aarjaneiro/planetmath huggingface dataset:
Dynkin diagram

Dynkin diagram

Dynkin diagrams

are a combinatorial way of representing the information in a

root
system

. Their primary advantage is that they are easier to write down, remember,
and analyze than explicit

representations

of a root system. They are an important
tool in the

classification

of

simple Lie algebras

.

Given a

reduced root system

R

⊂

E

, with

E

an inner-product space, choose a base
or

simple roots

Π

(or equivalently, a set of

positive roots

R

+

). The
Dynkin diagram associated to

R

is a graph whose vertices are

Π

. If

π

i

and

π

j

are distinct elements of the root system, we add

m

i

⁢

j

=

-

4

⁢

(

π

i

,

π

j

)

2

(

π

i

,

π

i

)

⁢

(

π

j

,

π

j

)

lines between them. This
number is obivously

positive

, and an integer since it is the product of 2
quantities that the axioms of a root system require to be integers. By the

Cauchy-Schwartz inequality

, and the fact that simple roots are never
anti-parallel (they are all

strictly

contained in some half space),

m

i

⁢

j

∈

{

0

,

1

,

2

,

3

}

. Thus Dynkin diagrams are

finite graphs

, with single,
double or triple edges. Fact, the criteria are much stronger than this:
if the

multiple edges

are counted as single edges, all Dynkin diagrams are
trees, and have at most one multiple edge. In fact, all Dynkin diagrams
fall into 4 infinite families, and 5 exceptional cases, in exact

parallel

to the
classification of simple Lie algebras.

(

Does anyone have good Dynkin diagram pictures? I’d love to put some up, but am decidedly lacking.

)

Title

Dynkin diagram

Canonical

name

DynkinDiagram

Date of creation

2013-03-22 13:28:05

Last modified on

2013-03-22 13:28:05

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 18:49:06 2018 by

LaTeXML
nilpotent cone

nilpotent cone

Let

𝔤

be a

finite dimensional

semisimple Lie algebra

. The

nilpotent cone

𝒩

of

𝔤

is the set of elements that
act nilpotently in all

representations

of

𝔤

. In other words,

𝒩

=

{

a

∈

𝔤

:

ρ

⁢

(

a

)

⁢

is nilpotent for all representations

⁢

ρ

:

𝔤

→

End

⁡

(

V

)

}

The nilpotent cone is an

irreducible

(

http://planetmath.org/IrreducibleClosedSet

)

subvariety

(

http://planetmath.org/AffineVariety

) of

𝔤

(considered as a

k

-

vector space

), and is

invariant

under the

adjoint action

of

𝔤

on itself.

Example:

if

𝔤

=

sl

2

, then the nilpotent cone
is the

variety

of all matrices in

𝔤

with rank

1

.

Title

nilpotent cone

Canonical name

NilpotentCone

Date of creation

2013-03-22 13:58:36

Last modified on

2013-03-22 13:58:36

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

9

Author

rmilson (146)

Entry type

Definition

Classification

msc 17B20

Synonym

nilcone

Generated on Fri Feb 9 18:49:07 2018 by

LaTeXML
parabolic subgroup

parabolic subgroup

Let

G

be a complex

semi-simple

Lie group. Then any subgroup

P

of

G

containg
a

Borel subgroup

B

is called

parabolic

. Parabolics are classified in the
following manner. Let

𝔤

be the

Lie algebra

of

G

,

𝔥

the unique

Cartan
subalgebra

contained in

𝔟

, the

algebra

of

B

,

R

the set of roots corresponding
to this choice of Cartan, and

R

+

the set of

positive roots

whose

root spaces

are
contained in

𝔟

and let

𝔭

be the Lie
algebra of

P

. Then there exists a unique subset

Π

P

of

Π

, the base of

simple
roots

associated to this choice of positive roots, such that

{

𝔟

,

𝔤

-

α

}

α

∈

Π

P

generates

𝔭

. In other words,
parabolics containing a single Borel subgroup are classified by subsets of the

Dynkin diagram

, with the empty set corresponding to the Borel, and the whole graph
corresponding to the group

G

.

Title

parabolic subgroup

Canonical name

ParabolicSubgroup

Date of creation

2013-03-22 13:28:02

Last modified on

2013-03-22 13:28:02

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 18:49:09 2018 by

LaTeXML
positive root

positive root

If

R

⊂

E

is a

root system

, with

E

an

Euclidean vector space

, then a subset

R

+

⊂

R

is called a set of

positive roots

if there is a vector

v

∈

E

such that

(

α

,

v

)

>

0

if

α

∈

R

+

, and

(

α

,

v

)

<

0

if

α

∈

R

\

R

+

.

http://planetmath.org/node/3645

Roots which are not

positive

are called

negative

. Since

-

α

is negative if and only if

α

is positive, exactly half the must be positive.

Title

positive root

Canonical

name

PositiveRoot

Date of creation

2013-03-22 13:11:46

Last modified on

2013-03-22 13:11:46

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

8

Author

mathwizard (128)

Entry type

Definition

Classification

msc 17B20

Defines

negative root

Generated on Fri Feb 9 18:49:20 2018 by

LaTeXML
rank (Lie algebra)

rank (Lie algebra)

Let

𝔤

be a

finite dimensional

Lie algebra

.
One can show that all

Cartan subalgebras

𝔥

⊂

𝔤

have the same

dimension

. The

rank

of

𝔤

is defined to be this dimension.

Title

rank (Lie algebra)

Canonical

name

RankLieAlgebra

Date of creation

2013-03-22 12:25:07

Last modified on

2013-03-22 12:25:07

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

12

Author

rmilson (146)

Entry type

Definition

Classification

msc 17B20

Defines

rank

Generated on Fri Feb 9 18:49:22 2018 by

LaTeXML
root lattice

root lattice

If

R

⊂

E

is a

root system

, and

E

a

Euclidean vector space

, then the

root lattice

Λ

R

of

R

is the subset of

E

generated by

R

as an

abelian group

. In fact, this
group is free on the

simple roots

, and is thus a full sublattice of

E

.

Title

root lattice

Canonical

name

RootLattice

Date of creation

2013-03-22 13:11:54

Last modified on

2013-03-22 13:11:54

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 18:49:24 2018 by

LaTeXML
root system

root system

A

root system

is a key notion in the classification and the

representation theory

of reflection groups and of

semi-simple Lie
algebras

. Let

E

be a

Euclidean vector space

with

inner product

(

⋅

,

⋅

)

. A root system is a finite

spanning set

R

⊂

E

such that for every

u

∈

R

, the

orthogonal

reflection

v

↦

v

-

2

⁢

(

u

,

v

)

(

u

,

u

)

⁢

u

,

v

∈

E

preserves

R

.

A root system is called

crystallographic

if

2

⁢

(

u

,

v

)

(

u

,

u

)

is an integer for all

u

,

v

∈

R

.

A root system is called

reduced

if for all

u

∈

R

, we have

k

⁢

u

∈

R

for

k

=

±

1

only.

We call a root system

indecomposable

if there is no proper

decomposition

R

=

R

′

∪

R

′′

such that every vector in

R

′

is orthogonal to
every vector in

R

′′

.

Title

root system

Canonical

name

RootSystem

Date of creation

2013-03-22 13:11:30

Last modified on

2013-03-22 13:11:30

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

13

Author

rmilson (146)

Entry type

Definition

Classification

msc 17B20

Related topic

SimpleAndSemiSimpleLieAlgebras2

Related topic

LieAlgebra

Defines

reduced root system

Defines

root

Defines

root space

Defines

root decomposition

Defines

indecomposable

Defines

reduced

Defines

crystallographic

Generated on Fri Feb 9 18:49:25 2018 by

LaTeXML
root system underlying a semi-simple Lie algebra

root system underlying a semi-simple Lie algebra

Crystallographic

,

reduced root systems

are in

one-to-one
correspondence

with

semi-simple

, complex Lie algebras. First, let us
describe how one passes from a

Lie algebra

to a

root system

. Let

𝔤

be a semi-simple, complex Lie algebra and let

𝔥

be a

Cartan
subalgebra

. Since

𝔤

is semi-simple,

𝔥

is abelian. Moreover,

𝔥

acts on

𝔤

(via the

adjoint representation

) by commuting,
simultaneously

diagonalizable

linear maps

. The simultaneous

eigenspaces

of this

𝔥

action are called

root spaces

, and the

decomposition

of

𝔤

into

𝔥

and the root spaces is called a

root decomposition

of

𝔤

. To be more precise, for

λ

∈

𝔥

*

, set

𝔤

λ

=

{

a

∈

𝔤

:

[

h

,

a

]

=

λ

⁢

(

h

)

⁢

a

⁢

for all

⁢

h

∈

𝔥

}

.

We call a non-zero

λ

∈

𝔥

*

a root if

𝔤

λ

is non-trivial, in which
case

𝔤

λ

is called a root space. It is possible to show
that that

𝔤

0

is just the Cartan subalgebra

𝔥

, and that

dim

⁡

𝔤

λ

=

1

for each root

λ

. Letting

R

⊂

𝔥

*

denote the set of all roots, we have

𝔤

=

𝔥

⊕

⊕

λ

∈

R

𝔤

λ

.

The Cartan subalgebra

𝔥

has a natural

inner product

, called the

Killing form

, which in turn induces an inner product on

𝔥

*

. It is
possible to show that, with respect to this inner product,

R

is a

reduced

, crystallographic root system.

Conversely

, let

R

⊂

E

be a reduced, crystallographic root
system. Let

Δ

be a base of

positive roots

. We define a Lie
algebra by taking

generators

H

λ

,

X

λ

,

Y

λ

,

λ

∈

Δ

,

subject to the following

relations

:

[

H

λ

,

H

μ

]

=

0

,

=

(

λ

,

μ

)

⁢

X

λ

,

=

-

(

λ

,

μ

)

⁢

Y

λ

,

=

H

λ

,

=

0

,

λ

≠

μ

;

(

ad

⁡

X

λ

)

-

(

λ

,

μ

)

+

1

⁢

(

X

μ

)

=

0

,

λ

≠

μ

,

(

ad

⁡

Y

λ

)

-

(

λ

,

μ

)

+

1

⁢

(

Y

μ

)

=

0

,

λ

≠

μ

,

The above are known as the

Chevalley-Serre relations

The resulting Lie
algebra turns out to be semi-simple, with a root system

isomorphic

to
the given

R

.

Thanks to the above

isomorphism

, to the difficult task of classifying
complex semi-simple Lie algebras is transformed into the somewhat
easier task of classifying crystallographic, reduced roots systems.
Furthermore, a complex Lie algebra is simple if and only if the
corresponding root system is

indecomposable

. Thus, we only need to
classify indecomposable root systems, since all other root systems and
semi-simple Lie algebras are built out of these.

Title

root system underlying a semi-simple Lie algebra

Canonical name

RootSystemUnderlyingASemisimpleLieAlgebra

Date of creation

2013-03-22 15:28:59

Last modified on

2013-03-22 15:28:59

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Result

Classification

msc 17B20

Related topic

SimpleAndSemiSimpleLieAlgebras2

Defines

Serre relations

Defines

Chevalley-Serre relations

Generated on Fri Feb 9 18:49:28 2018 by

LaTeXML
simple and semi-simple Lie algebras

simple and semi-simple Lie algebras

A

Lie algebra

is called

simple

if it has no

proper ideals

and is not

abelian

. A Lie algebra
is called

semi-simple

if it has no proper

solvable

ideals and is not abelian.

Let

k

=

ℝ

or

ℂ

. Examples of

simple algebras

are

𝔰

⁢

𝔩

n

⁢

k

, the Lie algebra
of the

special linear group

(traceless matrices),

𝔰

⁢

𝔬

n

⁢

k

, the Lie algebra of the

special
orthogonal group

(

skew-symmetric matrices

), and

𝔰

⁢

𝔭

2

⁢

n

⁢

k

the Lie algebra of the

symplectic group

. Over

ℝ

, there are other simple Lie algebas, such as

𝔰

⁢

𝔲

n

, the Lie algebra of the

special unitary group

(

skew-Hermitian matrices

). Any
semi-simple Lie algebra is a

direct product

of

simple Lie algebras

.

Simple and semi-simple Lie algebras are one of the most widely studied

classes of algebras

for a number of reasons. First of all, many of the most interesting Lie groups have semi-simple
Lie algebras. Secondly, their

representation theory

is very well understood. Finally, there is
a beautiful

classification

of simple Lie algebras.

Over

ℂ

, there are 3 infinite series of simple Lie algebras:

𝔰

⁢

𝔩

n

,

𝔰

⁢

𝔬

n

and

𝔰

⁢

𝔭

2

⁢

n

, and 5 exceptional simple Lie algebras

𝔤

2

,

𝔣

4

,

𝔢

6

,

𝔢

7

, and

𝔢

8

.
Over

ℝ

the picture is more complicated, as several different Lie algebras can have the same

complexification

(for example,

𝔰

⁢

𝔲

n

and

𝔰

⁢

𝔩

n

⁢

ℝ

both have complexification

𝔰

⁢

𝔩

n

⁢

ℂ

).

Title

simple and semi-simple Lie algebras

Canonical name

SimpleAndSemisimpleLieAlgebras

Date of creation

2013-03-22 13:11:28

Last modified on

2013-03-22 13:11:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 17B20

Related topic

LieAlgebra

Related topic

LieGroup

Related topic

RootSystem

Related topic

RootSystemUnderlyingASemiSimpleLieAlgebra

Defines

simple Lie algebra

Defines

semi-simple Lie algebra

Defines

semisimple Lie algebra

Defines

simple

Defines

semi-simple

Defines

semisimple

Generated on Fri Feb 9 18:49:30 2018 by

LaTeXML
simple root

simple root

\PMlinkescapephrase

vector space

Let

R

⊆

E

be a

root system

, with

E

a

Euclidean vector space

(

http://planetmath.org/VectorSpace

). If

R

+

is a set of

positive roots

, then a root is called

simple

if it is positive, and not the sum of any two
positive roots. The

simple roots

form a basis of the vector space

E

, and any positive root
is a positive integer

linear combination

of simple roots.

A set of roots which is simple with respect to some choice of a set of positive roots is called a

base

. The

Weyl group

of the root system acts simply transitively on the set of bases.

Title

simple root

Canonical name

SimpleRoot

Date of creation

2013-03-22 13:11:49

Last modified on

2013-03-22 13:11:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 17B20

Defines

base

Generated on Fri Feb 9 18:49:31 2018 by

LaTeXML
Verma module

Verma module

Let

𝔤

be a

semi-simple Lie algebra

,

𝔥

a

Cartan subalgebra

, and

𝔟

a

Borel subalgebra

. We work over a field

F

. Given a weight

λ

∈

𝔥

*

, let

F

λ

be the 1-d dimensional

𝔟

-module on which

𝔥

acts by

multiplication

by

λ

, and the

positive root

spaces act trivially. Now, the

Verma module

M

λ

of the weight

λ

is the

𝔤

-module induced from

F

λ

, i.e.

M

λ

=

𝒰

⁢

(

𝔤

)

⊗

𝒰

⁢

(

𝔟

)

F

λ

.

Using the PoincarÃÂ©-Birkhoff-Witt theorem we see that as a

vector space

M

λ

is

isomorphic

to

𝒰

⁢

(

𝔫

¯

)

, where

𝔫

¯

is the sum of the

negative

weight spaces

(so

𝔤

=

𝔟

⊕

𝔫

¯

)

. In particular

M

λ

is

infinite dimensional

.

We say a

𝔤

-module

V

is a highest weight module if it has a weight

μ

∈

𝔥

*

and a non-zero vector

v

∈

V

μ

with

X

⁢

v

=

0

for any

X

in a positive root space and such that

V

is generated as a

𝔤

-module by

v

. The Verma module

M

λ

is a highest weight module and we fix a

generator

1

⊗

1

.

The most important property of Verma modules is that they are

universal

amongst highest weight modules, in the following sense. If

V

is a highest weight module generated by

v

which has weight

λ

then there is a unique

surjective

homomorphism

M

λ

→

V

which sends

1

⊗

1

to

v

. That is, all highest weight modules with

highest weight

λ

are

quotients

of

M

λ

. Also,

M

λ

has a unique

maximal

submodule

, so there is a unique

irreducible representation

with highest weight

λ

. If

λ

is dominant and

integral

then this module is finite dimensional.

Title

Verma module

Canonical name

VermaModule

Date of creation

2013-03-22 13:12:13

Last modified on

2013-03-22 13:12:13

Owner

owenjonesuk (12024)

Last modified by

owenjonesuk (12024)

Numerical id

8

Author

owenjonesuk (12024)

Entry type

Definition

Classification

msc 17B20

Defines

highest weight module

Generated on Fri Feb 9 18:49:33 2018 by

LaTeXML
weight lattice

weight lattice

The

weight lattice

Λ

W

of a

root system

R

⊂

E

is the lattice

Λ

W

=

{

e

∈

E

|

(

e

,

α

)

(

α

,

α

)

∈

ℤ

⁢

for all

⁢

r

∈

R

}

.

Weights which lie in the weight lattice are called . If

R

⊂

𝔥

is the root system of a

semi-simple Lie algebra

𝔤

with

Cartan subalgebra

𝔥

, then

Λ

W

is exactly the set of weights appearing in

finite dimensional

representations

of

𝔤

.

Title

weight lattice

Canonical name

WeightLattice

Date of creation

2013-03-22 13:11:57

Last modified on

2013-03-22 13:11:57

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

9

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B20

Defines

integral weight

Generated on Fri Feb 9 18:49:35 2018 by

LaTeXML
weight (Lie algebras)

weight (Lie algebras)

Let

𝔥

be an

abelian Lie algebra

,

V

a

vector space

and

ρ

:

𝔥

→

End

⁡

V

a

representation

. Then the representation
is said to be

diagonalisable

, if

V

can be written as a

direct
sum

V

=

⊕

λ

∈

𝔥

*

V

λ

where

𝔥

*

is the

dual space

of

𝔥

and

V

λ

=

{

v

∈

V

∣

ρ

⁢

(

h

)

⁢

v

=

λ

⁢

(

h

)

⁢

v

⁢

for all

⁢

h

∈

𝔥

}

.

Now let

𝔤

be a

semi-simple Lie algebra

. Fix a

Cartan subalgebra

𝔥

, then

𝔥

is abelian. Let

ρ

:

𝔤

→

End

⁡

V

be a representation whose restriction to

𝔥

is
diagonalisable. Then for any

λ

∈

𝔥

*

, the space

V

λ

is the

weight space

of

λ

with respect to

ρ

. The

multiplicity

of

λ

with respect to

ρ

is the

dimension

of

V

λ

:

mult

ρ

⁡

(

λ

)

:=

dim

⁡

V

λ

.

If the multiplicity of

λ

is

greater than zero

, then

λ

is called a

weight

of the representation

ρ

.

A representation of a semi-simple Lie algebra is determined by the
multiplicities of its weights.

Title

weight (

Lie algebras

)

Canonical

name

WeightLieAlgebras

Date of creation

2013-03-22 13:11:42

Last modified on

2013-03-22 13:11:42

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

7

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 17B20

Synonym

weight

Defines

diagonalisable

Defines

diagonalizable

Defines

multiplicity

Defines

weight space

Generated on Fri Feb 9 18:49:38 2018 by

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Weyl chamber

Weyl chamber

Let

E

be a

Euclidean vector space

,

R

⊂

E

a

root system

, and

R

+

⊂

R

a choice of

positive roots

. We define the

positive Weyl chamber

(relative to

R

+

) to be the closed set

𝒞

=

{

u

∈

E

∣

(

u

,

α

)

≥

0

⁢

for all

⁢

α

∈

R

+

}

.

A weight which lies inside the positive Weyl chamber is
called

dominant

.

The interior of

𝒞

is a

fundamental domain

for the action
of the

Weyl group

on

E

. The image

w

⁢

(

𝒞

)

of

𝒞

under the any element

w

of the Weyl group is called a

Weyl
chamber

. The Weyl group

W

acts simply transitively on the set of
Weyl chambers.

Title

Weyl chamber

Canonical name

WeylChamber

Date of creation

2013-03-22 13:12:00

Last modified on

2013-03-22 13:12:00

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

8

Author

rmilson (146)

Entry type

Definition

Classification

msc 17B20

Defines

positive Weyl chamber

Defines

dominant weight

Generated on Fri Feb 9 18:49:39 2018 by

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Weyl group

Weyl group

The

Weyl group

W

R

of a

root system

R

⊂

E

, where

E

is a

Euclidean vector space

,
is the

subgroup

of

GL

⁢

(

E

)

generated by

reflection

in the

hyperplanes

perpendicular

to the roots. The map of reflection in a root

α

is given by

r

α

⁢

(

v

)

=

v

-

2

⁢

(

α

,

v

)

(

α

,

α

)

⁢

α

.

The Weyl group is generated by reflections in the

simple roots

for any choice of a set of

positive roots

.
There is a well-defined

length function

ℓ

:

W

R

→

ℤ

, where

ℓ

⁢

(

w

)

is the minimal
number of reflections in simple roots that

w

can be written as. This is also the number of positive
roots that

w

takes to negative roots.

Title

Weyl group

Canonical name

WeylGroup

Date of creation

2013-03-22 13:11:52

Last modified on

2013-03-22 13:11:52

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 18:49:41 2018 by

LaTeXML
Weyl’s theorem

Weyl’s theorem

Let

𝔤

be a

finite dimensional

semi-simple Lie algebra

. Then any finite dimensional

representation

of

𝔤

is

completely reducible

.

Title

Weyl’s theorem

Canonical name

WeylsTheorem

Date of creation

2013-03-22 13:21:24

Last modified on

2013-03-22 13:21:24

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 18:49:43 2018 by

LaTeXML
Lie’s theorem

Lie’s theorem

Let

𝔤

be a

finite dimensional

complex

solvable Lie algebra

, and

V

a repesentation of

𝔤

. Then there exists an element of

V

which is a simultaneous

eigenvector

for all elements of

𝔤

.

Applying this result inductively, we find that there is a basis of

V

with respect to which all elements of

𝔤

are

upper triangular

.

Title

Lie’s theorem

Canonical name

LiesTheorem

Date of creation

2013-03-22 13:20:40

Last modified on

2013-03-22 13:20:40

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Theorem

Classification

msc 17B30

Generated on Fri Feb 9 18:49:48 2018 by

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solvable Lie algebra

solvable Lie algebra

Let

𝔤

be a

Lie algebra

. The

lower central series

of

𝔤

is the

filtration

of

subalgebras

𝒟

1

⁢

𝔤

⊃

𝒟

2

⁢

𝔤

⊃

𝒟

3

⁢

𝔤

⊃

⋯

⊃

𝒟

k

⁢

𝔤

⊃

⋯

of

𝔤

, inductively defined for every

natural number

k

as follows:

𝒟

1

⁢

𝔤

:=

[

𝔤

,

𝔤

]

𝒟

k

⁢

𝔤

:=

[

𝔤

,

𝒟

k

-

1

⁢

𝔤

]

The

upper central series

of

𝔤

is the filtration

𝒟

1

⁢

𝔤

⊃

𝒟

2

⁢

𝔤

⊃

𝒟

3

⁢

𝔤

⊃

⋯

⊃

𝒟

k

⁢

𝔤

⊃

⋯

defined inductively by

𝒟

1

⁢

𝔤

:=

[

𝔤

,

𝔤

]

𝒟

k

⁢

𝔤

:=

[

𝒟

k

-

1

⁢

𝔤

,

𝒟

k

-

1

⁢

𝔤

]

In fact both

𝒟

k

⁢

𝔤

and

𝒟

k

⁢

𝔤

are ideals of

𝔤

, and

𝒟

k

⁢

𝔤

⊂

𝒟

k

⁢

𝔤

for all

k

. The Lie algebra

𝔤

is defined to be

nilpotent

if

𝒟

k

⁢

𝔤

=

0

for some

k

∈

ℕ

, and

solvable

if

𝒟

k

⁢

𝔤

=

0

for some

k

∈

ℕ

.

A subalgebra

𝔥

of

𝔤

is said to be

nilpotent

or

solvable

if

𝔥

is nilpotent or solvable when considered as a Lie algebra in its own right. The terms may also be applied to ideals of

𝔤

, since every ideal of

𝔤

is also a subalgebra.

Title

solvable Lie algebra

Canonical name

SolvableLieAlgebra

Date of creation

2013-03-22 12:41:06

Last modified on

2013-03-22 12:41:06

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 17B30

Defines

nilpotent Lie algebra

Defines

solvable

Defines

nilpotent

Defines

lower central series

Defines

upper central series

Generated on Fri Feb 9 18:49:50 2018 by

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Lie element

Lie element

Given a

Lie algebra

𝔤

and its univeral enveloping algebra

𝔘

⁢

(

𝔤

)

, then an element

x

∈

𝔘

⁢

(

𝔤

)

is called
a

Lie element

if

x

is in the image of

𝔤

in

𝔘

⁢

(

𝔤

)

.

This term is most often used for

free Lie algebras

and specifically in
the famous

Baker-Campbell-Hausdorff formula

(theorem).

Title

Lie element

Canonical name

LieElement

Date of creation

2013-03-22 16:51:14

Last modified on

2013-03-22 16:51:14

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

4

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 17B35

Classification

msc 16S30

Defines

Lie element

Generated on Fri Feb 9 18:49:52 2018 by

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