7→ g × g from G to G × G. Similarly there is an action of G on a 1-pointset, induced by a map g 7→ ∗ from G to a 1-point set. These extra maps arenot usually mentioned in the definition of a group, because they are uniquelydetermined. There is a unique diagonal map from G to G × G so that the mapsG 7→ G × G 7→ G are identities. So we have a coassociative coproduct mapG 7→ G × G and a counit G 7→ 1 making G into a “cogroup”, but this structureis boring because it is uniquely determined for any set. The reason it is uniqueis that we define the product of a group using the categorical product of sets.Similarly we can define groups in any category with products, and again the“coalgebra” structure is uniquely determined.Now