6 1. INTRODUCTION

g

❍

❍

❍

❍ ❥ ι

U(g)

✟

✟

✟

✟ ✯

(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)

Xi1

(Xizi2 · · · zim ) + [ Xi, Xi1 ]zi2 · · · zim (i i1)