Lie algebra
A vector space
g
over a field
F
with a binary operation [·, ·] (called the
Lie bracket or abbr.
bracket) , which satisfies the following conditions:
∀
a
,
b
∈
F
,
x
,
y
,
z
∈
g
,
-
[
a
x
+
b
y
,
z
]
=
a
[
x
,
z
]
+
b
[
y
,
z
]
(bilinearity)
-
[
x
,
x
]
=
0
(alternating)
-
[
[
x
,
y
]
,
z
]
+
[
[
y
,
z
]
,
x
]
+
[
[
z
,
x
]
,
y
]
=
0
(Jacobi identity)
associative algebra
An associative algebra
A
can be made to a Lie algebra by defining the bracket
[
x
,
y
]
=
x
y
−
y
x
(the commutator of
x
,
y
)
∀
x
,
y
∈
A
.
homomorphism
A vector space homomorphism
ϕ
:
g
1
→
g
2
is said to be a Lie algebra homomorphism if
ϕ
(
[
x
,
y
]
)
=
[
ϕ
(
x
)
,
ϕ
(
y
)
]
∀
x
,
y
∈
g
1
.
adjoint representation
Given
x
∈
g
, define map
ad
x
by
ad
x
:
g
→
g
y
↦
[
x
,
y
]
ad
x
is a Lie algebra derivation. The map
ad
:
g
→
g
l
(
g
)
x
↦
a
d
x
thus defined is a Lie algebra homomorphism.
ad
:
g
→
End
(
g
)
is called adjoint representation.
Jacobi identity
The identity [[
x,
y],
z] + [[
y,
z],
x] + [[
z,
x],
y] = 0.
To say Jacobi identity holds in a vector space is equivalent to say adjoint of all elements are derivations :
ad
x
(
[
y
,
z
]
)
=
[
ad
x
(
y
)
,
z
]
+
[
y
,
ad
x
(
z
)
]
.
subalgebra
A subspace
g
′
of a Lie algebra
g
is called the subalgebra of
g
if it is closed under bracket, i.e.
[
g
′
,
g
′
]
⊆
g
′
.
ideal
A subspace
g
′
of a Lie algebra
g
is the ideal of
g
if
[
g
′
,
g
]
⊆
g
′
.
In particular, every ideal is also a subalgebra. Every kernel of a Lie algebra homomorphism is an ideal. Unlike in ring theory, there is no distinguishability of left ideal and right ideal.
derived algebra
The derived algebra of a Lie algebra
g
is
[
g
,
g
]
. It is a subalgebra.
normalizer
The normalizer of a subspace
K
of a Lie algebra
g
is
N
g
(
K
)
:=
{
x
∈
g
|
[
x
,
K
]
⊆
K
}
.
centralizer
The centralizer of a subset
X
of a Lie algebra
g
is
C
g
(
X
)
:=
{
x
∈
g
|
[
x
,
X
]
=
{
0
}
}
.
center
The center of a Lie algebra is the centralizer of itself :
Z
(
L
)
:=
{
x
∈
g
|
[
x
,
g
]
=
0
}
radical
The radical
Rad
(
g
)
is the maximum solvable ideal of
g
.
abelian
A Lie algebra is said to be abelian if its derived algebra is zero.
nilpotent Lie algebra
A Lie algebra
L
is said to be nilpotent if
C
N
(
L
)
=
{
0
}
for some positive integer
N
.
The following conditions are equivalent:
C
N
(
L
)
=
{
0
}
for some positive integer
N
, i.e. the descending central series eventually terminates to
{
0
}
.
C
N
(
L
)
=
L
for some positive integer N, i.e. the ascending central series eventually terminates to L.
There exists a chain of ideals of
L
,
L
=
I
1
⊇
I
2
⊇
I
3
⊇
⋯
⊇
I
n
=
{
0
}
, such that
[
L
,
I
k
]
⊆
I
k
+
1
.
There exists chain of ideals of
L
,
L
=
I
1
⊇
I
2
⊇
I
3
⋯
⊇
I
n
=
{
0
}
, such that
I
k
/
I
k
+
1
⊆
Z
(
L
/
I
k
+
1
)
.
ad
x
is nilpotent
∀
x
∈
L
. (Engel's theorem)
ad
L
is a nilpotent Lie algebra.
In particular, every nilpotent Lie algebra is solvable.
If
L
is nilpotent, any subalgebra and quotient of
L
are nilpotent.
solvable Lie algebra
A Lie algebra
L
is said to be solvable if
L
(
N
)
=
0
for some positive integer
N
, i.e. the derived series eventually terminates to
{
0
}
.
The following condition is equivalent to solvability:
* There exists chain of ideals of
L
,
L
=
I
1
⊇
I
2
⊇
I
3
⋯
⊇
I
n
=
{
0
}
, such that
[
I
k
,
I
k
]
⊆
I
k
+
1
.
If
L
is solvable, any subalgebra and quotient of
L
are solvable.
Let
I
is an ideal of a Lie algebra
L
. If
L
/
I
,
I
are solvable,
L
is solvable.
simple
A Lie algebra is said to be simple if it is non-abelian and has only two ideals, itself and
{
0
}
.
semisimple Lie algebra
A Lie algebra is said to be semisimple if its radical is
{
0
}
.
semisimple element in a Lie algebra
split Lie algebra
free Lie algebra
toral Lie algebra
Lie's theorem
Let
g
be a finite-dimensional complex solvable Lie algebra over algebraically closed field of characteristic
0
, and let
V
be a nonzero finite dimensional representation of
g
. Then there exists an element of
V
which is a simultaneous eigenvector for all elements of
g
.
Corollary: There exists a basis of
V
with respect to which all elements of
g
are upper triangular.
Killing form
The Killing form on a Lie algebra
g
is a symmetric, associative, bilinear form defined by
κ
(
x
,
y
)
:=
Tr
(
ad
x
ad
y
)
∀
x
,
y
∈
g
.
Cartan criterion for solvability
A Lie algebra
g
is solvable iff
κ
(
g
,
[
g
,
g
]
)
=
0
.
Cartan criterion for semisimplity
If
κ
(
⋅
,
⋅
)
is nondegenerate, then
g
is semisimple.
If
g
is semisimple and the underlying field
F
has characteristic 0 , then
κ
(
⋅
,
⋅
)
is nondegenerate.
lower central series
synonymous to "descending central series".
upper central series
synonymous to "ascending central series".
Cartan subalgebra
A Cartan subalgebra
h
of a Lie algebra
g
is a nilpotent subalgebra satisfying
N
g
(
h
)
=
h
.
regular element of a Lie algebra
maximal toral subalgebra
Borel subalgebra
root of a semisimple Lie algebra
Let
g
be a semisimple Lie algebra,
h
be a Cartan subalgebra of
g
. For
α
∈
h
∗
, let
g
α
:=
{
x
∈
g
|
[
h
,
x
]
=
α
(
h
)
x
∀
h
∈
h
}
. alpha is called a root of
g
if it is nonzero and
g
α
≠
{
0
}
The set of all roots is denoted by
Φ
; it forms a root system.
Root system
A subset
Φ
of the Euclidean space
E
is called a root system if it satisfies the following conditions:
Φ
is finite,
span
(
Φ
)
=
E
and
0
∉
Φ
.
For all
α
∈
Φ
and
c
∈
R
,
c
α
∈
Φ
iff
c
=
±
1
.
For all
α
,
β
∈
Φ
,
<
α
,
β
>
is an integer.
For all
α
,
β
∈
Φ
,
S
α
(
β
)
∈
Φ
, where
S
α
is reflection through hyperplane normal to
α
i.e.
S
α
(
x
)
=
x
−
<
x
,
α
>
α
.
Cartan matrix
Cartan matrix of root system
Φ
is matrix
(
<
α
i
,
α
j
>
)
i
,
j
=
1
n
where
Δ
=
{
α
1
…
α
n
}
is a set of simple roots of
Φ
.
Dynkin diagrams
Simple Roots
A subset
Δ
of a root system
Φ
is called a set of simple roots if it satisfies the following conditions:
Δ
is linear basis of
E
.
Each element of
Φ
is a linear combination of elements of
Δ
with coefficients which are either all nonnegative or all nonpositive.
a partial order on the Eucliean space E defined by the set of simple root
∀
λ
,
μ
∈
E
,
λ
>
μ
⟺
λ
−
μ
>
0
⟺
∃
k
1
,
k
2
,
.
.
.
,
k
n
∈
Z
+
,
α
1
,
α
2
,
.
.
.
,
α
n
∈
Δ
,
λ
−
μ
=
∑
i
k
i
α
i
regular element with respect to a root system
Let
Φ
be a root system.
γ
∈
E
is called regular if
(
γ
,
α
)
≠
0
∀
γ
∈
Φ
.
For each set of simple roots
Δ
of
Φ
, there exists a regular element
γ
∈
E
such that
(
γ
,
α
)
>
0
∀
γ
∈
Δ
, conversely for each regular
γ
there exist a unique set of base roots
Δ
(
γ
)
such that the previous condition holds for
Δ
=
Δ
(
γ
)
. It can be determined in following way: let
Φ
+
(
γ
)
=
{
α
∈
Φ
|
(
α
,
γ
)
>
0
}
. Call an element
α
of
Φ
+
(
γ
)
decomposable if
α
=
α
′
+
α
″
where
α
′
,
α
″
∈
Φ
+
(
γ
)
, then
Δ
(
γ
)
is the set of all indecomposable elements of
Φ
+
(
γ
)
positive roots
Positive root of root system
Φ
with respect to a set of simple roots
Δ
is a root of
Φ
which is a linear combination of elements of
Δ
with nonnegative coefficients.
negative roots
Negative root of root system
Φ
with respect to a set of simple roots
Δ
is a root of
Φ
which is a linear combination of elements of
Δ
with nonpositive coefficients.
long root
short root
Weyl group
Weyl group of a root system
Φ
is a (necessarily finite) group of orthogonal linear transformations of
E
which is generated by reflections through hyperplanes normal to roots of
Φ
inverse of a root system
Given a root system
Φ
. Define
α
v
=
2
α
(
α
,
α
)
,
Φ
v
=
{
α
v
|
α
∈
Φ
}
is called the inverse of a root system.
Φ
v
is again a root system and have the identical Weyl group as
Φ
.
base of a root system
synonymous to "set of simple roots"
dual of a root system
synonymous to "inverse of a root system"
<--
weight in a root system
λ
∈
E
is called a weight if
<
λ
,
α
>∈
Z
∀
α
∈
Φ
. -->
weight lattice
weight space
dominant weight
A weight lambda is dominant if
<
λ
,
α
>∈
Z
+
for some
α
∈
Φ
fundamental dominant weight
Given a set of simple roots
Δ
=
{
α
1
,
α
2
,
.
.
.
,
α
n
}
, it is a basis of
E
.
α
1
v
,
α
2
v
,
.
.
.
,
α
n
v
∈
Φ
v
is a basis of
E
too; the dual basis
λ
1
,
λ
2
,
.
.
.
,
λ
n
defined by
(
λ
i
,
α
j
v
)
=
δ
i
j
, is called the fundamental dominant weights.
highest weight
minimal weight
multiplicity (of weight)
radical weight
strongly dominant weight
module
Define an action of
g
on a vector space
V
( i.e. an operation
g
×
V
→
V
,
(
x
,
v
)
↦
x
v
) such that:
∀
a
,
b
∈
F
,
x
,
y
∈
g
,
v
,
w
∈
V
satisfy
#
(
a
x
+
b
y
)
v
=
a
(
x
v
)
+
b
(
y
v
)
#
x
(
a
v
+
b
w
)
=
a
(
x
v
)
+
b
(
x
w
)
#
[
x
,
y
]
v
=
x
(
y
v
)
−
y
(
x
v
)
Then
V
is called a
g
-module. (Remark:
V
,
g
have the same underlying field
F
.)
Each
g
-module corresponds to a representation
g
→
g
l
V
.
A subspace W is a
submodule (more precisely, sub
g
-module) of
V
if
g
-module
W
⊂
V
.
representation
For a vector space
V
, if there is a Lie algebra homomorphism
π
:
g
→
g
l
V
, then
π
is called a representation of
g
.
Each representation
g
→
g
l
V
corresponds to a
g
-module
V
.
A subrepresentation is the representation corresponding to a submodule.
homomorphism
Given two
g
-module V, W, a
g
-module homomorphism
ϕ
is a vector space homomorphism satisfying
ϕ
(
x
v
)
=
x
ϕ
(
v
)
∀
x
∈
g
,
v
∈
V
.
trivial representation
A representation is said to be trivial if the image of
g
is the zero vector space. It corresponds to the action of
g
on module
V
by
x
v
=
0
∀
x
∈
g
,
v
∈
V
.
faithful representation
If the representation
g
→
g
l
V
is injective, it is said to be faithful.
tautology representation
If a Lie algebra
g
is defined as a subalgebra of
g
l
(
n
,
F
)
, like
s
l
(
n
,
F
)
,
o
(
2
l
,
F
)
,
t
(
n
,
F
)
(the upper triangular matrices), the tautology representation is the imbedding
g
→
g
l
(
n
,
F
)
. It corresponds to the action on module
F
n
by the matrix multiplication.
adjoint representation
The representation
ad
:
g
→
gl
g
x
↦
ad
x
. It corresponds viewing
g
as a
g
-module - the action on the module is given by the adjoint endomorphism.
irreducible modules
A module is said to be irreducible if it has only two submodules, itself and zero.
indecomposable module
A module is said to be indecomposable if it cannot be written as direct sum of two non-zero submodules.
An irreducible module need not be indecomposable but the converse is not true.
completely reducible module
A module is said to be completely reducible if it can be written as direct sum of irreducible modules.
simple module
Synonymous as irreducible module.
quotient module / quotient representation
Given a
g
-module V and its submodule W, an action
g
on V/W can be defined by
x
(
v
+
W
)
=
x
v
+
W
∀
x
∈
g
,
v
∈
V
. V/W is said to be a quotient module in this case.
Schur's lemma
Statement in the language of module theory: Given V an irreducible
g
-module,
ϕ
V
→
V
is a
g
-module homomorphism iff
ϕ
=
λ
1
V
for some
λ
∈
F
.
Statement in the language of representation theory: Given an irreducible representation
ϕ
:
L
→
g
l
(
V
)
, for
θ
∈
End
(
V
)
,
θ
ϕ
(
x
)
=
ϕ
(
x
)
θ
iff
θ
=
λ
1
V
for some
λ
∈
F
.
simple module
synonymous to "irreducible module".
factor module
synonymous to "quotient module".
PBW theorem (Poincaré–Birkhoff–Witt theorem)
BGG category mathcal{O}
a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups.
The generators of a Lie group are split into the generators H and E such that:
[
H
α
i
,
H
α
j
]
=
0
[
H
α
i
,
E
α
j
]
=
A
i
j
E
α
j
[
E
α
i
,
E
α
j
]
=
H
α
j
[
E
β
,
E
γ
]
=
±
(
p
+
1
)
E
β
+
γ
where p = m if β + γ is a root and m is the greatest positive integer such that γ − mβ is a root.
general linear algebra
g
l
(
n
,
F
)
Ado's theorem
Any finite-dimensional Lie algebra is isomorphic to a subalgebra of
g
l
V
for some finite-dimensional vector space V.
Classical Lie algebras:
Exceptional Lie algebras:
Poisson algebra
Kac–Moody algebra
Lie group
Glossary of semisimple groups
Linear algebraic group
Particle physics and representation theory
