Lecture 02
Suppose
k
is an ideal of
g
. The corresponding quotient algebra is constructed as follows:
Let
g
/
k
be the quotient vector space. Define a bracket [
·
,
·
] on
g
/
k
by
[
x
+
k
, y
+
k
] := [
x, y
] +
k
.
This bracket is welldefined. Indeed, suppose
x
0
+
k
=
x
+
k
and
y
0
+
k
=
y
+
k
. Then
[
x
0
, y
0
]

[
x, y
]
=
[
x
0

x, y
0
] + [
x, y
0

y
]
.
Since
k
is an ideal, each of [
x
0

x, y
0
] and [
x, y
0

y
] is in
k
. Since
k
is a subspace,
[
x
0

x, y
0
] + [
x, y
0

y
]
∈
k
.
It is an easy exercise to verify that [
·
,
·
] defined as above on
g
/
k
is bilinear, alternating and
satisfies the Jacobi identity.
Example 1.
The quotient of
g
by its derived algebra
[
g
,
g
]
is abelian (this is obvious from the
definition of derived algebra). In fact,
[
g
,
g
]
is the smallest ideal
k
for which
g
/
k
is abelian:
If
g
/
k
is abelian, then for each
x, y
∈
g
,
[
x, y
] +
k
= [
x
+
k
, y
+
k
] = 0
so that
[
x, y
]
∈
k
. By considering the linear span of all
x, y
∈
g
, we see that
[
g
,
g
]
⊂
k
.
Example 2.
Z
(
g
/Z
(
g
))
need not be zero.
Example? See the Heisenberg algebra (next lecture).
Definition 1.
A
homomorphism
from
g
to
h
is a linear map
φ
:
g
→
h
that preserves
brackets:
x, y
∈
g
⇒
φ
([
x, y
]) = [
φ
(
x
)
, φ
(
y
)]
.
An
isomorphism
from
g
to
h
is a homomorphism
φ
:
g
→
h
for which there is a homomor
phism
ψ
:
h
→
g
such that both
ψ
◦
φ
= id
g
and
φ
◦
ψ
= id
h
(alternatively,
φ
is a bijective homomorphism).
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 Summer '10
 Staff
 Linear Algebra, Algebra, Vector Space, basis, Algebraic structure, Homomorphism, LIE ALGEBRAS

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