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 =
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] =
Then the aim of the exercise is to show
that the algebra defined by generators X1,...,XN and relations XiXj XjXi =
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
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
= 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.
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