aarjaneiro/planetmath huggingface dataset:
Poincaré-Birkhoff-Witt theorem

Poincaré-Birkhoff-Witt theorem

Let

𝔤

be a

Lie algebra

over a field

k

, and let

B

be a

k

-basis of

𝔤

equipped with a linear
order

≤

. The

Poincaré-Birkhoff-Witt-theorem

(often
abbreviated to

PBW-theorem

) states that the

monomials

x

1

⁢

x

2

⁢

⋯

⁢

x

n

⁢

with

⁢

x

1

≤

x

2

≤

⋯

≤

x

n

⁢

elements of

⁢

B

constitute a

k

-basis of the

universal enveloping algebra

U

⁢

(

𝔤

)

of

𝔤

. Such monomials are often called

ordered monomials

or

PBW-monomials

.

It is easy to see that they span

U

⁢

(

𝔤

)

: for all

n

∈

ℕ

, let

M

n

denote the set

M

n

=

{

(

x

1

,

…

,

x

n

)

∣

x

1

≤

⋯

≤

x

n

}

⊂

B

n

,

and denote by

π

:

⋃

n

=

0

∞

B

n

→

U

⁢

(

𝔤

)

the

multiplication

map. Clearly it suffices to prove that

π

⁢

(

B

n

)

⊆

∑

i

=

0

n

π

⁢

(

M

i

)

for all

n

∈

ℕ

; to this end, we proceed by

induction

. For

n

=

0

the statement is clear. Assume that it holds for

n

-

1

≥

0

, and consider a
list

(

x

1

,

…

,

x

n

)

∈

B

n

. If it is an element of

M

n

, then we are
done. Otherwise, there exists an index

i

such that

x

i

>

x

i

+

1

.
Now we have

π

⁢

(

x

1

,

…

,

x

n

)

=

π

⁢

(

x

1

,

…

,

x

i

-

1

,

x

i

+

1

,

x

i

,

x

i

+

2

,

…

,

x

n

)

+

x

1

⁢

⋯

⁢

x

i

-

1

⁢

[

x

i

,

x

i

+

1

]

⁢

x

i

+

1

⁢

⋯

⁢

x

n

.

As

B

is a basis of

𝔨

,

[

x

i

,

x

i

+

1

]

is a

linear
combination

of

B

. Using this to expand the second term above, we find
that it is in

∑

i

=

0

n

-

1

π

⁢

(

M

i

)

by the

induction hypothesis

.
The

argument

of

π

in the first term, on the other hand, is
lexicographically smaller than

(

x

1

,

…

,

x

n

)

, but contains the
same entries. Clearly this rewriting proces must end, and this
concludes the induction step.

The proof of

linear independence

of the PBW-monomials is slightly more
difficult, but can be found in most introductory texts on Lie algebras, such as the classic below.

References

1

N. Jacobson.

. Dover Publications, New York, 1979

Title

Poincaré-Birkhoff-Witt theorem

Canonical

name

PoincareBirkhoffWittTheorem

Date of creation

2013-03-22 13:03:38

Last modified on

2013-03-22 13:03:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 17B35

Synonym

PBW-theorem

Related topic

LieAlgebra

Related topic

UniversalEnvelopingAlgebra

Related topic

FreeLieAlgebra

Generated on Fri Feb 9 18:49:54 2018 by

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universal enveloping algebra

universal enveloping algebra

A

universal enveloping algebra

of a

Lie algebra

𝔤

over
a field

k

is an

associative

http://planetmath.org/node/

Algebra

algebra

U

(with unity) over

k

, together
with a Lie algebra

homomorphism

ι

:

𝔤

→

U

(where
the Lie algebra structure on

U

is given by the

commutator

), such that
if

A

is a another

associative algebra

over

k

and

ϕ

:

𝔤

→

A

is another Lie algebra homomorphism, then there exists a
unique homomorphism

ψ

:

U

→

A

of associative algebras such
that the

diagram

Generated on Fri Feb 9 18:49:59 2018 by

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extended Cartan matrix

extended Cartan matrix

Let

A

be the

Cartan matrix

of a complex,

semi-simple

,

finite dimensional

,

Lie algebra

𝔤

. Recall that

A

=

(

a

i

⁢

j

)

where

a

i

⁢

j

=

⟨

α

i

,

α

j

∨

⟩

where the

α

i

are

simple roots

for

𝔤

and the

α

j

∨

are simple coroots.
The

extended Cartan matrix

denoted

A

^

is obtained from

A

by adding a zero-th row and column corresponding to adding a new simple root

α

0

:=

-

θ

where

θ

is the

maximal

(relative to

{

α

1

,

…

,

α

n

}

) root for

𝔤

.

θ

can be defined as a root of

𝔤

such that when written in terms of simple roots

θ

=

∑

i

m

i

⁢

α

i

the coefficient sum

∑

i

m

i

is maximal (i.e. it has maximal height). Such a root can be shown to be unique.

The matrix

A

^

is an example of a

generalized Cartan matrix

. The corresponding Kac-Moody Lie algerba is said to be of affine type.

For example if

𝔤

=

𝔰

⁢

𝔩

n

⁢

ℂ

then

A

^

is obtained from

A

by adding a zero-th row:

(

2

,

-

1

,

0

,

…

,

0

,

-

1

)

and zero-th column

(

2

,

-

1

,

0

,

…

,

0

,

-

1

)

simultaneously to the Cartan matrix for

𝔰

⁢

𝔩

n

⁢

ℂ

.

References

1

Victor Kac,

Infinite Dimensional Lie Algebras

, Third edition. Cambridge University Press, Cambridge, 1990.

Title

extended Cartan matrix

Canonical name

ExtendedCartanMatrix

Date of creation

2013-03-22 15:30:14

Last modified on

2013-03-22 15:30:14

Owner

benjaminfjones (879)

Last modified by

benjaminfjones (879)

Numerical id

8

Author

benjaminfjones (879)

Entry type

Definition

Classification

msc 17B67

Related topic

GeneralizedCartanMatrix

Defines

extended Cartan matrix

Generated on Fri Feb 9 18:50:01 2018 by

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generalized Cartan matrix

generalized Cartan matrix

A

generalized Cartan matrix

is a matrix

A

whose

diagonal

entries are all 2, and whose

off-diagonal entries

are nonpositive integers, such that

a

i

⁢

j

=

0

if and only if

a

j

⁢

i

=

0

. Such a matrix is called

symmetrizable

if there is a

diagonal matrix

B

such that

A

⁢

B

is

symmetric

.

Title

generalized Cartan matrix

Canonical name

GeneralizedCartanMatrix

Date of creation

2013-03-22 13:52:56

Last modified on

2013-03-22 13:52:56

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B67

Related topic

ExtendedCartanMatrix

Defines

symmetrizable

Generated on Fri Feb 9 18:50:02 2018 by

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Kac-Moody algebra

Kac-Moody algebra

Let

A

be an

n

×

n

generalized Cartan matrix

. If

n

-

r

is the rank of

A

, then let

𝔥

be a

n

+

r

dimensional

complex vector space

. Choose

n

linearly independent

elements

α

1

,

…

,

α

n

∈

𝔥

*

(called

roots

), and

α

ˇ

1

,

…

,

α

ˇ

n

∈

𝔥

(called

coroots

) such that

⟨

α

i

,

α

j

ˇ

⟩

=

a

i

⁢

j

, where

⟨

⋅

,

⋅

⟩

is the natural pairing of

𝔥

*

and

𝔥

. This choice is unique up to

automorphisms

of

𝔥

.

Then the

Kac-Moody algebra

associated to

𝔤

⁢

(

A

)

is the

Lie algebra

generated by elements

X

1

,

…

,

X

n

,

Y

1

,

…

,

Y

n

and the elements of

𝔥

, with the

relations

[

X

i

,

Y

i

]

=

α

i

ˇ

[

X

i

,

Y

j

]

=

0

=

α

i

⁢

(

h

)

⁢

X

i

[

Y

i

,

h

]

=

-

α

i

⁢

(

h

)

⁢

Y

i

[

X

i

,

[

X

i

,

⋯

,

[

X

i

⏟

1

-

a

i

⁢

j

⁢

times

,

X

j

]

⋯

]

]

=

0

[

Y

i

,

[

Y

i

,

⋯

,

[

Y

i

⏟

1

-

a

i

⁢

j

⁢

times

,

Y

j

]

⋯

]

]

=

0

for any

h

∈

𝔥

.

If the matrix

A

is positive-definite, we obtain a

finite dimensional

semi-simple Lie algebra

, and

A

is the

Cartan matrix

associated to a

Dynkin diagram

. Otherwise, the

algebra

we obtain is infinite dimensional and has an

r

-dimensional center.

Title

Kac-Moody algebra

Canonical name

KacMoodyAlgebra

Date of creation

2013-03-22 13:31:52

Last modified on

2013-03-22 13:31:52

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B67

Generated on Fri Feb 9 18:50:05 2018 by

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Euclid’s lemma proof

Euclid’s lemma proof

We have

a

|

b

c

, so

b

⁢

c

=

n

⁢

a

, with

n

an

integer

. Dividing both sides by

a

, we have

b

⁢

c

a

=

n

But

gcd

⁡

(

a

,

b

)

=

1

implies

b

/

a

is only an integer if

a

=

1

. So

b

⁢

c

a

=

b

⁢

c

a

=

n

which means

a

must divide

c

.

Note that this proof relies on the

Fundamental Theorem of Arithmetic

. The alternative proof of Euclid’s lemma avoids this.

Title

Euclid’s lemma proof

Canonical name

EuclidsLemmaProof

Date of creation

2013-03-22 11:47:11

Last modified on

2013-03-22 11:47:11

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Proof

Classification

msc 17B80

Classification

msc 81T30

Classification

msc 11A05

Classification

msc 81-00

Generated on Fri Feb 9 18:50:07 2018 by

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Heisenberg algebra

Heisenberg algebra

Let

R

be a

commutative ring

. Let

M

be a

http://planetmath.org/node/5420

module
over

R

http://planetmath.org/node/5420

freely generated

by two sets

{

P

i

}

i

∈

I

and

{

Q

i

}

i

∈

I

, where

I

is an

index set

, and a further

element

c

. Define a

product

[

⋅

,

⋅

]

:

M

×

M

→

M

by bilinear extension by setting

[

c

,

c

]

=

[

c

,

P

i

]

=

[

P

i

,

c

]

=

[

c

,

Q

i

]

=

[

Q

i

,

c

]

=

[

P

i

,

P

j

]

=

[

Q

i

,

Q

j

]

=

0

⁢

for
all

⁢

i

,

j

∈

I

,

=

[

Q

i

,

P

j

]

=

0

⁢

for all distinct

⁢

i

,

j

∈

I

,

=

-

[

Q

i

,

P

i

]

=

c

⁢

for all

⁢

i

∈

I

.

The module

M

together with this product is called a

Heisenberg
algebra

. The element

c

is called the

central element

.

It is easy to see that the product

[

⋅

,

⋅

]

also fulfills the

Jacobi identity

, so a Heisenberg algebra is actually a

Lie algebra

of
rank

|

I

|

+

1

(opposed to the rank of

M

as

free module

, which is

2

⁢

|

I

|

+

1

) with one-dimensional center generated by

c

.

Heisenberg algebras arise in quantum mechanics with

R

=

ℂ

and
typically

I

=

{

1

,

2

,

3

}

, but also in the theory of vertex
with

I

=

ℤ

.

In the case where

R

is a field, the Heisenberg algebra is related to
a

Weyl algebra

: let

U

be the

universal enveloping algebra

of

M

, then the

quotient

U

/

⟨

c

-

1

⟩

is

isomorphic

to the

|

I

|

-th Weyl
algebra over

R

.

Title

Heisenberg algebra

Canonical name

HeisenbergAlgebra

Date of creation

2013-03-22 15:24:57

Last modified on

2013-03-22 15:24:57

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

6

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 17B99

Related topic

WeylAlgebra

Defines

central element

Generated on Fri Feb 9 18:50:09 2018 by

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Jacobi identity interpretations

Jacobi identity interpretations

The

Jacobi identity

in a

Lie algebra

𝔤

has various

interpretations

that are more transparent, whence easier to remember, than the usual form

[

x

,

[

y

,

z

]

]

+

[

y

,

[

z

,

x

]

]

+

[

z

,

[

x

,

y

]

]

=

0

.

One is the fact that the

adjoint representation

1

1

Here, “

𝔤

⁢

𝔩

⁢

(

𝔤

)

” means the space o

endomorphisms

of

𝔤

, viewed as a

vector space

, with Lie
bracket on

𝔤

⁢

𝔩

⁢

(

𝔤

)

being

commutator

.

ad

:

𝔤

→

𝔤

⁢

𝔩

⁢

(

𝔤

)

really is a

representation

. Yet another way to formulate the

identity

is

ad

⁡

(

x

)

⁢

[

y

,

z

]

=

[

ad

⁡

(

x

)

⁢

y

,

z

]

+

[

y

,

ad

⁡

(

x

)

⁢

z

]

,

i.e.,

ad

⁡

(

x

)

is a

derivation

on

𝔤

for all

x

∈

𝔤

.

Title

Jacobi identity interpretations

Canonical name

JacobiIdentityInterpretations

Date of creation

2013-03-22 13:03:42

Last modified on

2013-03-22 13:03:42

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 17B99

Generated on Fri Feb 9 18:50:11 2018 by

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local Nagano theorem

local Nagano theorem

Theorem

(Local Nagano Theorem).

Let

Ω

⊂

R

n

be an open neighbourhood of a point

x

0

. Further let

g

be a

Lie subalgebra

of the

Lie algebra

of
real analytic real vector fields on

Ω

which is also a

C

ω

⁢

(

Ω

;

R

)

-module. Then there exists a

real analytic
submanifold

M

⊂

Ω

with

x

0

∈

M

, such that for all

x

∈

M

we have

T

x

⁢

(

M

)

=

𝔤

⁢

(

x

)

.

Furthermore the germ of

M

at

x

is the unique germ of a

submanifold

with
this property.

Here note that

T

x

⁢

(

M

)

is the

tangent space

of

M

at

x

,

C

ω

⁢

(

Ω

;

ℝ

)

are the real analytic real valued functions
on

Ω

. Also real analytic real vector fields on

Ω

are the
real analytic

sections

of

T

⁢

(

Ω

)

, the real

tangent bundle

of

Ω

.

Definition

.

The germ of the

manifold

M

is called the

local Nagano leaf

of

𝔤

at

x

0

.

Definition

.

The union of all connected real analytic embedded submanifolds of

Ω

whose
germ at

x

0

coincides with the germ of

M

at

x

0

is called the

global Nagano leaf

.

The global Nagano leaf turns out to be a connected immersed real analytic submanifold which may however not be an embedded submanifold of

Ω

.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

Title

local Nagano theorem

Canonical name

LocalNaganoTheorem

Date of creation

2013-03-22 14:48:27

Last modified on

2013-03-22 14:48:27

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Theorem

Classification

msc 17B99

Classification

msc 53B25

Synonym

Nagano’s theorem

Defines

local Nagano leaf

Defines

Nagano leaf

Defines

global Nagano leaf

Generated on Fri Feb 9 18:50:21 2018 by

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real form

real form

Let

G

be a complex Lie group. A real Lie group

K

called a

real form

of

G

if

𝔤

≅

ℂ

⊗

ℝ

𝔨

, where

𝔤

and

𝔨

are the

Lie algebras

of

G

and

K

, respectively.

Title

real form

Canonical name

RealForm

Date of creation

2013-03-22 13:23:34

Last modified on

2013-03-22 13:23:34

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 17B99

Generated on Fri Feb 9 18:50:25 2018 by

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(special) unitary Lie algebra

(special) unitary Lie algebra

Let

V

be a

vector space

over a field

K

admitting an

involution

σ

:

K

→

K

, and let

B

:

V

×

V

→

𝔽

be a

http://planetmath.org/node/SesquilinearFormsOverGeneralFields

hermitian form

relative to

σ

. Then the

unitary Lie algebra

𝔲

⁢

(

V

,

B

)

, or just

𝔲

⁢

(

V

)

, consists of the

linear transformations

T

satisfying

B

⁢

(

T

⁢

x

,

y

)

+

B

⁢

(

x

,

T

⁢

y

)

=

0

,

for all

x

,

y

∈

V

. This is a

Lie algebra

over

k

=

{

α

∈

K

∣

α

σ

=

α

}

, but not over

K

in the case that

K

≠

k

(because

B

is linear in the first, but not in the second variable).

The

special unitary Lie algebra

𝔰

⁢

u

⁢

(

V

,

B

)

, or just

𝔰

⁢

u

⁢

(

V

)

, consists of those linear transformations in

𝔲

⁢

(

V

,

B

)

with trace zero.

Title

(special) unitary Lie algebra

Canonical name

specialUnitaryLieAlgebra

Date of creation

2013-03-22 18:45:21

Last modified on

2013-03-22 18:45:21

Owner

pan (17366)

Last modified by

pan (17366)

Numerical id

6

Author

pan (17366)

Entry type

Definition

Classification

msc 17B99

Synonym

unitary algebra

Synonym

special unitary algebra

Defines

unitary algebra

Defines

special unitary algebra

Generated on Fri Feb 9 18:50:29 2018 by

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Jordan algebra

Jordan algebra

Let

R

be a commutative ring with

1

≠

0

. An

R

-

algebra

A

with multiplication not assumed to be

associative

is called a (

commutative

)

Jordan algebra

if

1.

A

is commutative:

a

⁢

b

=

b

⁢

a

, and

2.

A

satisfies the

Jordan identity

:

(

a

2

⁢

b

)

⁢

a

=

a

2

⁢

(

b

⁢

a

)

,

for any

a

,

b

∈

A

.

The above can be restated as

1.

[

A

,

A

]

=

0

, where

[

,

]

is the

commutator bracket

, and

2.

for any

a

∈

A

,

[

a

2

,

A

,

a

]

=

0

, where

[

,

,

]

is the

associator

bracket.

If

A

is a Jordan algebra, a subset

B

⊆

A

is called a

Jordan subalgebra

if

B

⁢

B

⊆

B

. Let

A

and

B

be two Jordan algebras. A

Jordan algebra homomorphism

, or simply

Jordan homomorphism

, from

A

to

B

is an algebra homomorphism that respects the above two laws. A

Jordan algebra isomorphism

is just a bijective Jordan algebra homomorphism.

Remarks

.

•

If

A

is a Jordan algebra such that

char

⁡

(

A

)

≠

2

, then

A

is power-associative (

http://planetmath.org/PowerAssociativeAlgebra

).

•

If in addition

2

=

1

+

1

≠

char

⁡

(

A

)

, then by replacing

a

with

a

+

1

in the Jordan identity and simplifying,

A

is

flexible

(

http://planetmath.org/FlexibleAlgebra

).

•

Given any

associative algebra

A

, we can define a Jordan algebra

A

+

. To see this, let

A

be an associative algebra with associative multiplication

⋅

and suppose

2

=

1

+

1

is

invertible

in

R

. Define a new multiplication given by

a

⁢

b

=

1

2

⁢

(

a

⋅

b

+

b

⋅

a

)

.

(1)

It is readily checked that this new multiplication satisifies both the commutative law and the Jordan identity. Thus

A

with the new multiplication is a Jordan algebra and we denote it by

A

+

. However, unlike

Lie algebras

, not every Jordan algebra is embeddable in an associative algebra. Any Jordan algebra that is

isomorphic

to a Jordan subalgebra of

A

+

for some associative algebra

A

is called a

special Jordan algebra

. Otherwise, it is called an

exceptional Jordan algebra

. As a side note, the

right hand side

of Equation (1) is called the

Jordan product

.

•

An example of an exceptional Jordan algebra is

H

3

⁢

(

𝕆

)

, the algebra of

3

×

3

Hermitian matrices

over the

octonions

.

Title

Jordan algebra

Canonical name

JordanAlgebra

Date of creation

2013-03-22 14:52:15

Last modified on

2013-03-22 14:52:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17C05

Synonym

Jordan homomorphism

Synonym

Jordan isomorphism

Defines

Jordan identity

Defines

special Jordan algebra

Defines

exceptional Jordan algebra

Defines

Jordan algebra homomorphism

Defines

Jordan subalgebra

Defines

Jordan algebra isomorphism

Generated on Fri Feb 9 18:50:31 2018 by

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Jordan triple product

Jordan triple product

Given a

Jordan algebra

J

we define a ternary product on

J

by

{

x

y

z

}

=

(

x

.

y

)

.

z

+

(

y

.

z

)

.

x

-

(

z

.

x

)

.

y

.

When

J

is a

special Jordan algebra

of

characteristic

not 2, we know the product

x

.

y

=

1

2

⁢

(

x

⁢

y

+

y

⁢

x

)

. In this

context

a

computation

shows

{

x

⁢

y

⁢

z

}

=

1

2

⁢

(

x

⁢

y

⁢

z

+

z

⁢

y

⁢

x

)

.

This gives a simple method by which to compute the triple product in a special
Jordan algebra.

A key instance of the

Jordan triple product

is the case when

x

=

z

(setting

x

.2

=

x

.

x

for notation). Here we get

{

x

y

x

}

=

2

(

x

.

y

)

.

x

-

x

.2

.

y

.

In a special Jordan algebra this becomes

{

x

⁢

y

⁢

x

}

=

x

⁢

y

⁢

x

in the

associative

product. To treat a Jordan algebra as a

quadratic Jordan algebra

this product plays the role of one of the two quadratic operations:

U

y

:

y

↦

{

x

⁢

y

⁢

x

}

.
The other is the usual squaring product

y

↦

y

.2

.

To establish uniform proof for Jordan algebra in characteristic 2 and also include

exceptional Jordan algebras

it is often preferable to encode computations using the unary product:

x

.2

and the triple product

{

x

⁢

y

⁢

z

}

. The connection between the triple product and the quadratic unary product is found in the Jordan identity:

(

a

.2

.

b

)

.

a

=

a

.2

.

(

b

.

a

)

This idea was exploited by McCrimmon to establish quadratic Jordan algebras and many uniform and previously unknown results on Jordan algebras.

The triple product can be compared to the Jacobi product on an

algebra

with a

[

,

]

multiplication

#

⁢

x

⁢

y

⁢

z

⁢

#

=

[

[

x

,

y

]

,

z

]

+

[

[

y

,

z

]

,

x

]

+

[

[

z

,

x

]

,

y

]

.

To make a closer parallel use the typical assumption that

[

z

,

x

]

=

-

[

x

,

z

]

(outside of characteristic 0) then we can write:

#

⁢

x

⁢

y

⁢

z

⁢

#

=

[

[

x

,

y

]

,

z

]

+

[

[

y

,

z

]

,

x

]

-

[

[

x

,

z

]

,

y

]

.

Since Jordan algebras are

commutative

we can also write

{

x

y

z

}

=

(

x

.

y

)

.

z

+

(

y

.

z

)

.

x

-

(

x

.

z

)

.

y

.

However, unlike

Lie algebras

where

#

⁢

x

⁢

y

⁢

z

⁢

#

=

0

, in Jordan algebras the triple product is almost never 0.

Title

Jordan triple product

Canonical name

JordanTripleProduct

Date of creation

2013-03-22 16:27:27

Last modified on

2013-03-22 16:27:27

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

7

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 17C05

Defines

Jordan triple product

Generated on Fri Feb 9 18:50:34 2018 by

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alternative algebra

alternative algebra

A

non-associative algebra

A

is

alternative

if

1.

(

left alternative laws

)

[

a

,

a

,

b

]

=

0

, and

2.

(right alternative laws)

[

b

,

a

,

a

]

=

0

,

for any

a

,

b

∈

A

, where

[

,

,

]

is the

associator

on

A

.

Remarks

•

Let

A

be alternative and suppose

char

⁡

(

A

)

≠

2

. From the fact that

[

a

+

b

,

a

+

b

,

c

]

=

0

, we can deduce that the associator

[

,

,

]

is

anti-commutative

, when one of the three coordinates is held fixed. That is, for any

a

,

b

,

c

∈

A

,

(a)

[

a

,

b

,

c

]

=

-

[

b

,

a

,

c

]

(b)

[

a

,

b

,

c

]

=

-

[

a

,

c

,

b

]

(c)

[

a

,

b

,

c

]

=

-

[

c

,

b

,

a

]

Put more succinctly,

[

a

1

,

a

2

,

a

3

]

=

sgn

⁡

(

π

)

⁢

[

a

π

⁢

(

1

)

,

a

π

⁢

(

2

)

,

a

π

⁢

(

3

)

]

,

where

π

∈

S

3

, the

symmetric group on three letters

, and

sgn

⁡

(

π

)

is the sign (

http://planetmath.org/SignatureOfAPermutation

) of

π

.

•

An

alternative algebra

is a

flexible algebra

, provided that the

algebra

is not

Boolean

(

http://planetmath.org/BooleanLattice

) (

characteristic

(

http://planetmath.org/Characteristic

)

≠

2

). To see this, replace

c

in the first anti-commutative

identities

above with

a

and the result follows.

•

Artin’s Theorem

: If a non-associative algebra

A

is not Boolean, then

A

is alternative iff every

subalgebra

of

A

generated by two elements is

associative

. The proof is clear from the above discussion.

•

A

commutative

alternative algebra

A

is a

Jordan algebra

. This is true since

a

2

⁢

(

b

⁢

a

)

=

a

2

⁢

(

a

⁢

b

)

=

(

a

⁢

b

)

⁢

a

2

=

(

(

a

⁢

b

)

⁢

a

)

⁢

a

=

(

a

⁢

(

a

⁢

b

)

)

⁢

a

=

(

a

2

⁢

b

)

⁢

a

shows that the Jordan identity is satisfied.

•

Alternativity can be defined for a general ring

R

: it is a non-associative ring such that for any

a

,

b

∈

R

,

(

a

⁢

a

)

⁢

b

=

a

⁢

(

a

⁢

b

)

and

(

a

⁢

b

)

⁢

b

=

a

⁢

(

b

⁢

b

)

. Equivalently, an alternative ring is an alternative algebra over

ℤ

.

Title

alternative algebra

Canonical name

AlternativeAlgebra

Date of creation

2013-03-22 14:43:24

Last modified on

2013-03-22 14:43:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17D05

Related topic

Associator

Related topic

FlexibleAlgebra

Defines

Artin’s theorem on alternative algebras

Defines

alternative ring

Defines

left alternative law

Defines

right alternative law

Generated on Fri Feb 9 18:50:39 2018 by

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examples of algebraic K-theory groups

examples of algebraic K-theory groups

R

K

0

⁢

(

R

)

K

1

⁢

(

R

)

K

2

⁢

(

R

)

K

3

⁢

(

R

)

K

4

⁢

(

R

)

ℤ

ℤ

ℤ

/

2

ℤ

/

2

ℤ

/

48

0

ℝ

ℤ

ℝ

×

ℂ

ℤ

ℂ

×

Algebraic K-theory

of some common rings.

Title

examples of algebraic K-theory groups

Canonical name

ExamplesOfAlgebraicKtheoryGroups

Date of creation

2013-03-22 13:31:35

Last modified on

2013-03-22 13:31:35

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

7

Author

mhale (572)

Entry type

Example

Classification

msc 19-00

Classification

msc 18F25

Related topic

ExamplesOfGroups

Generated on Fri Feb 9 19:14:31 2018 by

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