4 1. INTRODUCTION

The way of defining an algebra as in Exercise1.7 is called definition by gen-

erators and relations. Since we will meet this kind of definition repeatedly in

later sections, I shall give its precise definition here. The field C can be replaced

by any commutative ring.

Definition 1.8. We take a C-vector space V and its basis X1,...,XN . For

elements R1,...,RM ∈ T (V ) = ⊕n≥0V ⊗n, we set I =

∑M

i=1

T (V )RiT (V ), which is

the minimal two-sided ideal of T (V ) containing R1,...,RM . The algebra T (V )/I

is the algebra defined by generators X1,...,XN and relations R1 = 0,...,RM = 0.

In the above exercise, we let X , . . . , XN be a basis of g, and let cij k be complex

numbers defined by [Xi, Xj] =

∑N1

k=1

cijXk.

k

Then the aim of the exercise is to show

that the algebra defined by generators X1,...,XN and relations XiXj − XjXi =

∑N

k=1

cijXk k (1 ≤ i j ≤ N) is isomorphic to the enveloping algebra of g. Here,

Xi1 · · · XiN stands for Xi1 ⊗ · · · ⊗ XiN .

Let ρ : g → End(V ) be a representation of g. By the universality property of

the enveloping algebra, there exists a unique algebra homomorphism φ : U(g) →

End(V ) satisfying φ ◦ ι = ρ. This implies that we can extend the action of g on V

to the action of U(g) in a unique way. Using this correspondence (which associates

a representation (φ, V ) of U(g) with a representation (ρ, V ) of g), we can interpret

results about representations of a Lie algebra as results about representations of

the corresponding enveloping algebra.

Proposition 1.9. Let V1,V2 be g-modules, f : V1 → V2 be a g-module homo-

morphism, i.e. a linear map for which Xf(m) = f(Xm) (X ∈ g, m ∈ V1) holds.

Then f is a U(g)-module homomorphism, i.e. af(m) = f(am) holds for a ∈ U(g)

and m ∈ V1.

Proof. Let ρi : g → End(Vi) be representations on the spaces Vi, and let φi

be their unique extensions φi : U(g) → End(Vi) (i = 1, 2). We set W = V1 ⊕ V2

and define a linear operator A ∈ End(W ) by

A(v1 ⊕ v2) = v1 ⊕ (f(v1) + v2).

In matrix form, A is given by

I 0

f I

.

Let ρ = ρ1 ⊕ ρ2 : g → End(W ). Then

A−1ρ(X)A

=

I 0

−f I

ρ1(X) 0

0 ρ2(X)

I 0

f I

=

ρ1(X) 0

−f ◦ ρ1(X) + ρ2(X) ◦ f ρ2(X)

= ρ(X).

By the uniqueness property of φ, we have

A−1φ(a)A

= φ(a) for φ = φ1 ⊕ φ2. By

expanding the equation φ(a)A = Aφ(a), we obtain f ◦φ1(a) = φ2(a)◦f (a ∈ U(g)),

which implies af(m) = f(am) for m ∈ V1.

As a corollary, we have the following proposition. By definition, the objects of

the category g − Mod are representations of a Lie algebra g and the morphisms of

the category are g-module homomorphisms.