(φ1 φ2) Δ
End(V1 V2)

ρ1 1 + 1 ρ2
We call Δ the coproduct of U(g).
Proof. We set A = U(g) U(g) in the commutative diagram to define the
universality of U(g), and consider g A given by X ι(X) 1 + 1 ι(X). Then
the existence and uniqueness of Δ follows from the universality of U(g) and the
fact that ι ([X, Y ]) 1 + 1 ι ([X, Y ]) equals
[ι(X) 1 + 1 ι(X), ι(Y ) 1 + 1 ι(Y )] .
Next set A = End(V1 V2). Then we can check that the required map φ for the
tensor product representation is given by (φ1 φ2) Δ. In fact, the commutativity
of the diagram is verified by
(φ1 φ2) Δ ι(X) = (φ1 φ2) (ι(X) 1 + 1 ι(X))
= φ1 ι(X) 1 + 1 φ2 ι(X)
= ρ1(X) 1 + 1 ρ2(X) = ρ(X).
Hence the uniqueness of φ implies that φ1 φ2 is the map for the tensor product
representation ρ1 ρ2.
For enveloping algebras, the following result, the PBW(Poincare-Birkhoff-Witt)
theorem, is fundamental.
Proposition 1.13. If {Xi}i∈I is a basis of a Lie algebra g, then the following
set is a basis of U(g).
{ Xi1 · · · Xim | i1 · · · im,m = 0, 1,... }
Proof. We first show that U(g) is spanned by these elements. If we con-
sider those elements Xi1 · · · Xim whose indices i1,...,im are not necessarily non-
decreasing, it is obvious that these span U(g). Choose the minimal index among
i1,...,im, and move this to the left end using the relation XjXk = XkXj +[Xj, Xk].
Next choose the minimal index among the remaining indices and move this to
the second to the leftmost position. Continue this procedure to reorder the in-
dices i1,...,im in non-decreasing order. Since newly appearing terms have smaller
length, we apply the same procedure to these terms and after a finite number of
steps, we can rewrite the original monomial as a linear combination of monomials
with non-decreasing indices.
Next we show that these monomials are linearly independent. To do this, we
consider indeterminates {zi}i∈I which are in bijection with {Xi}i∈I , and denote
the polynomial ring generated by these indeterminates by S.
Since { zi1 · · · zim | i1 · · · im} is a basis of S, we can define an action of g
on S as follows. (If m = 0, we set Xi1 = zi.)
Xizi1 · · · zim =
zizi1 · · · zim (i i1)
(Xizi2 · · · zim ) + [ Xi, Xi1 ]zi2 · · · zim (i i1)
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