From Wikipedia, the free encyclopedia
This is a glossary for the terminology applied in the mathematical theories of Lie algebras . The statements in this glossary mainly focus on the algebraic sides of the concepts, without referring to Lie groups or other related subjects.
Definition
Lie algebra
A vector space
g
{\displaystyle {\mathfrak {g}}}
over a field
F
{\displaystyle F}
with a binary operation [·, ·] (called the Lie bracket or abbr. bracket ) , which satisfies the following conditions:
∀
a
,
b
∈
F
,
x
,
y
,
z
∈
g
{\displaystyle \forall a,b\in F,x,y,z\in {\mathfrak {g}}}
,
[
a
x
+
b
y
,
z
]
=
a
[
x
,
z
]
+
b
[
y
,
z
]
{\displaystyle [ax+by,z]=a[x,z]+b[y,z]}
(bilinearity )
[
x
,
x
]
=
0
{\displaystyle [x,x]=0}
(alternating )
[
[
x
,
y
]
,
z
]
+
[
[
y
,
z
]
,
x
]
+
[
[
z
,
x
]
,
y
]
=
0
{\displaystyle [[x,y],z]+[[y,z],x]+[[z,x],y]=0}
(Jacobi identity )
associative algebra
An associative algebra
A
{\displaystyle A}
can be made to a Lie algebra by defining the bracket
[
x
,
y
]
=
x
y
−
y
x
{\displaystyle [x,y]=xy-yx}
(the commutator of
x
,
y
{\displaystyle x,y}
)
∀
x
,
y
∈
A
{\displaystyle \forall x,y\in A}
.
homomorphism
A vector space homomorphism
ϕ
:
g
1
→
g
2
{\displaystyle \phi :{\mathfrak {g}}_{1}\to {\mathfrak {g}}_{2}}
is said to be a Lie algebra homomorphism if
ϕ
(
[
x
,
y
]
)
=
[
ϕ
(
x
)
,
ϕ
(
y
)
]
∀
x
,
y
∈
g
1
.
{\displaystyle \phi ([x,y])=[\phi (x),\phi (y)]\,\forall x,y\in {\mathfrak {g}}_{1}.}
adjoint representation
Given
x
∈
g
{\displaystyle x\in {\mathfrak {g}}}
, define map
ad
x
{\displaystyle {\textrm {ad}}_{x}}
by
ad
x
:
g
→
g
y
↦
[
x
,
y
]
{\displaystyle {\begin{aligned}{\textrm {ad}}_{x}:&{\mathfrak {g}}\to {\mathfrak {g}}\\&y\mapsto [x,y]\end{aligned}}}
ad
x
{\displaystyle {\textrm {ad}}_{x}}
is a Lie algebra derivation . The map
ad
:
g
→
g
l
(
g
)
x
↦
a
d
x
{\displaystyle {\begin{aligned}{\textrm {ad}}:&{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})\\&x\mapsto \mathrm {ad} _{x}\end{aligned}}}
thus defined is a Lie algebra homomorphism .
ad
:
g
→
End
(
g
)
{\displaystyle {\textrm {ad}}:{\mathfrak {g}}\to {\textrm {End}}({\mathfrak {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
)
]
{\displaystyle {\textrm {ad}}_{x}([y,z])=[{\textrm {ad}}_{x}(y),z]+[y,{\textrm {ad}}_{x}(z)]}
.
subalgebras
subalgebra
A subspace
g
′
{\displaystyle {\mathfrak {g'}}}
of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is called the subalgebra of
g
{\displaystyle {\mathfrak {g}}}
if it is closed under bracket, i.e.
[
g
′
,
g
′
]
⊆
g
′
.
{\displaystyle [{\mathfrak {g'}},{\mathfrak {g'}}]\subseteq {\mathfrak {g'}}.}
ideal
A subspace
g
′
{\displaystyle {\mathfrak {g'}}}
of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is the ideal of
g
{\displaystyle {\mathfrak {g}}}
if
[
g
′
,
g
]
⊆
g
′
.
{\displaystyle [{\mathfrak {g'}},{\mathfrak {g}}]\subseteq {\mathfrak {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
{\displaystyle {\mathfrak {g}}}
is
[
g
,
g
]
{\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]}
. It is a subalgebra.
normalizer
The normalizer of a subspace
K
{\displaystyle K}
of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is
N
g
(
K
)
:=
{
x
∈
g
|
[
x
,
K
]
⊆
K
}
{\displaystyle N_{\mathfrak {g}}(K):=\{x\in {\mathfrak {g}}|[x,K]\subseteq K\}}
.
centralizer
The centralizer of a subset
X
{\displaystyle X}
of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is
C
g
(
X
)
:=
{
x
∈
g
|
[
x
,
X
]
=
{
0
}
}
{\displaystyle C_{\mathfrak {g}}(X):=\{x\in {\mathfrak {g}}|[x,X]=\{0\}\}}
.
center
The center of a Lie algebra is the centralizer of itself :
Z
(
L
)
:=
{
x
∈
g
|
[
x
,
g
]
=
0
}
{\displaystyle Z(L):=\{x\in {\mathfrak {g}}|[x,{\mathfrak {g}}]=0\}}
radical
The radical
Rad
(
g
)
{\displaystyle {\textrm {Rad}}({\mathfrak {g}})}
is the maximum solvable ideal of
g
{\displaystyle {\mathfrak {g}}}
.
Solvability, nilpotency, Jordan decomposition, semisimplicity
abelian
A Lie algebra is said to be abelian if its derived algebra is zero.
nilpotent Lie algebra
A Lie algebra
L
{\displaystyle L}
is said to be nilpotent if
C
N
(
L
)
=
{
0
}
{\displaystyle C^{N}(L)=\{0\}}
for some positive integer
N
{\displaystyle N}
.
The following conditions are equivalent:
C
N
(
L
)
=
{
0
}
{\displaystyle C^{N}(L)=\{0\}}
for some positive integer
N
{\displaystyle N}
, i.e. the descending central series eventually terminates to
{
0
}
{\displaystyle \{0\}}
.
C
N
(
L
)
=
L
{\displaystyle 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
{\displaystyle L}
,
L
=
I
1
⊇
I
2
⊇
I
3
⊇
⋯
⊇
I
n
=
{
0
}
{\displaystyle L=I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq \cdots \supseteq I_{n}=\{0\}}
, such that
[
L
,
I
k
]
⊆
I
k
+
1
{\displaystyle [L,I_{k}]\subseteq I_{k+1}}
.
There exists chain of ideals of
L
{\displaystyle L}
,
L
=
I
1
⊇
I
2
⊇
I
3
⋯
⊇
I
n
=
{
0
}
{\displaystyle L=I_{1}\supseteq I_{2}\supseteq I_{3}\cdots \supseteq I_{n}=\{0\}}
, such that
I
k
/
I
k
+
1
⊆
Z
(
L
/
I
k
+
1
)
{\displaystyle I_{k}/I_{k+1}\subseteq Z(L/I_{k+1})}
.
ad
x
{\displaystyle {\textrm {ad}}\,x}
is nilpotent
∀
x
∈
L
{\displaystyle \forall x\in L}
. (Engel's theorem )
ad
L
{\displaystyle {\textrm {ad}}\,L}
is a nilpotent Lie algebra.
In particular, every nilpotent Lie algebra is solvable.
If
L
{\displaystyle L}
is nilpotent, any subalgebra and quotient of
L
{\displaystyle L}
are nilpotent.
nilpotent element in a Lie algebra
An element
x
∈
L
{\displaystyle x\in L}
is said to be nilpotent in
L
{\displaystyle L}
if
a
d
x
{\displaystyle ad_{x}}
is a nilpotent endomorphism, i.e. viewing
ad
x
{\displaystyle {\textrm {ad}}_{x}}
as a matrix in
g
l
g
{\displaystyle {\mathfrak {gl}}_{\mathfrak {g}}}
,
∃
N
∈
Z
+
,
(
ad
x
)
N
=
0
{\displaystyle \exists N\in \mathbb {Z} ^{+},({\textrm {ad}}_{x})^{N}=0}
. It is equivalent to
(
ad
x
)
N
y
=
[
x
[
x
…
[
x
[
x
,
y
]
…
]
=
0
∀
y
∈
L
{\displaystyle ({\textrm {ad}}_{x})^{N}y=[x[x\ldots [x[x,y]\ldots ]=0\ \forall y\in L}
descending central series
a sequence of ideals of a Lie algebra
L
{\displaystyle L}
defined by
C
0
(
L
)
=
L
,
C
1
(
L
)
=
[
L
,
L
]
,
C
n
+
1
(
L
)
=
[
L
,
C
n
(
L
)
]
{\displaystyle C^{0}(L)=L,\,C^{1}(L)=[L,L],\,C^{n+1}(L)=[L,C^{n}(L)]}
ascending central series
a sequence of ideals of a Lie algebra
L
{\displaystyle L}
defined by
C
0
(
L
)
=
{
0
}
,
C
1
(
L
)
=
Z
(
L
)
{\displaystyle C_{0}(L)=\{0\},\,C_{1}(L)=Z(L)}
(center of L) ,
C
n
+
1
(
L
)
=
π
n
−
1
(
Z
(
L
/
C
n
(
L
)
)
)
{\displaystyle C_{n+1}(L)=\pi _{n}^{-1}(Z(L/C_{n}(L)))}
, where
π
i
{\displaystyle \pi _{i}}
is the natural homomorphism
L
→
L
/
C
n
(
L
)
{\displaystyle L\to L/C_{n}(L)}
solvable Lie algebra
A Lie algebra
L
{\displaystyle L}
is said to be solvable if
L
(
N
)
=
0
{\displaystyle L^{(N)}=0}
for some positive integer
N
{\displaystyle N}
, i.e. the derived series eventually terminates to
{
0
}
{\displaystyle \{0\}}
.
The following condition is equivalent to solvability:
* There exists chain of ideals of
L
{\displaystyle L}
,
L
=
I
1
⊇
I
2
⊇
I
3
⋯
⊇
I
n
=
{
0
}
{\displaystyle L=I_{1}\supseteq I_{2}\supseteq I_{3}\cdots \supseteq I_{n}=\{0\}}
, such that
[
I
k
,
I
k
]
⊆
I
k
+
1
{\displaystyle [I_{k},I_{k}]\subseteq I_{k+1}}
.
If
L
{\displaystyle L}
is solvable, any subalgebra and quotient of
L
{\displaystyle L}
are solvable.
Let
I
{\displaystyle I}
is an ideal of a Lie algebra
L
{\displaystyle L}
. If
L
/
I
,
I
{\displaystyle L/I,I}
are solvable,
L
{\displaystyle L}
is solvable.
derived series
a sequence of ideals of a Lie algebra L defined by
L
(
0
)
=
L
,
L
(
1
)
=
[
L
,
L
]
,
L
(
n
+
1
)
=
[
L
(
n
)
,
L
(
n
)
]
{\displaystyle L^{(0)}=L,\,L^{(1)}=[L,L],\,L^{(n+1)}=[L^{(n)},L^{(n)}]}
simple
A Lie algebra is said to be simple if it is non-abelian and has only two ideals, itself and
{
0
}
{\displaystyle \{0\}}
.
semisimple Lie algebra
A Lie algebra is said to be semisimple if its radical is
{
0
}
{\displaystyle \{0\}}
.
semisimple element in a Lie algebra
split Lie algebra
free Lie algebra
toral Lie algebra
Lie's theorem
Let
g
{\displaystyle {\mathfrak {g}}}
be a finite-dimensional complex solvable Lie algebra over algebraically closed field of characteristic
0
{\displaystyle 0}
, and let
V
{\displaystyle V}
be a nonzero finite dimensional representation of
g
{\displaystyle {\mathfrak {g}}}
. Then there exists an element of
V
{\displaystyle V}
which is a simultaneous eigenvector for all elements of
g
{\displaystyle {\mathfrak {g}}}
.
Corollary: There exists a basis of
V
{\displaystyle V}
with respect to which all elements of
g
{\displaystyle {\mathfrak {g}}}
are upper triangular .
Killing form
The Killing form on a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is a symmetric, associative, bilinear form defined by
κ
(
x
,
y
)
:=
Tr
(
ad
x
ad
y
)
∀
x
,
y
∈
g
{\displaystyle \kappa (x,y):={\textrm {Tr}}({\textrm {ad}}\,x\,{\textrm {ad}}\,y)\ \forall x,y\in {\mathfrak {g}}}
.
Cartan criterion for solvability
A Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is solvable iff
κ
(
g
,
[
g
,
g
]
)
=
0
{\displaystyle \kappa ({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0}
.
Cartan criterion for semisimplity
If
κ
(
⋅
,
⋅
)
{\displaystyle \kappa (\cdot ,\cdot )}
is nondegenerate, then
g
{\displaystyle {\mathfrak {g}}}
is semisimple.
If
g
{\displaystyle {\mathfrak {g}}}
is semisimple and the underlying field
F
{\displaystyle F}
has characteristic 0 , then
κ
(
⋅
,
⋅
)
{\displaystyle \kappa (\cdot ,\cdot )}
is nondegenerate.
lower central series
synonymous to "descending central series".
upper central series
synonymous to "ascending central series".
Semisimple Lie algebra
This section is empty. You can help by
adding to it .
(July 2010 )
Root System (for classification of semisimple Lie algebra)
In the below section, denote
(
⋅
,
⋅
)
{\displaystyle (\cdot ,\cdot )}
as the inner product of a Euclidean space E.
In the below section,
<
⋅
,
⋅
>
{\displaystyle <\cdot ,\cdot >}
denoted the function defined as
<
β
,
α
>=
(
β
,
α
)
(
α
,
α
)
∀
α
,
β
∈
E
{\displaystyle <\beta ,\alpha >={\frac {(\beta ,\alpha )}{(\alpha ,\alpha )}}\,\forall \alpha ,\beta \in E}
.
Cartan subalgebra
A Cartan subalgebra
h
{\displaystyle {\mathfrak {h}}}
of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is a nilpotent subalgebra satisfying
N
g
(
h
)
=
h
{\displaystyle N_{\mathfrak {g}}({\mathfrak {h}})={\mathfrak {h}}}
.
regular element of a Lie algebra
maximal toral subalgebra
Borel subalgebra
root of a semisimple Lie algebra
Let
g
{\displaystyle {\mathfrak {g}}}
be a semisimple Lie algebra,
h
{\displaystyle {\mathfrak {h}}}
be a Cartan subalgebra of
g
{\displaystyle {\mathfrak {g}}}
. For
α
∈
h
∗
{\displaystyle \alpha \in {\mathfrak {h}}^{*}}
, let
g
α
:=
{
x
∈
g
|
[
h
,
x
]
=
α
(
h
)
x
∀
h
∈
h
}
{\displaystyle {\mathfrak {g_{\alpha }}}:=\{x\in {\mathfrak {g}}|[h,x]=\alpha (h)x\,\forall h\in {\mathfrak {h}}\}}
. \alpha is called a root of
g
{\displaystyle {\mathfrak {g}}}
if it is nonzero and
g
α
≠
{
0
}
{\displaystyle {\mathfrak {g_{\alpha }}}\neq \{0\}}
The set of all roots is denoted by
Φ
{\displaystyle \Phi }
; it forms a root system.
Root system
A subset
Φ
{\displaystyle \Phi }
of the Euclidean space
E
{\displaystyle E}
is called a root system if it satisfies the following conditions:
Φ
{\displaystyle \Phi }
is finite,
span
(
Φ
)
=
E
{\displaystyle {\textrm {span}}(\Phi )=E}
and
0
∉
Φ
{\displaystyle 0\notin \Phi }
.
For all
α
∈
Φ
{\displaystyle \alpha \in \Phi }
and
c
∈
R
{\displaystyle c\in \mathbb {R} }
,
c
α
∈
Φ
{\displaystyle c\alpha \in \Phi }
iff
c
=
±
1
{\displaystyle c=\pm 1}
.
For all
α
,
β
∈
Φ
{\displaystyle \alpha ,\beta \in \Phi }
,
<
α
,
β
>
{\displaystyle <\alpha ,\beta >}
is an integer.
For all
α
,
β
∈
Φ
{\displaystyle \alpha ,\beta \in \Phi }
,
S
α
(
β
)
∈
Φ
{\displaystyle S_{\alpha }(\beta )\in \Phi }
, where
S
α
{\displaystyle S_{\alpha }}
is reflection through hyperplane normal to
α
{\displaystyle \alpha }
i.e.
S
α
(
x
)
=
x
−
<
x
,
α
>
α
{\displaystyle S_{\alpha }(x)=x-<x,\alpha >\alpha }
.
Cartan matrix
Cartan matrix of root system
Φ
{\displaystyle \Phi }
is matrix
(
<
α
i
,
α
j
>
)
i
,
j
=
1
n
{\displaystyle (<\alpha _{i},\alpha _{j}>)_{i,j=1}^{n}}
where
Δ
=
{
α
1
…
α
n
}
{\displaystyle \Delta =\{\alpha _{1}\ldots \alpha _{n}\}}
is a set of simple roots of
Φ
{\displaystyle \Phi }
.
Dynkin diagrams
Simple Roots
A subset
Δ
{\displaystyle \Delta }
of a root system
Φ
{\displaystyle \Phi }
is called a set of simple roots if it satisfies the following conditions:
Δ
{\displaystyle \Delta }
is linear basis of
E
{\displaystyle E}
.
Each element of
Φ
{\displaystyle \Phi }
is a linear combination of elements of
Δ
{\displaystyle \Delta }
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
{\displaystyle \forall \lambda ,\mu \in E,\,\,\lambda >\mu \iff \lambda -\mu >0\iff \,\,\,\exists k_{1},k_{2},...,k_{n}\in \mathbb {Z} ^{+},\,\alpha _{1},\alpha _{2},...,\alpha _{n}\in \Delta ,\,\,\lambda -\mu =\sum _{i}k_{i}\alpha _{i}}
regular element with respect to a root system
Let
Φ
{\displaystyle \Phi }
be a root system.
γ
∈
E
{\displaystyle \gamma \in E}
is called regular if
(
γ
,
α
)
≠
0
∀
γ
∈
Φ
{\displaystyle (\gamma ,\alpha )\neq 0\,\forall \gamma \in \Phi }
.
For each set of simple roots
Δ
{\displaystyle \Delta }
of
Φ
{\displaystyle \Phi }
, there exists a regular element
γ
∈
E
{\displaystyle \gamma \in E}
such that
(
γ
,
α
)
>
0
∀
γ
∈
Δ
{\displaystyle (\gamma ,\alpha )>0\,\forall \gamma \in \Delta }
, conversely for each regular
γ
{\displaystyle \gamma }
there exist a unique set of base roots
Δ
(
γ
)
{\displaystyle \Delta (\gamma )}
such that the previous condition holds for
Δ
=
Δ
(
γ
)
{\displaystyle \Delta =\Delta (\gamma )}
. It can be determined in following way: let
Φ
+
(
γ
)
=
{
α
∈
Φ
|
(
α
,
γ
)
>
0
}
{\displaystyle \Phi ^{+}(\gamma )=\{\alpha \in \Phi |(\alpha ,\gamma )>0\}}
. Call an element
α
{\displaystyle \alpha }
of
Φ
+
(
γ
)
{\displaystyle \Phi ^{+}(\gamma )}
decomposable if
α
=
α
′
+
α
″
{\displaystyle \alpha =\alpha '+\alpha ''}
where
α
′
,
α
″
∈
Φ
+
(
γ
)
{\displaystyle \alpha ',\alpha ''\in \Phi ^{+}(\gamma )}
, then
Δ
(
γ
)
{\displaystyle \Delta (\gamma )}
is the set of all indecomposable elements of
Φ
+
(
γ
)
{\displaystyle \Phi ^{+}(\gamma )}
positive roots
Positive root of root system
Φ
{\displaystyle \Phi }
with respect to a set of simple roots
Δ
{\displaystyle \Delta }
is a root of
Φ
{\displaystyle \Phi }
which is a linear combination of elements of
Δ
{\displaystyle \Delta }
with nonnegative coefficients.
negative roots
Negative root of root system
Φ
{\displaystyle \Phi }
with respect to a set of simple roots
Δ
{\displaystyle \Delta }
is a root of
Φ
{\displaystyle \Phi }
which is a linear combination of elements of
Δ
{\displaystyle \Delta }
with nonpositive coefficients.
long root
short root
Weyl group
Weyl group of a root system
Φ
{\displaystyle \Phi }
is a (necessarily finite) group of orthogonal linear transformations of
E
{\displaystyle E}
which is generated by reflections through hyperplanes normal to roots of
Φ
{\displaystyle \Phi }
inverse of a root system
Given a root system
Φ
{\displaystyle \Phi }
. Define
α
v
=
2
α
(
α
,
α
)
{\displaystyle \alpha ^{v}={\frac {2\alpha }{(\alpha ,\alpha )}}}
,
Φ
v
=
{
α
v
|
α
∈
Φ
}
{\displaystyle \Phi ^{v}=\{\alpha ^{v}|\alpha \in \Phi \}}
is called the inverse of a root system.
Φ
v
{\displaystyle \Phi ^{v}}
is again a root system and have the identical Weyl group as
Φ
{\displaystyle \Phi }
.
base of a root system
synonymous to "set of simple roots"
dual of a root system
synonymous to "inverse of a root system"
theory of weights
<--
weight in a root system
λ
∈
E
{\displaystyle \lambda \in E}
is called a weight if
<
λ
,
α
>∈
Z
∀
α
∈
Φ
{\displaystyle <\lambda ,\alpha >\in \mathbb {Z} \,\forall \alpha \in \Phi }
. -->
weight lattice
weight space
dominant weight
A weight \lambda is dominant if
<
λ
,
α
>∈
Z
+
{\displaystyle <\lambda ,\alpha >\in \mathbb {Z} ^{+}}
for some
α
∈
Φ
{\displaystyle \alpha \in \Phi }
fundamental dominant weight
Given a set of simple roots
Δ
=
{
α
1
,
α
2
,
.
.
.
,
α
n
}
{\displaystyle \Delta =\{\alpha _{1},\alpha _{2},...,\alpha _{n}\}}
, it is a basis of
E
{\displaystyle E}
.
α
1
v
,
α
2
v
,
.
.
.
,
α
n
v
∈
Φ
v
{\displaystyle \alpha _{1}^{v},\alpha _{2}^{v},...,\alpha _{n}^{v}\in \Phi ^{v}}
is a basis of
E
{\displaystyle E}
too; the dual basis
λ
1
,
λ
2
,
.
.
.
,
λ
n
{\displaystyle \lambda _{1},\lambda _{2},...,\lambda _{n}}
defined by
(
λ
i
,
α
j
v
)
=
δ
i
j
{\displaystyle (\lambda _{i},\alpha _{j}^{v})=\delta _{ij}}
, is called the fundamental dominant weights.
highest weight
minimal weight
multiplicity (of weight)
radical weight
strongly dominant weight
Representation theory
module
Define an action of
g
{\displaystyle {\mathfrak {g}}}
on a vector space
V
{\displaystyle V}
( i.e. an operation
g
×
V
→
V
,
(
x
,
v
)
↦
x
v
{\displaystyle {\mathfrak {g}}\times V\to V,\,(x,v)\mapsto xv}
) such that:
∀
a
,
b
∈
F
,
x
,
y
∈
g
,
v
,
w
∈
V
{\displaystyle \forall a,b\in F,x,y\in {\mathfrak {g}},v,w\in V}
satisfy
#
(
a
x
+
b
y
)
v
=
a
(
x
v
)
+
b
(
y
v
)
{\displaystyle (ax+by)v=a(xv)+b(yv)}
#
x
(
a
v
+
b
w
)
=
a
(
x
v
)
+
b
(
x
w
)
{\displaystyle x(av+bw)=a(xv)+b(xw)}
#
[
x
,
y
]
v
=
x
(
y
v
)
−
y
(
x
v
)
{\displaystyle [x,y]v=x(yv)-y(xv)}
Then
V
{\displaystyle V}
is called a
g
{\displaystyle {\mathfrak {g}}}
-module. (Remark:
V
,
g
{\displaystyle V,{\mathfrak {g}}}
have the same underlying field
F
{\displaystyle F}
.)
Each
g
{\displaystyle {\mathfrak {g}}}
-module corresponds to a representation
g
→
g
l
V
{\displaystyle {\mathfrak {g}}\to {\mathfrak {gl}}_{V}}
.
A subspace W is a submodule (more precisely, sub
g
{\displaystyle {\mathfrak {g}}}
-module) of
V
{\displaystyle V}
if
g
{\displaystyle {\mathfrak {g}}}
-module
W
⊂
V
{\displaystyle W\subset V}
.
representation
For a vector space
V
{\displaystyle V}
, if there is a Lie algebra homomorphism
π
:
g
→
g
l
V
{\displaystyle \pi :{\mathfrak {g}}\to {\mathfrak {gl}}_{V}}
, then
π
{\displaystyle \pi }
is called a representation of
g
{\displaystyle {\mathfrak {g}}}
.
Each representation
g
→
g
l
V
{\displaystyle {\mathfrak {g}}\to {\mathfrak {gl}}_{V}}
corresponds to a
g
{\displaystyle {\mathfrak {g}}}
-module
V
{\displaystyle V}
.
A subrepresentation is the representation corresponding to a submodule.
homomorphism
Given two
g
{\displaystyle {\mathfrak {g}}}
-module V, W, a
g
{\displaystyle {\mathfrak {g}}}
-module homomorphism
ϕ
{\displaystyle \phi }
is a vector space homomorphism satisfying
ϕ
(
x
v
)
=
x
ϕ
(
v
)
∀
x
∈
g
,
v
∈
V
{\displaystyle \phi (xv)=x\phi (v)\forall x\in {\mathfrak {g}},v\in V}
.
trivial representation
A representation is said to be trivial if the image of
g
{\displaystyle {\mathfrak {g}}}
is the zero vector space. It corresponds to the action of
g
{\displaystyle {\mathfrak {g}}}
on module
V
{\displaystyle V}
by
x
v
=
0
∀
x
∈
g
,
v
∈
V
{\displaystyle xv=0\forall x\in {\mathfrak {g}},v\in V}
.
faithful representation
If the representation
g
→
g
l
V
{\displaystyle {\mathfrak {g}}\to {\mathfrak {gl}}_{V}}
is injective, it is said to be faithful.
tautology representation
If a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
is defined as a subalgebra of
g
l
(
n
,
F
)
{\displaystyle {\mathfrak {gl}}(n,F)}
, like
s
l
(
n
,
F
)
,
o
(
2
l
,
F
)
,
t
(
n
,
F
)
{\displaystyle {\mathfrak {sl}}(n,F),{\mathfrak {o}}(2l,F),{\mathfrak {t}}(n,F)}
(the upper triangular matrices), the tautology representation is the imbedding
g
→
g
l
(
n
,
F
)
{\displaystyle {\mathfrak {g}}\to {\mathfrak {gl}}(n,F)}
. It corresponds to the action on module
F
n
{\displaystyle F^{n}}
by the matrix multiplication.
adjoint representation
The representation
ad
:
g
→
gl
g
x
↦
ad
x
{\displaystyle {\begin{aligned}{\textrm {ad}}:&{\mathfrak {g}}\to {\textrm {gl}}_{\mathfrak {g}}\\&x\mapsto {\textrm {ad}}_{x}\end{aligned}}}
. It corresponds viewing
g
{\displaystyle {\mathfrak {g}}}
as a
g
{\displaystyle {\mathfrak {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 indecmoposable 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 irreduible module.
quotient module / quotient representation
Given a
g
{\displaystyle {\mathfrak {g}}}
-module V and its submodule W, an action
g
{\displaystyle {\mathfrak {g}}}
on V/W can be defined by
x
(
v
+
W
)
=
x
v
+
W
∀
x
∈
g
,
v
∈
V
{\displaystyle x(v+W)=xv+W\,\forall x\in {\mathfrak {g}},v\in 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
{\displaystyle {\mathfrak {g}}}
-module,
ϕ
V
→
V
{\displaystyle \phi V\to V}
is a
g
{\displaystyle {\mathfrak {g}}}
-module homomorphism iff
ϕ
=
λ
1
V
{\displaystyle \phi =\lambda 1_{V}}
for some
λ
∈
F
{\displaystyle \lambda \in F}
.
Statement in the language of representation theory: Given an irreducible representation
ϕ
:
L
→
g
l
(
V
)
{\displaystyle \phi \colon L\to {\mathfrak {gl}}(V)}
, for
θ
∈
End
(
V
)
{\displaystyle \theta \in {\textrm {End}}(V)}
,
θ
ϕ
(
x
)
=
ϕ
(
x
)
θ
{\displaystyle \theta \phi (x)=\phi (x)\theta }
iff
θ
=
λ
1
V
{\displaystyle \theta =\lambda 1_{V}}
for some
λ
∈
F
{\displaystyle \lambda \in F}
.
simple module
synonymous to "irreduible module".
factor module
synonymous to "quotient module".
Universal enveloping algebras
PBW theorem (Poincaré–Birkhoff–Witt theorem)
Verma modules
BGG category \mathcal{O}
cohomology
This section is empty. You can help by
adding to it .
(July 2010 )
Chevalley basis
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
{\displaystyle [H_{\alpha _{i}},H_{\alpha _{j}}]=0}
[
H
α
i
,
E
α
j
]
=
A
i
j
E
α
j
{\displaystyle [H_{\alpha _{i}},E_{\alpha _{j}}]=A_{ij}E_{\alpha _{j}}}
[
E
α
i
,
E
α
j
]
=
H
α
j
{\displaystyle [E_{\alpha _{i}},E_{\alpha _{j}}]=H_{\alpha _{j}}}
[
E
β
,
E
γ
]
=
±
(
p
+
1
)
E
β
+
γ
{\displaystyle [E_{\beta },E_{\gamma }]=\pm (p+1)E_{\beta +\gamma }}
where p = m if β + γ is a root and m is the greatest positive integer such that γ − m β is a root.
Examples of Lie algebra
general linear algebra
g
l
(
n
,
F
)
{\displaystyle gl(n,F)}
Ado's theorem
Any finite-dimensional Lie algebra is isomorphic to a subalgebra of
g
l
V
{\displaystyle {\mathfrak {gl}}_{V}}
for some finite-dimensional vector space V.
complex Lie algebras of 1D, 2D, 3D
This section is empty. You can help by
adding to it .
(January 2011 )
Simple Algebras
Classical Lie algebras :
Name
Root System
dimension
construction as subalgebra of
g
l
(
n
,
F
)
{\displaystyle {\mathfrak {gl}}(n,F)}
Special linear algebra
A
l
(
l
≥
1
)
{\displaystyle A_{l}\ (l\geq 1)}
l
2
+
2
l
{\displaystyle l^{2}+2l}
s
l
(
l
+
1
,
F
)
=
{
x
∈
g
l
(
l
+
1
,
F
)
|
T
r
(
x
)
=
0
}
{\displaystyle {\mathfrak {sl}}(l+1,F)=\{x\in {\mathfrak {gl}}(l+1,F)|Tr(x)=0\}}
(traceless matrices)
Orthogonal algebra
B
l
(
l
≥
1
)
{\displaystyle B_{l}\ (l\geq 1)}
2
l
2
+
l
{\displaystyle 2l^{2}+l}
o
(
2
l
+
1
,
F
)
=
{
x
∈
g
l
(
2
l
+
1
,
F
)
|
s
x
=
−
x
t
s
,
s
=
(
1
0
0
0
0
I
l
0
I
l
0
)
}
{\displaystyle {\mathfrak {o}}(2l+1,F)=\{x\in {\mathfrak {gl}}(2l+1,F)|sx=-x^{t}s,s={\begin{pmatrix}1&0&0\\0&0&I_{l}\\0&I_{l}&0\end{pmatrix}}\}}
Symplectic algebra
C
l
(
l
≥
2
)
{\displaystyle C_{l}\ (l\geq 2)}
2
l
2
−
l
{\displaystyle 2l^{2}-l}
s
p
(
2
l
,
F
)
=
{
x
∈
g
l
(
2
l
,
F
)
|
s
x
=
−
x
t
s
,
s
=
(
0
I
l
−
I
l
0
)
}
{\displaystyle {\mathfrak {sp}}(2l,F)=\{x\in {\mathfrak {gl}}(2l,F)|sx=-x^{t}s,s={\begin{pmatrix}0&I_{l}\\-I_{l}&0\end{pmatrix}}\}}
Orthogonal algebra
D
l
(
l
≥
1
)
{\displaystyle D_{l}(l\geq 1)}
2
l
2
+
l
{\displaystyle 2l^{2}+l}
o
(
2
l
,
F
)
=
{
x
∈
g
l
(
2
l
,
F
)
|
s
x
=
−
x
t
s
,
s
=
(
0
I
l
I
l
0
)
}
{\displaystyle {\mathfrak {o}}(2l,F)=\{x\in {\mathfrak {gl}}(2l,F)|sx=-x^{t}s,s={\begin{pmatrix}0&I_{l}\\I_{l}&0\end{pmatrix}}\}}
Exceptional Lie algebras :
Root System
dimension
G2
14
F4
52
E6
78
E7
133
E8
248
Miscellaneous
Other discipline related
References
Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras , 1st edition, Springer, 2006. ISBN 1-84628-040-0
Humphreys, James E. Introduction to Lie Algebras and Representation Theory , Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5