aarjaneiro/planetmath huggingface dataset:
radical

radical

Let

𝔤

be a

Lie algebra

. Since the sum of any two

solvable

ideals of

𝔤

is in turn solvable, there is a unique maximal solvable ideal of any Lie algebra. This ideal is called the

radical

of

𝔤

. Note that

𝔤

/

rad

⁢

𝔤

has no solvable ideals, and is thus

semi-simple

. Thus, every Lie algebra is an extension (

http://planetmath.org/Extension

) of a semi-simple algebra by a solvable one.

Title

radical

Canonical

name

Radical

Date of creation

2013-03-22 13:51:18

Last modified on

2013-03-22 13:51:18

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B05

Generated on Fri Feb 9 18:48:32 2018 by

LaTeXML
regular element of a Lie algebra

regular element of a Lie algebra

An element

X

∈

𝔤

of a

Lie algebra

is called

regular

if the

dimension

of its

centralizer

ζ

𝔤

⁢

(

X

)

=

{

Y

∈

𝔤

∣

[

X

,

Y

]

=

0

}

is minimal among all centralizers of elements in

𝔤

.

Regular elements

clearly exist and moreover they are Zariski dense in

𝔤

. The function

X

↦

dim

⁡

ζ

𝔤

⁢

(

X

)

is an upper semi-continuous function

𝔤

→

ℤ

≥

0

. Indeed, it is a constant minus

rank

⁡

(

a

⁢

d

X

)

and

X

↦

rank

⁡

(

a

⁢

d

X

)

is lower semi-continuous. Thus the set of elements whose centralizer dimension is (greater than or) equal to that of any given regular element is Zariski open and non-empty.

If

𝔤

is reductive then the minimal centralizer dimension is equal to the rank of

𝔤

.

More generally if

V

is a

representation

for a Lie algebra

𝔤

,
an element

v

∈

V

is called

regular

if the dimension of its

stabilizer

is minimal among all stabilizers of elements in

V

.

Examples

1.

In

𝔰

⁢

𝔩

n

⁢

ℂ

a

diagonal matrix

X

=

diag

⁡

(

s

1

,

…

,

s

n

)

is regular iff

(

s

i

-

s

j

)

≠

0

for all pairs

1

≤

i

<

j

≤

n

. Any

conjugate

of such a matrix is also obviously regular.

2.

In

𝔰

⁢

𝔩

n

⁢

ℂ

the

nilpotent matrix

(

0

1

⋯

0

0

0

1

⋮

⋱

⋱

1

0

⋯

0

)

is regular. Moreover, it’s

adjoint

orbit contains the set of all regular

nilpotent elements

. The centralizer of this matrix is the full

subalgebra

of
trace zero,

diagonal

matricies.

Title

regular element of a Lie algebra

Canonical

name

RegularElementOfALieAlgebra

Date of creation

2013-03-22 15:30:53

Last modified on

2013-03-22 15:30:53

Owner

benjaminfjones (879)

Last modified by

benjaminfjones (879)

Numerical id

6

Author

benjaminfjones (879)

Entry type

Definition

Classification

msc 17B05

Defines

regular element

Generated on Fri Feb 9 18:48:34 2018 by

LaTeXML
adjoint representation

adjoint representation

\DeclareMathOperator

\ad

ad

\DeclareMathOperator

\End

End

Let

\mathfrak

⁢

g

be a

Lie algebra

. For every

a

∈

\mathfrak

⁢

g

we define the
, a.k.a. the

adjoint action

,

\ad

⁢

(

a

)

:

\mathfrak

⁢

g

→

\mathfrak

⁢

g

to be the

linear transformation

with
action

\ad

⁢

(

a

)

:

b

↦

[

a

,

b

]

,

b

∈

\mathfrak

⁢

g

.

For any

vector space

V

, we use

\mathfrak

⁢

g

⁢

l

⁢

(

V

)

to denote the Lie algebra
of

\End

⁢

V

determined by the

commutator bracket

. So

\mathfrak

⁢

g

⁢

l

⁢

(

V

)

=

\End

⁢

V

as vector spaces, only the

multiplications

are different.

In this notation, treating

\mathfrak

⁢

g

as a vector space, the linear mapping

\ad

:

\mathfrak

⁢

g

→

\mathfrak

⁢

g

⁢

l

⁢

(

\mathfrak

⁢

g

)

with action

a

↦

\ad

⁢

(

a

)

,

a

∈

\mathfrak

⁢

g

is called the

adjoint representation

of

\mathfrak

⁢

g

. The fact that

\ad

defines a

representation

is a straight-forward

consequence

of
the

Jacobi identity

axiom. Indeed, let

a

,

b

∈

\mathfrak

⁢

g

be given. We
wish to show that

\ad

⁢

(

[

a

,

b

]

)

=

[

\ad

⁢

(

a

)

,

\ad

⁢

(

b

)

]

,

where the bracket on the left is the

\mathfrak

⁢

g

multiplication

structure

, and the bracket on the right is the
commutator bracket. For all

c

∈

\mathfrak

⁢

g

the

left hand side

maps

c

to

[

[

a

,

b

]

,

c

]

,

while the right hand side maps

c

to

[

a

,

[

b

,

c

]

]

+

[

b

,

[

a

,

c

]

]

.

Taking skew-symmetry of the bracket as a
given, the equality of these two expressions is

logically equivalent

to the Jacobi identity:

[

a

,

[

b

,

c

]

]

+

[

b

,

[

c

,

a

]

]

+

[

c

,

[

a

,

b

]

]

=

0

.

Title

adjoint representation

Canonical name

AdjointRepresentation

Date of creation

2015-10-05 17:38:19

Last modified on

2015-10-05 17:38:19

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Definition

Classification

msc 17B10

Related topic

IsotropyRepresentation

Defines

adjoint action

Defines

gl

Defines

general linear Lie algebra

Generated on Fri Feb 9 18:48:36 2018 by

LaTeXML
Ado’s theorem

Ado’s theorem

Every

finite dimensional

Lie algebra

has a faithful finite dimensional representation. In other words, every finite dimensional Lie algebra is a matrix algebra.

This result is not true for Lie groups.

Title

Ado’s theorem

Canonical name

AdosTheorem

Date of creation

2013-03-22 13:20:42

Last modified on

2013-03-22 13:20:42

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Theorem

Classification

msc 17B10

Related topic

ExamplesOfNonMatrixLieGroup

Generated on Fri Feb 9 18:48:37 2018 by

LaTeXML
examples of non-matrix Lie groups

examples of non-matrix Lie groups

While most well-known Lie groups are

matrix groups

, there do in fact exist Lie groups which are not matrix groups. That is, they have no

faithful

finite dimensional representations.

For example, let

H

be the real

Heisenberg group

H

=

{

(

1

a

b

0

1

c

0

0

1

)

∣

a

,

b

,

c

∈

ℝ

}

,

and

Γ

the discrete subgroup

Γ

=

{

(

1

0

n

0

1

0

0

0

1

)

∣

n

∈

ℤ

}

.

The

subgroup

Γ

is

central

, and thus normal. The Lie group

H

/

Γ

has no faithful finite dimensional representations over

ℝ

or

ℂ

.

Another example is the

universal cover

of

SL

2

⁢

ℝ

.

SL

2

⁢

ℝ

is

homotopy equivalent

to a circle, and thus

π

⁢

(

SL

2

⁢

ℝ

)

≅

ℤ

, and thus has an infinite-sheeted cover. Any real or complex representation of this group factors through the projection map to

SL

2

⁢

ℝ

.

Title

examples of non-matrix Lie groups

Canonical name

ExamplesOfNonmatrixLieGroups

Date of creation

2013-03-22 13:20:48

Last modified on

2013-03-22 13:20:48

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Example

Classification

msc 17B10

Related topic

AdosTheorem

Generated on Fri Feb 9 18:48:39 2018 by

LaTeXML
isotropy representation

isotropy representation

Let

𝔤

be a

Lie algebra

, and

𝔥

⊂

𝔤

a

subalgebra

. The

isotropy representation

of

𝔥

relative to

𝔤

is the naturally
defined action of

𝔥

on the

quotient vector space

𝔤

/

𝔥

.

Here is a synopsis of the technical details. As is customary, we will
use

b

+

𝔥

,

b

∈

𝔤

to denote the coset elements of

𝔤

/

𝔥

.
Let

a

∈

𝔥

be given. Since

𝔥

is invariant with respect to

ad

𝔤

(

a

)

, the

adjoint action

factors through the

quotient

to
give a well defined

endomorphism

of

𝔤

/

𝔥

. The action is given
by

b

+

𝔥

↦

[

a

,

b

]

+

𝔥

,

b

∈

𝔤

.

This is the action
alluded to in the first paragraph.

Title

isotropy representation

Canonical name

IsotropyRepresentation

Date of creation

2013-03-22 12:42:28

Last modified on

2013-03-22 12:42:28

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Definition

Classification

msc 17B10

Related topic

AdjointRepresentation

Generated on Fri Feb 9 18:48:41 2018 by

LaTeXML
Lie algebra representation

Lie algebra representation

A

representation

of a

Lie algebra

𝔤

is a Lie algebra

homomorphism

ρ

:

𝔤

→

End

V

,

where

End

V

is the

commutator Lie
algebra

of some

vector space

V

. In other words,

ρ

is a

linear
mapping

that satisfies

ρ

⁢

(

[

a

,

b

]

)

=

ρ

⁢

(

a

)

⁢

ρ

⁢

(

b

)

-

ρ

⁢

(

b

)

⁢

ρ

⁢

(

a

)

,

a

,

b

∈

𝔤

Alternatively, one calls

V

a

𝔤

-module, and calls

ρ

⁢

(

a

)

,

a

∈

𝔤

the action of

a

on

V

.

We call the representation

faithful

if

ρ

is injective.

A

invariant subspace

or sub-module

W

⊂

V

is a

subspace

of

V

satisfying

ρ

⁢

(

a

)

⁢

(

W

)

⊂

W

for all

a

∈

𝔤

. A representation is
called

irreducible

or simple if its only invariant subspaces are

{

0

}

and the whole representation.

The

dimension

of

V

is called the dimension of the representation.
If

V

is infinite-dimensional, then one speaks of an
infinite-dimensional representation.

Given a pair of representations, we can define a new representation, called the

direct sum

of the two given representations:

If

ρ

:

𝔤

→

End

(

V

)

and

σ

:

𝔤

→

End

(

W

)

are representations, then

V

⊕

W

has the obvious Lie algebra action, by the embedding

End

(

V

)

×

End

(

W

)

↪

End

(

V

⊕

W

)

.

Title

Lie algebra representation

Canonical name

LieAlgebraRepresentation

Date of creation

2013-03-22 12:41:13

Last modified on

2013-03-22 12:41:13

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Definition

Classification

msc 17B10

Synonym

representation

Related topic

Dimension3

Defines

irreducible

Defines

module

Defines

dimension

Defines

finite dimensional

Defines

finite-dimensional

Defines

infinite dimensional

Defines

infinite-dimensional

Defines

faithful

Defines

direct sum of representations

Generated on Fri Feb 9 18:48:43 2018 by

LaTeXML
quadratic Lie algebra

quadratic Lie algebra

A

Lie algebra

𝔤

is said to be

quadratic

if

𝔤

as a

representation

(under the

adjoint action

) admits a

non-degenerate

,

invariant scalar product

(

⋅

∣

⋅

)

.

𝔤

being quadratic implies that the

adjoint

and co-adjoint representations of

𝔤

are

isomorphic

.

Indeed, the non-degeneracy of

(

⋅

∣

⋅

)

implies that the induced map

ϕ

:

𝔤

→

𝔤

*

given by

ϕ

(

X

)

(

Z

)

=

(

X

∣

Z

)

is an

isomorphism

of

vector spaces

. Invariance of the

scalar product

means that

(

[

X

,

Y

]

∣

Z

)

=

-

(

Y

∣

[

X

,

Z

]

)

=

(

Y

∣

[

Z

,

X

]

)

. This implies that

ϕ

is a map of representations:

ϕ

(

a

d

X

(

Y

)

)

(

Z

)

=

ϕ

(

[

X

,

Y

]

)

(

Z

)

=

(

[

X

,

Y

]

∣

Z

)

=

(

Y

∣

[

Z

,

X

]

)

=

a

d

X

*

(

ϕ

(

Y

)

(

Z

)

)

Title

quadratic Lie algebra

Canonical name

QuadraticLieAlgebra

Date of creation

2013-03-22 15:30:44

Last modified on

2013-03-22 15:30:44

Owner

benjaminfjones (879)

Last modified by

benjaminfjones (879)

Numerical id

6

Author

benjaminfjones (879)

Entry type

Definition

Classification

msc 17B10

Classification

msc 17B01

Related topic

quadraticAlgebra

Defines

quadratic Lie algebra

Generated on Fri Feb 9 18:48:45 2018 by

LaTeXML
invariant form (Lie algebras)

invariant form (Lie algebras)

Let

V

be a

representation

of a

Lie algebra

𝔤

over a field

k

. Then a

bilinear form

B

:

V

×

V

→

k

is

invariant

if

B

⁢

(

X

⁢

v

,

w

)

+

B

⁢

(

v

,

X

⁢

w

)

=

0

.

for all

X

∈

𝔤

,

v

,

w

∈

V

. This criterion seems a little odd, but in the context of Lie algebras,
it makes sense. For example, the map

B

~

:

V

→

V

*

given by

v

↦

B

⁢

(

⋅

,

v

)

is

equivariant

if and only if

B

is an invariant form.

Title

invariant form (Lie algebras)

Canonical name

InvariantFormLieAlgebras

Date of creation

2013-03-22 13:15:45

Last modified on

2013-03-22 13:15:45

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B15

Generated on Fri Feb 9 18:48:47 2018 by

LaTeXML
Borel subalgebra

Borel subalgebra

Let

𝔤

be a

semi-simple

Lie group

,

𝔥

a

Cartan subalgebra

,

R

the associated

root system

, and

R

+

⊂

R

a set of

positive roots

. We have a root decomposition into the Cartan subalgebra and the root spaces

𝔤

α

𝔤

=

𝔥

⊕

(

⊕

α

∈

R

𝔤

α

)

.

Now let

𝔟

be the

direct sum

of the Cartan subalgebra and the positive root spaces.

𝔟

=

𝔥

⊕

(

⊕

β

∈

R

+

𝔤

β

)

.

This is called a

Borel subalgebra

.

Title

Borel subalgebra

Canonical name

BorelSubalgebra

Date of creation

2013-03-22 13:12:16

Last modified on

2013-03-22 13:12:16

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 17B20

Generated on Fri Feb 9 18:48:48 2018 by

LaTeXML
Borel subgroup

Borel subgroup

Let

G

=

GL

n

⁢

ℂ

, the group of all

automorphisms

of the

n

-dimensional

vector space

over the field of

complex numbers

ℂ

, and

H

≤

G

a

subgroup

of

G

. The

standard

Borel subgroup

of

H

is the subgroup of

H

consisting of all

upper triangular matrices

(in

H

). A

Borel subgroup

of

H

is a

conjugate

(in

H

) of the standard Borel subgroup of

H

.

The notion of a Borel subgroup can be generalized. Let

G

be a complex

semi-simple

Lie group

. Then any

maximal

solvable

subgroup

B

≤

G

is called a Borel subgroup. All Borel subgroups of a given group are
conjugate. Any

Borel group

is

connected

and equal to its own

normalizer

, and contains a
unique

Cartan subgroup

. The intersection of

B

with a maximal compact subgroup

K

of

G

is the

maximal torus

of

K

.

Title

Borel subgroup

Canonical name

BorelSubgroup

Date of creation

2013-03-22 13:27:58

Last modified on

2013-03-22 13:27:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17B20

Generated on Fri Feb 9 18:48:50 2018 by

LaTeXML
Cartan matrix

Cartan matrix

Let

R

⊂

E

be a

reduced root system

, with

E

a

Euclidean vector space

, with

inner product

(

⋅

,

⋅

)

, and let

Π

=

{

α

1

,

⋯

,

α

n

}

be a base of this

root system

. Then the

Cartan matrix

of the root system is the matrix

C

i

,

j

=

(

2

⁢

(

α

i

,

α

j

)

(

α

i

,

α

i

)

)

.

The Cartan matrix uniquely determines the root system, and is unique up to simultaneous

permutation

of the rows and columns. It is also the

basis change

matrix from the basis
of fundamental weights to the basis of

simple roots

in

E

.

Title

Cartan matrix

Canonical name

CartanMatrix

Date of creation

2013-03-22 13:17:56

Last modified on

2013-03-22 13:17:56

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B20

Generated on Fri Feb 9 18:48:52 2018 by

LaTeXML
Cartan subalgebra

Cartan subalgebra

Let

𝔤

be a

Lie algebra

. Then a

Cartan subalgebra

is a

maximal

subalgebra

of

𝔤

which is

self-normalizing

, that is, if

[

g

,

h

]

∈

𝔥

for all

h

∈

𝔥

, then

g

∈

𝔥

as well. Any Cartan subalgebra

𝔥

is

nilpotent

, and if

𝔤

is

semi-simple

, it is

abelian

. All Cartan subalgebras of a Lie algebra are

conjugate

by the

adjoint action

of any Lie group with

algebra

𝔤

.

The

dimension

of

𝔥

is called the

rank

of

𝔤

.

Title

Cartan subalgebra

Canonical name

CartanSubalgebra

Date of creation

2013-03-22 13:20:09

Last modified on

2013-03-22 13:20:09

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B20

Defines

rank of a Lie algebra

Generated on Fri Feb 9 18:48:54 2018 by

LaTeXML
Cartan’s criterion

Cartan’s criterion

A

Lie algebra

𝔤

is

semi-simple

if and only if its

Killing form

B

𝔤

is

nondegenerate

.

Title

Cartan’s criterion

Canonical name

CartansCriterion

Date of creation

2013-03-22 13:15:50

Last modified on

2013-03-22 13:15:50

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Theorem

Classification

msc 17B20

Generated on Fri Feb 9 18:48:55 2018 by

LaTeXML
Casimir operator

Casimir operator

Let

𝔤

be a

semisimple Lie algebra

, and let

(

⋅

,

⋅

)

denote the

Killing form

. If

{

g

i

}

is a basis of

𝔤

, then there is a

dual basis

{

g

i

}

with respect to the Killing form, i.e.,

(

g

i

,

g

j

)

=

δ

i

⁢

j

.
Consider the element

Ω

=

∑

g

i

⁢

g

i

of the

universal enveloping algebra

of

𝔤

. This element, called the

Casimir operator

is

central

in the enveloping algebra, and thus commutes with the

𝔤

action on any

representation

.

Title

Casimir operator

Canonical name

CasimirOperator

Date of creation

2013-03-22 13:52:53

Last modified on

2013-03-22 13:52:53

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B20

Generated on Fri Feb 9 18:48:57 2018 by

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classification of finite-dimensional representations of semi-simple Lie algebras

classification of finite-dimensional representations of semi-simple Lie algebras

If

𝔤

is a

semi-simple Lie algebra

, then we say that an

representation

V

has

highest weight

λ

, if there is a vector

v

∈

V

λ

, the

weight space

of

λ

, such that

X

⁢

v

=

0

for

X

in any

positive root

space, and

v

is called a

highest
vector

, or

vector of highest weight

.

There is a unique (up to

isomorphism

)

irreducible

finite dimensional representation of

𝔤

with highest weight

λ

for any

dominant
weight

λ

∈

Λ

W

, where

Λ

W

is the

weight lattice

of

𝔤

, and
every

irreducible representation

of

𝔤

is of this type.

Title

classification of finite-dimensional representations of semi-simple Lie algebras

Canonical name

ClassificationOfFinitedimensionalRepresentationsOfSemisimpleLieAlgebras

Date of creation

2013-03-22 13:11:40

Last modified on

2013-03-22 13:11:40

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B20

Defines

highest weight

Defines

highest vector

Defines

vector of highest weight

Defines

highest weight representation

Generated on Fri Feb 9 18:48:59 2018 by

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classification of indecomposable root systems

classification of indecomposable root systems

There are four infinite families of

indecomposable

root systems

:

A

n

=

{

±

e

i

∓

e

j

:

1

≤

i

<

j

≤

n

+

1

}

;

B

n

=

{

±

e

i

±

e

j

:

1

≤

i

<

j

≤

n

}

∪

{

±

e

i

:

1

≤

i

≤

n

}

;

C

n

=

{

±

e

i

±

e

j

:

1

≤

i

<

j

≤

n

}

∪

{

±

2

⁢

e

i

:

1

≤

i

≤

n

}

;

D

n

=

{

±

e

i

±

e

j

:

1

≤

i

<

j

≤

n

}

The subscript on the name of the root system is the

dimension

of

𝐄

,
the ambient

Euclidean space

containing the root system. In the case of

A

n

, the ambient

𝐄

is the

n

-dimensional

subspace

perpendicular

to

∑

i

=

1

n

e

i

. In the other 3 cases,

𝐄

=

ℝ

n

.
Throughout, we endow

ℝ

n

with the standard Euclidean

inner
product

, and let

e

i

,

1

≤

i

≤

n

denote the

standard basis

.

As well, there are 5 exceptional, crystallographic root systems:

G

2

=

A

3

∪

{

±

1

3

⁢

(

2

⁢

e

1

-

e

2

-

e

3

)

,

±

1

3

⁢

(

-

e

1

+

2

⁢

e

2

-

e

3

)

,

±

1

3

⁢

(

-

e

1

-

e

2

+

2

⁢

e

3

)

}

;

F

4

=

B

4

∪

{

1

2

⁢

(

±

e

1

±

e

2

±

e

3

±

e

4

)

}

;

E

6

=

A

6

∪

{

±

(

e

7

-

e

8

)

}

∪

{

1

2

⁢

(

∑

i

=

1

6

(

±

e

i

)

±

(

e

7

-

e

8

)

)

:

4 minus signs

}

;

E

7

=

A

8

∪

{

∑

i

=

1

8

(

±

e

i

)

:

4 minus signs

}

;

E

8

=

D

8

∪

{

∑

i

=

1

8

(

±

e

i

)

:

even number of minus signs

}

.

The following table indicates the cardinality of and the

Lie algebras

and

Dynkin diagrams

corresponding to the above root systems.

Figure 1:

Irreducible root systems and

simple Lie algebras

.

Title

classification of indecomposable root systems

Canonical name

ClassificationOfIndecomposableRootSystems

Date of creation

2013-03-22 15:28:56

Last modified on

2013-03-22 15:28:56

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Result

Classification

msc 17B20

Generated on Fri Feb 9 18:49:02 2018 by

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cohomology of semi-simple Lie algebras

cohomology of semi-simple Lie algebras

There are some important facts that make the

cohomology of semi-simple Lie algebras

easier to deal with than general

Lie algebra cohomology

. In particular, there are a number of vanishing theorems.

First of all, let

𝔤

be a

finite-dimensional

semi-simple Lie algebra

over a field

𝕂

of

characteristic

0

.

Theorem [Whitehead] -

Let

M

be an

irreducible

𝔤

-module (

http://planetmath.org/RepresentationLieAlgebra

) of

dimension

greater than

1

. Then all the

cohomology groups

with coefficients in

M

are trivial, i.e.

H

n

⁢

(

𝔤

,

M

)

=

0

for all

n

∈

ℕ

.

Thus, the only interesting cohomology groups with coefficients in an irreducible

𝔤

-module are

H

n

⁢

(

𝔤

,

𝕂

)

. For arbitrary

𝔤

-modules there are still two vanishing results, which are usually called

Whitehead’s lemmas

.

Whitehead’s Lemmas -

Let

M

be a finite-dimensional

𝔤

-module. Then

•

First Lemma :

H

1

⁢

(

𝔤

,

M

)

=

0

.

•

Second Lemma :

H

2

⁢

(

𝔤

,

M

)

=

0

.

Whitehead’s lemmas lead to two very important results. From the vanishing of

H

1

, we can derive Weyl’s theorem, the fact that representations of semi-simple Lie algebras are completely reducible, since

extensions

of

M

by

N

are classified by

H

1

⁢

(

𝔤

,

Hom

⁢

(

M

,

N

)

)

. And from the vanishing of

H

2

, we obtain Levi’s theorem, which that every

Lie algebra

is a

split extension

of a semi-simple algebra by a

solvable

algebra

since

H

2

⁢

(

𝔤

,

M

)

classifies extensions of

𝔤

by

M

with a specified action of

𝔤

on

M

.

Title

cohomology of semi-simple Lie algebras

Canonical name

CohomologyOfSemisimpleLieAlgebras

Date of creation

2013-03-22 13:51:13

Last modified on

2013-03-22 13:51:13

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Feature

Classification

msc 17B20

Classification

msc 17B56

Defines

Whitehead’s lemmas

Defines

Whitehead’s first lemma

Defines

Whitehead’s second lemma

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