aarjaneiro/planetmath huggingface dataset:
commutator bracket

commutator bracket

Let

A

be an

associative algebra

over a field

K

. For

a

,

b

∈

A

,
the element of

A

defined by

[

a

,

b

]

=

a

⁢

b

-

b

⁢

a

is called the

commutator

of

a

and

b

.
The corresponding

bilinear operation

[

-

,

-

]

:

A

×

A

→

A

is called the commutator bracket.

The commutator bracket is

bilinear

, skew-symmetric, and also

satisfies

the

Jacobi identity

. To wit, for

a

,

b

,

c

∈

A

we have

[

a

,

[

b

,

c

]

]

+

[

b

,

[

c

,

a

]

]

+

[

c

,

[

a

,

b

]

]

=

0

.

The proof of this assertion is straightforward. Each of the brackets in
the left-hand side expands to 4 terms, and then everything cancels.

In

categorical

terms, what we have here is a

functor

from the

category

of associative algebras to the category of

Lie algebras

over a fixed
field. The action of this functor is to turn an associative algebra

A

into a Lie algebra that has the same underlying

vector space

as

A

, but whose

multiplication

operation

is given by the commutator
bracket. It must be noted that this functor is right-adjoint to the

universal enveloping algebra

functor.

Examples

•

Let

V

be a vector space.

Composition

endows the vector space of

endomorphisms

End

⁢

V

with the

structure

of an associative algebra.
However, we could also regard

End

⁢

V

as a Lie algebra relative to
the commutator bracket:

[

X

,

Y

]

=

X

⁢

Y

-

Y

⁢

X

,

X

,

Y

∈

End

⁢

V

.

•

The

algebra

of differential operators has some interesting
properties when viewed as a Lie algebra. The fact is that even
though the composition of differential operators is a

non-commutative operation

, it is

commutative

when restricted to the
highest order terms of the involved

operators

. Thus, if

X

,

Y

are
differential operators of order

p

and

q

, respectively, the
compositions

X

⁢

Y

and

Y

⁢

X

have order

p

+

q

. Their highest order
term coincides, and hence the commutator

[

X

,

Y

]

has order

p

+

q

-

1

.

•

In light of the preceding comments, it is evident that the
vector space of first-order differential operators is closed with
respect to the commutator bracket. Specializing even further we
remark that, a vector field is just a

homogeneous

first-order
differential operator, and that the commutator bracket for vector
fields, when viewed as first-order operators, coincides with the
usual, geometrically motivated vector field bracket.

Title

commutator bracket

Canonical name

CommutatorBracket

Date of creation

2013-03-22 12:33:51

Last modified on

2013-03-22 12:33:51

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

8

Author

rmilson (146)

Entry type

Definition

Classification

msc 17A01

Classification

msc 17B05

Classification

msc 18A40

Related topic

LieAlgebra

Defines

commutator Lie algebra

Defines

commutator

Generated on Fri Feb 9 18:47:37 2018 by

LaTeXML
non-associative algebra

non-associative algebra

A

non-associative algebra

is an

algebra

in which the assumption of

multiplicative

associativity is dropped. From this definition, a non-associative algebra does not that the associativity fails. Rather, it enlarges the class of

associative algebras

, so that any associative algebra is a non-associative algebra.

In much of the literature concerning non-associative algebras, where the meaning of a “non-associative algebra” is clear, the word “non-associative” is dropped for simplicity and clarity.

Lie algebras

and

Jordan algebras

are two famous examples of non-associative algebras that are not associative.

If we substitute the word “algebra” with “ring” in the above paragraphs, then we arrive at the definition of a

non-associative ring

. Alternatively, a non-associative ring is just a non-associative algebra over the

integers

.

References

1

Richard D. Schafer,

An Introduction to

Nonassociative Algebras

, Dover Publications, (1995).

Title

non-associative algebra

Canonical name

NonassociativeAlgebra

Date of creation

2013-03-22 15:06:44

Last modified on

2013-03-22 15:06:44

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17A01

Related topic

Semifield

Related topic

Algebras

Defines

non-associative ring

Generated on Fri Feb 9 18:47:41 2018 by

LaTeXML
nucleus

nucleus

Let

A

be an

algebra

, not necessarily associative multiplicatively. The

nucleus

of

A

is:

𝒩

⁢

(

A

)

:=

{

a

∈

A

∣

[

a

,

A

,

A

]

=

[

A

,

a

,

A

]

=

[

A

,

A

,

a

]

=

0

}

,

where

[

,

,

]

is the

associator

bracket. In other words, the nucleus is the set of elements that multiplicatively

associate

with all elements of

A

. An element

a

∈

A

is

nuclear

if

a

∈

𝒩

⁢

(

A

)

.

𝒩

⁢

(

A

)

is a

Jordan subalgebra

of

A

. To see this, let

a

,

b

∈

𝒩

⁢

(

A

)

. Then for any

c

,

d

∈

A

,

[

a

⁢

b

,

c

,

d

]

=

(

(

a

⁢

b

)

⁢

c

)

⁢

d

-

(

a

⁢

b

)

⁢

(

c

⁢

d

)

=

(

a

⁢

(

b

⁢

c

)

)

⁢

d

-

(

a

⁢

b

)

⁢

(

c

⁢

d

)

(1)

=

a

⁢

(

(

b

⁢

c

)

⁢

d

)

-

(

a

⁢

b

)

⁢

(

c

⁢

d

)

=

a

⁢

(

b

⁢

(

c

⁢

d

)

)

-

(

a

⁢

b

)

⁢

(

c

⁢

d

)

(2)

=

a

⁢

(

b

⁢

(

c

⁢

d

)

)

-

a

⁢

(

b

⁢

(

c

⁢

d

)

)

=

0

(3)

Similarly,

[

c

,

a

⁢

b

,

d

]

=

[

c

,

d

,

a

⁢

b

]

=

0

and so

a

⁢

b

∈

𝒩

⁢

(

A

)

.

Accompanying the concept of a nucleus is that of the

center of a

nonassociative algebra

A

(which is slightly different from the

definition

of the center of an

associative algebra

):

𝒵

⁢

(

A

)

:=

{

a

∈

𝒩

⁢

(

A

)

∣

[

a

,

A

]

=

0

}

,

where

[

,

]

is the

commutator bracket

.

Hence elements in

𝒵

⁢

(

A

)

commute

as well as

associate with all elements of

A

. Like the nucleus, the center of

A

is also a Jordan subalgebra of

A

.

Title

nucleus

Canonical

name

Nucleus

Date of creation

2013-03-22 14:52:19

Last modified on

2013-03-22 14:52:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17A01

Defines

center of a nonassociative algebra

Defines

nuclear

Generated on Fri Feb 9 18:47:44 2018 by

LaTeXML
power-associative algebra

power-associative algebra

Let

A

be a

non-associative algebra

. A

subalgebra

B

of

A

is said to be

cyclic

if it is generated by one

element

.

A non-associative algebra is

power-associative

if,

[

B

,

B

,

B

]

=

0

for any cyclic subalgebra

B

of

A

, where

[

-

,

-

,

-

]

is the

associator

.

If we inductively define the powers of an element

a

∈

A

by

1.

(when

A

is unital with

1

≠

0

)

a

0

:=

1

,

2.

a

1

:=

a

, and

3.

a

n

:=

a

⁢

(

a

n

-

1

)

for

n

>

1

,

then power-associativity of

A

means that

[

a

i

,

a

j

,

a

k

]

=

0

for any non-negative integers

i

,

j

and

k

, since the associator is trilinear (linear in each of the three coordinates). This implies that

a

m

⁢

a

n

=

a

m

+

n

. In

addition

,

(

a

m

)

n

=

a

m

⁢

n

.

A

theorem

, due to A. Albert, states that any finite power-associative

division algebra

over the integers of

characteristic

not equal to 2, 3, or 5 is a field. This is a

generalization

of the Wedderburn’s Theorem on finite

division rings

.

References

1

R. D. Schafer,

An

Introduction

on

Nonassociative Algebras

, Dover, New York (1995).

Title

power-associative algebra

Canonical name

PowerassociativeAlgebra

Date of creation

2013-03-22 14:43:27

Last modified on

2013-03-22 14:43:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17A05

Synonym

di-associative

Synonym

diassociative

Related topic

Associator

Generated on Fri Feb 9 18:47:49 2018 by

LaTeXML
flexible algebra

flexible algebra

A

non-associative algebra

A

is

flexible

if

[

a

,

b

,

a

]

=

0

for all

a

,

b

∈

A

, where

[

,

,

]

is the

associator

on

A

. In other words, we have

(

a

⁢

b

)

⁢

a

=

a

⁢

(

b

⁢

a

)

for all

a

,

b

∈

A

. Any

associative algebra

is clearly flexible. Furthermore, any

alternative algebra

with

characteristic

≠

2

is flexible.

Given an

element

a

in a flexible algebra

A

, define the

left power

of

a

iteratively as follows:

1.

L

1

⁢

(

a

)

=

a

,

2.

L

n

⁢

(

a

)

=

a

⋅

L

n

-

1

⁢

(

a

)

.

Similarly, we can define the

right power

of

a

as:

1.

R

1

⁢

(

a

)

=

a

,

2.

R

n

⁢

(

a

)

=

R

n

-

1

⁢

(

a

)

⋅

a

.

Then, we can show that

L

n

⁢

(

a

)

=

R

n

⁢

(

a

)

for all

positive integers

n

. As a result, in a flexible algebra, one can define the (

multiplicative

) power of an element

a

as

a

n

unambiguously.

Title

flexible algebra

Canonical name

FlexibleAlgebra

Date of creation

2013-03-22 14:43:30

Last modified on

2013-03-22 14:43:30

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17A20

Related topic

Associator

Related topic

AlternativeAlgebra

Defines

left power

Defines

right power

Defines

flexible

Generated on Fri Feb 9 18:47:51 2018 by

LaTeXML
derivation

derivation

Let

R

be a

commutative ring

. A

derivation

d

on an

R

-

algebra

A

into an

A

-module

M

is an

R

-

linear transformation

d

:

A

→

M

satisfying the

properties

•

d

⁢

(

a

⁢

𝐱

+

b

⁢

𝐲

)

=

a

⁢

d

⁢

𝐱

+

b

⁢

d

⁢

𝐲

•

d

⁢

(

𝐱

⋅

𝐲

)

=

𝐱

⋅

d

⁢

𝐲

+

d

⁢

𝐱

⋅

𝐲

for all

a

,

b

∈

R

and

𝐱

,

𝐲

∈

A

.

Title

derivation

Canonical

name

Derivation

Date of creation

2013-03-22 12:02:41

Last modified on

2013-03-22 12:02:41

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 17A36

Classification

msc 16W25

Classification

msc 13N15

Generated on Fri Feb 9 18:47:53 2018 by

LaTeXML
quadratic algebra

quadratic algebra

A

non-associative algebra

A

(with unity

1

A

) over a commutative
ring

R

(with unity

1

R

) is called a

quadratic algebra

if

A

admits a

quadratic form

Q

:

A

→

R

such that

1.

Q

⁢

(

1

A

)

=

1

R

,

2.

the quadratic equation

x

2

-

b

⁢

(

1

A

,

x

)

⁢

x

+

Q

⁢

(

x

)

⁢

1

A

=

0

is satisfied
by all

x

∈

A

, where

b

is the associated

symmetric bilinear form

given by

b

⁢

(

x

,

y

)

:=

Q

⁢

(

x

+

y

)

-

Q

⁢

(

x

)

-

Q

⁢

(

y

)

.

Title

quadratic algebra

Canonical name

QuadraticAlgebra

Date of creation

2013-03-22 15:11:38

Last modified on

2013-03-22 15:11:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17A45

Related topic

QuadraticLieAlgebra

Generated on Fri Feb 9 18:47:58 2018 by

LaTeXML
composition algebra over algebaically closed fields

composition algebra over algebaically closed fields

Theorem 1

.

There are 4 non-isomorphic

composition algebras

over an algebraically closed field

k

:
one

division algebra

, the field itself, and the three split

algebras

.

1.

k

.

2.

The

exchange

algebra:

k

⊕

k

.

3.

2

×

2

matrices over

k

:

M

2

⁢

(

k

)

.

4.

The cross-product of

2

×

2

-matrices over

k

:

M

2

⁢

(

k

)

∘

M

2

⁢

(

k

)

.

Proof.

To see this recall that every composition algebra comes equipped with a

quadratic form

.
Any 2-dimensional anisotropic

subspace

arises from a

quadratic field

extension

. As
our field is

algebraically closed

the quadratic form has no anisotropic subspaces and
is therefore the unique quadratic form of maximal Witt index. Following Hurwitz’s theorem
we know the composition algebras come in

dimensions

1,2,4, and 8 and arise by the
Cayley-Dickson method. Thus we have the field itself and the three split
composition algebras.
∎

Title

composition algebra over algebaically closed fields

Canonical name

CompositionAlgebraOverAlgebaicallyClosedFields

Date of creation

2013-03-22 17:18:22

Last modified on

2013-03-22 17:18:22

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 17A75

Related topic

HurwitzsTheorem

Related topic

JacobsonsTheoremOnCompositionAlgebras

Generated on Fri Feb 9 18:48:07 2018 by

LaTeXML
composition algebras over finite fields

composition algebras over finite fields

Theorem 1

.

There are 5 non-isomorphic

composition algebras

over a

finite field

k

of

characteristic

not 2,
2

division algebras

and 3 split

algebras

.

1.

The field

k

.

2.

The unique

quadratic extension

field

K

/

k

.

3.

The

exchange

algebra:

k

⊕

k

.

4.

2

×

2

matrices over

k

:

M

2

⁢

(

k

)

.

5.

The split

Cayley algebra

.

Proof.

Following Hurwitz’s theorem every composition algebra is given by the Cayley-Dickson construction
and has

dimension

1,2, 4 or 8. Now we consider the possible

non-degenerate
quadratic forms

of these dimensions.

Since every anisotropic 2 space corresponds to a

quadratic field

extension

, and our field is finite,
it follows that there is at most one anisotropic 2

subspace

of our

quadratic form

. Therefore
if

dim

⁡

C

>

2

then the quadratic form is isotropic and so the algebra is a split.
Therefore in the Cayley-Dickson construction over a finite field there every

quaternion
algebra

is split, thus

M

2

⁢

(

k

)

. To build the non-associative

division

Cayley algebra of dimension 8 requires
we start the Cayley-Dickson construction with a

division ring

which is not a field, and thus there are no
Cayley division algebras over finite fields.
∎

This result also can be seen as a consequence of Wedderburn’s theorem that every finite division ring
is a field.
Likewise, a theorem of Artin and Zorn asserts that every finite alternative division ring is in fact associative,
thus excluding the Cayley algebras in a fashion

similar

to how Wedderburn’s theorem excludes division
quaternion algebras.

Title

composition algebras over finite fields

Canonical name

CompositionAlgebrasOverFiniteFields

Date of creation

2013-03-22 17:18:26

Last modified on

2013-03-22 17:18:26

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 17A75

Related topic

HurwitzsTheorem

Related topic

JacobsonsTheoremOnCompositionAlgebras

Generated on Fri Feb 9 18:48:08 2018 by

LaTeXML
composition algebras over ℚ

composition algebras over

ℚ

Theorem 1

.

There are infinitely many

composition algebras

over

Q

.

Proof.

Every

quadratic extension

of

ℚ

is a distinct composition algebra. For example,

(

p

ℚ

)

for

p

a

prime number

. This is sufficient to illustrate an

infinite

number of quadratic composition algebras.
∎

The other families of composition algebras also have an infinite number of non-isomorphic

division algebras

though the proofs are more involved. It suffices to show provide
an infinite family of non-isometric

quadratic forms

of the form:

N

p

,

q

⁢

(

a

,

b

,

c

,

d

)

=

a

2

-

b

2

⁢

p

-

c

2

⁢

q

+

d

2

⁢

p

⁢

q

for

rational numbers

p

and

q

. Such questions can involve

complex number

theory as
for instance, if

p

is a prime

congruent

to

1

modulo

4

then

N

-

1

,

-

p

is isometric to

N

-

1

,

-

1

and thus

N

-

1

,

-

p

is isometric to

N

-

1

,

-

q

for
any other prime

q

≡

1

(

mod

4

)

. But if

p

≡

3

(

mod

4

)

then this cannot be said.

Title

composition algebras over

ℚ

Canonical name

CompositionAlgebrasOvermathbbQ

Date of creation

2013-03-22 17:18:29

Last modified on

2013-03-22 17:18:29

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Example

Classification

msc 17A75

Related topic

HurwitzsTheorem

Related topic

JacobsonsTheoremOnCompositionAlgebras

Generated on Fri Feb 9 18:48:10 2018 by

LaTeXML
composition algebras over ℝ

composition algebras over

ℝ

There are 7 non-isomorphic

composition algebras

over

ℝ

, first 4

division algebras

and secondly
3 split

algebras

.

1.

The

real numbers

ℝ

.

2.

The

complex numbers

ℂ

.

3.

The

Hamiltonians

(also known as the

quaternions

)

ℍ

.

4.

The

octonions

(also known as the Cayley or

Cayley-Dickson algebra

)

𝕆

.

5.

The

exchange

algebra:

ℝ

⊕

ℝ

.

6.

2

×

2

matrices over

ℝ

:

M

2

⁢

(

ℝ

)

.

7.

The cross-product of

2

×

2

-matrices over

ℝ

:

M

2

⁢

(

ℝ

)

∘

M

2

⁢

(

ℝ

)

.

The proof can be seen as a consquence of a theorem of Hurwitz and a theorem of Jacobson.
In reality various authors contributed to the

solution

including Albert, Dickson and
Kaplansky.

Title

composition algebras over

ℝ

Canonical name

CompositionAlgebrasOvermathbbR

Date of creation

2013-03-22 17:18:17

Last modified on

2013-03-22 17:18:17

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

7

Author

Algeboy (12884)

Entry type

Example

Classification

msc 17A75

Related topic

HurwitzsTheorem

Related topic

JacobsonsTheoremOnCompositionAlgebras

Generated on Fri Feb 9 18:48:12 2018 by

LaTeXML
Hurwitz’s theorem on composition algebras

Hurwitz’s theorem on composition algebras

Theorem 1

(Hurwitz).

[

1

, Theorem 3.25]

Given a field

k

of characteristic not

2

, then every unital

composition algebra

C

over

k

is

isomorphic

to one of:

1.

k

,

2.

(

α

k

)

for

α

∈

k

,

3.

(

α

,

β

k

)

for

α

,

β

∈

k

,

4.

(

α

,

β

,

γ

k

)

for

α

,

β

,

γ

∈

k

.

In particular, all composition algebras over

k

are

finite dimensional

and of

dimension

1

,

2

,

4

or

8

.

References

1

Richard D. Schafer,

An introduction to

nonassociative algebras

, Pure and
Applied Mathematics, Vol. 22, Academic Press, New York, 1966.

Title

Hurwitz’s theorem on composition algebras

Canonical name

HurwitzsTheoremOnCompositionAlgebras

Date of creation

2013-03-22 17:18:20

Last modified on

2013-03-22 17:18:20

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

4

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 17A75

Related topic

JacobsonsTheoremOnCompositionAlgebras

Generated on Fri Feb 9 18:48:14 2018 by

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Jacobson’s theorem on composition algebras

Jacobson’s theorem on composition algebras

Recall that

composition algebra

C

over a field

k

is specified with a

quadratic form

q

:

C

→

k

.
Furthermore, two quadratic forms

q

:

C

→

k

and

r

:

D

→

k

are isometric if there exists an

invertible linear map

f

:

C

→

D

such that

r

⁢

(

f

⁢

(

x

)

)

=

q

⁢

(

x

)

for all

x

∈

C

.

Theorem 1

(Jacobson).

[

1

, Theorem 3.23]

Two unital

Cayley-Dickson algebras

C

and

D

over a field

k

of

characteristic

not

2

are

isomorphic

if, and only if, their quadratic forms are isometric.

A Cayley-Dickson algebra is split if the

algebra

has non-trivial zero-divisors.

Corollary 2

.

[

1

, Corollary 3.24]

Upto

isomorphism

there is only one split Cayley-Dickson algebra and the quadratic form
has Witt index 4.

Over the real numbers instead of Witt index, we say the

signature

of the quadratic form is

(

4

,

4

)

.

This result is often used together with a theorem of Hurwitz which limits the

dimensions

of composition algebras to dimensions 1,2, 4 or 8. Thus to classify the composition algebras
over a given field

k

of characteristic not 2, it suffices to classify the

non-degenerate
quadratic forms

q

:

k

n

→

k

with

n

=

1

,

2

,

4

or

8

.

References

1

Richard D. Schafer,

An introduction to

nonassociative algebras

, Pure and
Applied Mathematics, Vol. 22, Academic Press, New York, 1966.

Title

Jacobson’s theorem on composition algebras

Canonical name

JacobsonsTheoremOnCompositionAlgebras

Date of creation

2013-03-22 17:18:14

Last modified on

2013-03-22 17:18:14

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

4

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 17A75

Related topic

CompositionAlgebrasOverMathbbR

Related topic

HurwitzsTheoremOnCompositionAlgebras

Related topic

CompositionAlgebraOverAlgebaicallyClosedFields

Related topic

CompositionAlgebrasOverFiniteFields

Related topic

CompositionAlgebrasOverMathbbQ

Generated on Fri Feb 9 18:48:16 2018 by

LaTeXML
index of a Lie algebra

index of a Lie algebra

Let

𝔮

be a

Lie algebra

over

𝕂

and

𝔮

*

its

vector space

dual. For

ξ

∈

𝔮

*

let

𝔮

ξ

denote the

stabilizer

of

ξ

with respect to the co-adjoint

representation

.

The

index

of

𝔮

is defined to be

ind

⁡

𝔮

:=

min

ξ

∈

𝔤

*

⁢

dim

⁡

𝔮

ξ

Examples

1.

If

𝔮

is reductive then

ind

⁡

𝔮

=

rank

⁡

𝔮

. Indeed,

𝔮

and

𝔮

*

are

isomorphic

as
representations for

𝔮

and so the index is the

minimal

dimension among stabilizers of

elements

in

𝔮

. In particular the minimum is realized in the stabilizer of any

regular

element of

𝔮

. These elemtents have stabilizer dimension equal to the rank of

𝔮

.

2.

If

ind

⁡

𝔮

=

0

then

𝔮

is called a

Frobenius Lie algebra

. This is

equivalent

to condition that
the

Kirillov form

K

ξ

:

𝔮

×

𝔮

→

𝕂

given by

(

X

,

Y

)

↦

ξ

⁢

(

[

X

,

Y

]

)

is

non-singular

for some

ξ

∈

𝔮

*

. Another equivalent condition when

𝔮

is the Lie algebra of an algebraic group

Q

is that

𝔮

is Frobenius if and only if

Q

has an open orbit on

𝔮

*

.

Title

index of a Lie algebra

Canonical name

IndexOfALieAlgebra

Date of creation

2013-03-22 15:30:47

Last modified on

2013-03-22 15:30:47

Owner

benjaminfjones (879)

Last modified by

benjaminfjones (879)

Numerical id

6

Author

benjaminfjones (879)

Entry type

Definition

Classification

msc 17B05

Defines

index of a Lie algebra

Defines

Frobenius Lie algebra

Defines

Kirillov form

Generated on Fri Feb 9 18:48:25 2018 by

LaTeXML
Killing form

Killing form

Let

𝔤

be a

finite dimensional

Lie algebra

over a field

k

, and

X

,

Y

∈

𝔤

. Let

ad

X

:

𝔤

→

𝔤

be the

adjoint action

,

ad

X

⁢

Y

=

[

X

,

Y

]

.

Then the

Killing form

on

𝔤

is a bilinear map

B

𝔤

:

𝔤

×

𝔤

→

k

given by

B

𝔤

⁢

(

X

,

Y

)

=

tr

⁢

(

ad

X

∘

ad

Y

)

.

The Killing form is

invariant

(

http://planetmath.org/InvariantFormLieAlgebras

) under the adjoint action and

symmetric

(since trace is
symmetric).

Title

Killing form

Canonical name

KillingForm

Date of creation

2013-03-22 13:15:47

Last modified on

2013-03-22 13:15:47

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

8

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B05

Generated on Fri Feb 9 18:48:27 2018 by

LaTeXML
Levi’s theorem

Levi’s theorem

Let

𝔤

be a complex Lie algebra,

𝔯

its radical. Then the

extension

0

→

𝔯

→

𝔤

→

𝔤

/

𝔯

→

0

is split, i.e., there exists a

subalgebra

𝔥

of

𝔤

mapping

isomorphically to

𝔤

/

𝔯

under the

natural projection

.

Title

Levi’s

theorem

Canonical name

LevisTheorem

Date of creation

2013-03-22 13:51:15

Last modified on

2013-03-22 13:51:15

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Theorem

Classification

msc 17B05

Generated on Fri Feb 9 18:48:28 2018 by

LaTeXML
nilradical

nilradical

Let

𝔤

be a Lie algebra. Then the nilradical

𝔫

of

𝔤

is defined to be the intersection of the kernels of all the

irreducible representations

of

𝔤

. Equivalently,

𝔫

=

[

𝔤

,

𝔤

]

∩

rad

⁢

𝔤

, the intersection of the derived ideal and

radical

of

𝔤

.

Title

nilradical

Canonical

name

Nilradical1

Date of creation

2013-03-22 13:51:22

Last modified on

2013-03-22 13:51:22

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 18:48:30 2018 by

LaTeXML