24 2. Basic notions of representation theory

Example 2.9.8. Some examples of representations of Lie algebras

are:

(1) V = 0.

(2) Any vector space V with ρ = 0 (the trivial representation).

(3) The adjoint representation V = g with ρ(a)(b) := [a, b].

That this is a representation follows from equation (2.9.1).

Thus, the meaning of the Jacobi identity is that it is equiv-

alent to the existence of the adjoint representation.

It turns out that a representation of a Lie algebra g is the same

thing as a representation of a certain associative algebra U(g). Thus,

as with quivers, we can view the theory of representations of Lie alge-

bras as a part of the theory of representations of associative algebras.

Definition 2.9.9. Let g be a Lie algebra with basis xi and [ , ]

defined by [xi,xj] =

∑

k

cijxk. k The universal enveloping algebra

U(g) is the associative algebra generated by the xi’s with the defining

relations xixj − xjxi =

∑

k

cijxk.k

Remark 2.9.10. This is not a very good definition since it depends

on the choice of a basis. Later we will give an equivalent definition

which will be basis-independent.

Exercise 2.9.11. Explain why a representation of a Lie algebra is

the same thing as a representation of its universal enveloping algebra.

Example 2.9.12. The associative algebra U(sl(2)) is the algebra

generated by e, f, h, with relations

he − eh = 2e, hf − fh = −2f, ef − fe = h.

Example 2.9.13. The algebra U(H), where H is the Heisenberg Lie

algebra, is the algebra generated by x, y, c with the relations

yx − xy = c, yc − cy = 0, xc − cx = 0.

Note that the Weyl algebra is the quotient of U(H) by the relation

c = 1.

Remark 2.9.14. Lie algebras were introduced by Sophus Lie (see

Section 2.10) as an infinitesimal version of Lie groups (in early texts