2.9. Lie algebras 23

This provides a motivation for the notion of a Lie algebra. Namely,

we see that Lie algebras arise as spaces of infinitesimal automorphisms

(= derivations) of associative algebras. In fact, they similarly arise as

spaces of derivations of any kind of linear algebraic structures, such

as Lie algebras, Hopf algebras, etc., and for this reason play a very

important role in algebra.

Here are a few more concrete examples of Lie algebras:

(1) R3 with [u, v] = u × v, the cross-product of u and v.

(2) sl(n), the set of n × n matrices with trace 0.

For example, sl(2) has the basis

e =

0 1

0 0

, f =

0 0

1 0

, h =

1 0

0 −1

with relations

[h, e] = 2e, [h, f] = −2f, [e, f] = h.

(3) The Heisenberg Lie algebra H of matrices

0 ∗ ∗

0 0 ∗

0 0 0

.

It has the basis

x =

⎛

⎝0

0 0 0

0

1⎠

0 0 0

⎞

, y =

⎛

⎝0

0 1 0

0

0⎠

0 0 0

⎞

, c =

⎛

⎝0

0 0 1

0

0⎠

0 0 0

⎞

with relations [y, x] = c and [y, c] = [x, c] = 0.

(4) The algebra aff(1) of matrices (

∗ ∗

0 0

).

Its basis consists of X = ( 1 0

0 0

) and Y = ( 0 1

0 0

), with [X, Y ] =

Y .

(5) so(n), the space of skew-symmetric n × n matrices, with

[a, b] = ab − ba.

Exercise 2.9.5. Show that example (1) is a special case of example

(5) (for n = 3).

Definition 2.9.6. Let g1, g2 be Lie algebras. A homomorphism

of Lie algebras ϕ : g1 −→ g2 is a linear map such that ϕ([a, b]) =

[ϕ(a),ϕ(b)].

Definition 2.9.7. A representation of a Lie algebra g is a vector

space V with a homomorphism of Lie algebras ρ : g −→ End V .