Lie algebra

Lie algebra

A

Lie algebra

over a field

k

is a

vector space

𝔤

with a

bilinear map

[

,

]

:

𝔤

×

𝔤

→

𝔤

, called the

Lie bracket

and denoted

(

x

,

y

)

↦

[

x

,

y

]

. It is required to

satisfy

:

1.

[

x

,

x

]

=

0

for all

x

∈

𝔤

.

2.

The

Jacobi identity

:

[

x

,

[

y

,

z

]

]

+

[

y

,

[

z

,

x

]

]

+

[

z

,

[

x

,

y

]

]

=

0

for all

x

,

y

,

z

∈

𝔤

.

1

Subalgebras & Ideals

A

vector subspace

𝔥

of the Lie algebra

𝔤

is a

subalgebra

if

𝔥

is

closed under

the Lie bracket

operation

, or, equivalently, if

𝔥

itself is a Lie algebra under the same bracket operation as

𝔤

. An

ideal

of

𝔤

is a

subspace

𝔥

for which

[

x

,

y

]

∈

𝔥

whenever either

x

∈

𝔥

or

y

∈

𝔥

. Note that every ideal is also a subalgebra.

Some general examples of subalgebras:

•

The

center

of

𝔤

, defined by

Z

⁢

(

𝔤

)

:=

{

x

∈

𝔤

∣

[

x

,

y

]

=

0

⁢

for all

⁢

y

∈

𝔤

}

. It is an ideal of

𝔤

.

•

The

normalizer

of a subalgebra

𝔥

is the set

N

⁢

(

𝔥

)

:=

{

x

∈

𝔤

∣

[

x

,

𝔥

]

⊂

𝔥

}

. The Jacobi identity guarantees that

N

⁢

(

𝔥

)

is always a subalgebra of

𝔤

.

•

The

centralizer

of a subset

X

⊂

𝔤

is the set

C

⁢

(

X

)

:=

{

x

∈

𝔤

∣

[

x

,

X

]

=

0

}

. Again, the Jacobi identity implies that

C

⁢

(

X

)

is a subalgebra of

𝔤

.

2

Homomorphisms

Given two Lie algebras

𝔤

and

𝔤

′

over the field

k

, a

homomorphism

from

𝔤

to

𝔤

′

is a

linear transformation

ϕ

:

𝔤

→

𝔤

′

such that

ϕ

⁢

(

[

x

,

y

]

)

=

[

ϕ

⁢

(

x

)

,

ϕ

⁢

(

y

)

]

for all

x

,

y

∈

𝔤

. An

injective

homomorphism is called a

monomorphism

, and a

surjective

homomorphism is called an

epimorphism

.

The

kernel

of a homomorphism

ϕ

:

𝔤

→

𝔤

′

(considered as a linear transformation) is denoted

ker

⁡

(

ϕ

)

. It is always an ideal in

𝔤

.

3

Examples

•

Any vector space can be made into a Lie algebra simply by setting

[

x

,

y

]

=

0

for all vectors

x

,

y

. The resulting Lie algebra is called an

abelian

Lie algebra.

•

If

G

is a Lie group, then the tangent space at the

identity

forms a Lie algebra over the real numbers.

•

ℝ

3

with the

cross product

operation is a

nonabelian

three dimensional Lie algebra over

ℝ

.

4

Historical Note

Lie algebras are so-named in honour of Sophus Lie, a Norwegian
mathematician who pioneered the study of these mathematical objects.
Lie’s discovery was tied to his investigation of continuous
transformation groups and

symmetries

. One joint project with Felix
Klein called for the classification of all

finite-dimensional

groups
acting on the plane. The task seemed hopeless owing to the generally
non-linear nature of such

group actions

. However, Lie was able to
solve the problem by remarking that a

transformation group

can be
locally reconstructed from its corresponding “infinitesimal
generators”, that is to say

vector fields

corresponding to various
1-parameter subgroups. In terms of this geometric correspondence, the
group composition operation manifests itself as the bracket of vector
fields, and this is very much a linear operation. Thus the task of
classifying group actions in the plane became the task of classifying
all finite-dimensional Lie algebras of planar vector field; a project
that Lie brought to a successful

conclusion

.

This “

linearization

trick” proved to be incredibly fruitful and led
to great advances in geometry and differential equations. Such
advances are based, however, on various results from the theory of Lie
algebras. Lie was the first to make significant contributions to this
purely algebraic theory, but he was surely not the last.

Title

Lie algebra

Canonical name

LieAlgebra

Date of creation

2013-03-22 12:03:36

Last modified on

2013-03-22 12:03:36

Owner

djao (24)

Last modified by

djao (24)

Numerical id

18

Author

djao (24)

Entry type

Definition

Classification

msc 17B99

Related topic

CommutatorBracket

Related topic

LieGroup

Related topic

UniversalEnvelopingAlgebra

Related topic

RootSystem

Related topic

SimpleAndSemiSimpleLieAlgebras2

Defines

Jacobi identity

Defines

subalgebra

Defines

ideal

Defines

normalizer

Defines

centralizer

Defines

kernel

Defines

homomorphism

Defines

center

Defines

centre

Defines

abelian Lie algebra

Defines

abelian

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