MATH 223A NOTES 2011
LIE ALGEBRAS
Contents
1.
Basic Concepts
2
1.1.
Defnition oF Lie algebra
2
1.2.
Examples
2
1.3.
Derivations
4
1.4.
Abstract Lie algebras
4
Date
: September 6, 2011.
1
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MATH 223A NOTES 2011 LIE ALGEBRAS
1.
Basic Concepts
1.1.
Defnition oF Lie algebra.
Defnition 1.1.1.
By an
(nonassociative) algebra
over a feld
F
we mean a vector space
A
together with an
F
bilinear operation
A
×
A
→
A
which is usually written (
x, y
)
°→
xy
.
The adjective “nonassociative” means “not necessarily associative”. An
associative
algebra
is an algebra
A
whose multiplication rule is associative:
x
(
yz
)=(
xy
)
z
For all
x, y, z
∈
A
. The existence oF a unit 1 is not assumed.
Defnition 1.1.2.
Let
L
be a vector space over a feld
F
. Then a bilinear operation
[
··
]:
L
×
L
→
L
sending (
x, y
)to[
xy
] is called a
bracket
iF it satisfes the Following two
conditions.
[L1][
xx
] = 0 For all
x
∈
L
.
[L2](
Jacobi identity
)[
x
[
]] + [
y
[
zx
]] + [
z
[
xy
]] = 0 For all
x, y, z
∈
L
.
A vector space
L
with a bracket [
··
] is called a
Lie algebra
. This is an example oF a
nonassociative algebra.
Let us analyze the two conditions. Condition [L1] implies:
[L1
°
][
xy
]=
−
[
yx
] For all
x, y
∈
L
. (Bracket is
skew commutative
.)
ProoF: [(
x
+
y
)(
x
+
y
)] = 0 = [
xx
]+[
xy
yy
]=[
xy
].
Conversely, iF the characteristic oF the feld
F
is not equal to 2 then [L1
°
] implies that
2[
xx
] = 0 implies [L1]. So, [L1] and [L1
°
] are equivalent when
char F
±
=2
.
The second condition [L2] can be rewritten as Follows:
[
x
[
]] = [[
xy
]
z
] + [[
]
y
]
.
The term [[
]
y
] prevents
L
From being associative. Since
z,x,y
are arbitrary we obtain:
Proposition 1.1.3.
A Lie algebra is associative if and only if
[[
LL
]
L
]=0
.
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 Fall '11
 K.Igusa
 Algebra, Vector Space, Lie algebra

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