Lecture 01
Definition 1.
A
Lie algebra
is a vector space
g
equipped with a bilinear
product
[
•
,
•
] :
g
×
g
→
g
that is alternating:
w
∈
g
⇒
[
w, w
] = 0
and satisfies the Jacobi identity: if
x, y, z
∈
g
then
[
x,
[
y, z
]] + [
y,
[
z, x
]] + [
z,
[
x, y
]] = 0
.
Here, the underlying scalar field may be arbitrary, but we shall usually
take it to be the real numbers
R
or the complex numbers
C
. Notice that the
alternating property implies the property
x, y
∈
g
⇒
[
x, y
] + [
y, x
] = 0
and that the two properties are equivalent in characteristic other than two.
Example 1.
g
=
R
3
with bracket defined by crossproduct:
[
a, b
] :=
a
×
b.
Only the Jacobi identity is less than obvious: for this, use the familiar ‘vector
triple product identity’
c
×
(
a
×
b
) = (
b
·
c
)
a

(
c
·
a
)
b.
Example 2.
Let
A
be any algebra with associative product (indicated simply
by juxtaposition). The ‘commutator bracket’ defined by
X, Y
∈
A
⇒
[
X, Y
] =
XY

Y X
converts
A
into a Lie algebra. Again, only the Jacobi identity calls for any
work in verification; this should be carried out just once as an easy exercise.
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 Summer '10
 Staff
 Algebra, Vector Space, Ring, Verification, Lie algebra

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