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 .
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