aarjaneiro/planetmath huggingface dataset:
nichols-zoeller theorem

nichols-zoeller theorem

Let

H

be a

Hopf algebra

over a field

k

with an antipode

S

. We will say that

K

⊆

H

is a

Hopf

subalgebra

if

K

is both subalgebra and subcoalgebra of underlaying

algebra

and coalgebra

structures

of

H

, and additionaly

S

⁢

(

K

)

⊆

K

. In particular a Hopf subalgebra

K

⊆

H

is an algebra over

k

, so

H

may be regarded as a

K

-module.

The

Nichols-Zoeller Theorem

.

If

K

⊆

H

is a Hopf subalgebra of a Hopf algebra

H

, then

H

is free as a

K

-module. In particular, if

H

is

finite dimensional

, then

dim

k

⁢

K

divides

dim

k

⁢

H

.

Remark 1.

This

theorem

shows that Hopf algebras are very

similar

to groups, because this is a Hopf analogue of the Lagrange Theorem.

Remark 2.

Generally this theorem does not need to hold if

H

is only an algebra. For example, consider

H

=

𝕄

n

⁢

(

k

)

the matrix algebra, where

n

≥

2

and let

T

⊆

H

be the

upper triangular matrix

subalgebra. It is well known that

dim

k

⁢

H

=

n

2

and

dim

k

⁢

T

=

n

⁢

(

n

+

1

)

2

. Of course

n

⁢

(

n

+

1

)

2

does not divide

n

2

for

n

≥

2

. Thus the Nichols-Zoeller Theorem does not hold for algebras.

Title

nichols-zoeller theorem

Canonical name

NicholszoellerTheorem

Date of creation

2013-03-22 18:58:34

Last modified on

2013-03-22 18:58:34

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

6

Author

joking (16130)

Entry type

Theorem

Classification

msc 16W30

Generated on Fri Feb 9 18:45:38 2018 by

LaTeXML
subcoalgebras and coideals

subcoalgebras and coideals

Let

(

C

,

Δ

,

ε

)

be a coalgebra over a field

k

.

Definition.

Vector subspace

D

⊆

C

is called

subcoalgebra

iff

Δ

⁢

(

D

)

⊆

D

⊗

D

.

Definition.

Vector subspace

I

⊆

C

is is called

coideal

iff

Δ

⁢

(

I

)

⊆

I

⊗

C

+

C

⊗

I

and

ε

⁢

(

I

)

=

0

.

One can show that if

D

⊆

C

is a subcoalgebra, then

(

D

,

Δ

|

D

,

ε

|

D

)

is also a coalgebra. On the other hand, if

I

⊆

C

is a coideal, then we can cannoicaly introduce a coalgebra

structure

on the

quotient space

C

/

I

. More precisely, if

x

∈

C

and

Δ

⁢

(

x

)

=

∑

a

i

⊗

b

i

, then we define

Δ

′

:

C

/

I

→

(

C

/

I

)

⊗

(

C

/

I

)

;

Δ

′

⁢

(

x

+

I

)

=

∑

(

a

i

+

I

)

⊗

(

b

i

+

I

)

and

ε

′

:

C

/

I

→

k

as

ε

′

⁢

(

x

+

I

)

=

ε

⁢

(

x

)

. One can show that these two maps are well defined and

(

C

/

I

,

Δ

′

,

ε

′

)

is a coalgebra.

Title

subcoalgebras and coideals

Canonical name

SubcoalgebrasAndCoideals

Date of creation

2013-03-22 18:49:19

Last modified on

2013-03-22 18:49:19

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 16W30

Generated on Fri Feb 9 18:45:40 2018 by

LaTeXML
graded algebra

graded algebra

An

algebra

A

over a

graded ring

B

is

graded

if it is itself a graded ring and a

graded module

over

B

such that

A

p

⋅

A

q

⊆

A

p

+

q

where

A

i

,

i

∈

ℕ

, are

submodules

of

A

.
More generally, one can replace

ℕ

by a monoid or semigroup

G

.
In which case,

A

is called a

G

-

graded algebra

.
A graded algebra then is the same thing as an

ℕ

-graded algebra.

Examples of graded algebras include the

polynomial ring

k

⁢

[

X

]

being an

ℕ

-graded

k

-algebra, and the

exterior algebra

.

Title

graded algebra

Canonical name

GradedAlgebra

Date of creation

2013-03-22 12:45:47

Last modified on

2013-03-22 12:45:47

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

8

Author

mhale (572)

Entry type

Definition

Classification

msc 16W50

Related topic

GradedModule

Related topic

SuperAlgebra

Related topic

LieSuperalgebra

Related topic

LieSuperalgebra3

Generated on Fri Feb 9 18:45:42 2018 by

LaTeXML
supercommutative

supercommutative

Let

R

be a

ℤ

2

-

graded ring

(or more generally, an

associative algebra

). We say that

R

is

supercommutative

if for any

homogeneous elements

a

and

b

∈

R

:

a

⁢

b

=

(

-

1

)

deg

⁡

a

⁢

deg

⁡

b

⁢

b

⁢

a

.

In other words, even homogeneous elements are in the center of the ring, and odd homogeneous elements anti-commute.

Common examples of supercommutative rings are the

exterior algebra

of a module over a

commutative ring

(in particular, a

vector space

) and the cohomology ring of a topological space (both with the standard

grading

by degree reduced mod 2).

Title

supercommutative

Canonical

name

Supercommutative

Date of creation

2013-03-22 13:25:18

Last modified on

2013-03-22 13:25:18

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Definition

Classification

msc 16W50

Synonym

graded-commutative

Synonym

supercommutative algebra

Synonym

even element

Synonym

odd element

Related topic

SuperAlgebra

Generated on Fri Feb 9 18:45:46 2018 by

LaTeXML
graded tensor product

graded tensor product

If

A

and

B

are

ℤ

-

graded algebras

, we define the

graded tensor product

(or

super tensor product

)

A

⊗

s

⁢

u

B

to be the ordinary

tensor product

as

graded modules

, but with

multiplication

- called the

super

product

- defined by

(

a

⊗

b

)

⁢

(

a

′

⊗

b

′

)

=

(

-

1

)

(

deg

⁢

b

)

⁢

(

deg

⁢

a

′

)

⁢

a

⁢

a

′

⊗

b

⁢

b

′

where

a

,

a

′

,

b

,

b

′

are

homogeneous

. The super tensor product of

A

and

B

is itself a graded algebra, as we grade the super tensor product of

A

and

B

as follows:

(

A

⊗

s

⁢

u

B

)

n

=

∐

p

,

q

⁢

:

⁢

p

+

q

=

n

A

p

⊗

B

q

Title

graded tensor product

Canonical name

GradedTensorProduct

Date of creation

2013-03-22 12:45:44

Last modified on

2013-03-22 12:45:44

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 16W55

Related topic

TensorProduct

Defines

super tensor product

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

LaTeXML
superalgebra

superalgebra

A

graded algebra

A

is said to be a

superalgebra

if it has a

ℤ

/

2

⁢

ℤ

grading

.
As a

vector space

, a superalgebra has a

decomposition

into two

homogeneous

subspaces

,

A

=

A

0

⊕

A

1

.
The homogeneous subspace

A

0

is known as the space of

even

elements of

A

, and

A

1

is known as the space of

odd

elements

.
Let

|

a

|

denote the degree of a

homogeneous element

.
That is,

|

a

|

=

0

if

a

∈

A

0

and

|

a

|

=

1

if

a

∈

A

1

.
The degree

satisfies

|

a

⁢

b

|

=

|

a

|

+

|

b

|

.

Title

superalgebra

Canonical name

Superalgebra

Date of creation

2013-03-22 12:46:18

Last modified on

2013-03-22 12:46:18

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

7

Author

mhale (572)

Entry type

Definition

Classification

msc 16W55

Synonym

super algebra

Related topic

Supernumber

Related topic

Supercommutative

Related topic

LieSuperalgebra

Related topic

LieSuperalgebra3

Related topic

GradedAlgebra

Generated on Fri Feb 9 18:45:50 2018 by

LaTeXML
supernumber

supernumber

Supernumbers

are the generalisation of

complex numbers

to a

commutative

superalgebra

of commuting and anticommuting “numbers”.
They are primarily used in the description of in .

Let

Λ

N

be the

Grassmann algebra

generated by

θ

i

,

i

=

1

⁢

…

⁢

N

,
such that

θ

i

⁢

θ

j

=

-

θ

j

⁢

θ

i

and

(

θ

i

)

2

=

0

.
Denote by

Λ

∞

, the Grassmann algebra of an

infinite

number of

generators

θ

i

.
A

supernumber

is an element of

Λ

N

or

Λ

∞

.

Any supernumber

z

can be expressed uniquely in the form

z

=

z

0

+

z

i

⁢

θ

i

+

1

2

⁢

z

i

⁢

j

⁢

θ

i

⁢

θ

j

+

…

+

1

n

!

⁢

z

i

1

⁢

…

⁢

i

n

⁢

θ

i

1

⁢

…

⁢

θ

i

n

+

…

,

where the

coefficients

z

i

1

⁢

…

⁢

i

n

∈

ℂ

are

antisymmetric

in their indices.

1

Body and soul

The

body

of a supernumber

z

is defined as

z

B

=

z

0

,
and its

soul

is defined as

z

S

=

z

-

z

B

.
If

z

B

≠

0

then

z

has an

inverse

given by

z

-

1

=

1

z

B

⁢

∑

k

=

0

∞

(

-

z

S

z

B

)

k

.

2

Odd and even

A supernumber can be decomposed into the even and odd parts:

z

even

=

z

0

+

1

2

⁢

z

i

⁢

j

⁢

θ

i

⁢

θ

j

+

…

+

1

(

2

⁢

n

)

!

⁢

z

i

1

⁢

…

⁢

i

2

⁢

n

⁢

θ

i

1

⁢

…

⁢

θ

i

2

⁢

n

+

…

,

z

odd

=

z

i

⁢

θ

i

+

1

6

⁢

z

i

⁢

j

⁢

k

⁢

θ

i

⁢

θ

j

⁢

θ

k

+

…

+

1

(

2

⁢

n

+

1

)

!

⁢

z

i

1

⁢

…

⁢

i

2

⁢

n

+

1

⁢

θ

i

1

⁢

…

⁢

θ

i

2

⁢

n

+

1

+

…

.

Even supernumbers commute with each other and are called

c-numbers

,
while odd supernumbers anticommute with each other and are called

a-numbers

.
Note, the product of two c-numbers is even,
the product of a c-number and an a-number is odd,
and the product of two a-numbers is even.
The superalgebra

Λ

N

has the

vector space

decomposition

Λ

N

=

ℂ

c

⊕

ℂ

a

,
where

ℂ

c

is the space of c-numbers,
and

ℂ

a

is the space of a-numbers.

3

Conjugation and involution

There are two ways, one can define a

complex conjugation

for supernumbers.
The first is to define a linear

conjugation

in

complete

analogy with complex numbers:

(

z

1

⁢

z

2

)

¯

=

z

1

¯

⁢

z

2

¯

.

The second way is to define an anti-linear

involution

:

(

z

1

⁢

z

2

)

*

=

z

2

*

⁢

z

1

*

.

The comes down to whether the product of two real odd supernumbers is real or

imaginary

.

Title

supernumber

Canonical name

Supernumber

Date of creation

2013-03-22 13:03:27

Last modified on

2013-03-22 13:03:27

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

12

Author

mhale (572)

Entry type

Definition

Classification

msc 16W55

Related topic

SuperAlgebra

Defines

body

Defines

soul

Generated on Fri Feb 9 18:45:53 2018 by

LaTeXML
The Hamiltonian ring is not a complex algebra

The Hamiltonian ring is not a complex algebra

The

Hamiltonian

algebra

(

http://planetmath.org/QuaternionAlgebra2

)

ℍ

contains

isomorphic

copies of the real

ℝ

and

complex

ℂ

numbers. However, the reals are a central

subalgebra

of

ℍ

which makes

ℍ

into a real algebra. This makes identifying

ℝ

in

ℍ

canonical:

1

∈

ℍ

determines a unique

embedding

ℝ

→

ℍ

:

r

↦

r

⁢

1

. Yet

ℍ

is not a complex algebra. The goal presently is to outline some of the incongruities of

ℂ

=

⟨

1

,

i

⟩

and

ℍ

=

⟨

1

,

ı

^

,

ȷ

^

,

k

^

⟩

which may be obscured by the notational overlap of the letter

i

.

Proposition 1

.

There are no proper finite dimensional

division rings

over algebraically closed fields.

Proof.

Let

D

be a finite dimensional division ring over an algebraically closed field

K

. This means that

K

is a central subalgebra of

D

. Let

a

∈

D

and consider

K

⁢

(

a

)

. Since

K

is central in

D

,

K

⁢

(

a

)

is

commutative

, and so

K

⁢

(

a

)

is a

field extension

of

K

. But as

D

is a finite dimensional

K

space, so is

K

⁢

(

a

)

. As any finite dimensional

extension

of

K

is algebraic,

K

⁢

(

a

)

is an

algebraic extension

. Yet

K

is

algebraically closed

so

K

⁢

(

a

)

=

K

. Thus

a

∈

K

so in fact

D

=

K

.
∎

•

In particular, this

proposition

proves

ℍ

is not a complex algebra.

•

Alternatively, from the

Wedderburn-Artin theorem

we know the only

semisimple

complex algebra of

dimension

2 is

ℂ

⊕

ℂ

. This has

proper ideals

and so it cannot be the division ring

ℍ

.

•

It is also evident that the usual, notationally driven, embedding of

ℂ

into

ℍ

is non-central. That is,

ℂ

embeds as

a

+

b

⁢

i

↦

a

+

b

⁢

i

^

, into

ℍ

=

⟨

1

,

ı

^

,

ȷ

^

,

k

^

⟩

. This is not central:

(

1

+

ı

^

)

⁢

ȷ

^

=

ȷ

^

+

k

^

≠

ȷ

^

⁢

(

1

+

ı

^

)

=

ȷ

^

-

k

^

.

•

Further evidence of the incompatiblity of

ℍ

and

ℂ

comes from considering

polynomials

. If

x

2

+

1

is considered as a polynomial over

ℂ

⁢

[

x

]

then it has exactly two roots

i

,

-

i

as expected. However, if it is considered as a polynomial over

ℍ

⁢

[

x

]

we arrive at 6 obvious roots:

{

ı

^

,

-

ı

^

,

ȷ

^

,

-

ȷ

^

,

k

^

,

-

k

^

}

. But indeed, given any

q

∈

ℍ

,

q

≠

0

, then

q

⁢

ı

^

⁢

q

-

1

is also a root. Thus there are an

infinite

number of roots to

x

2

+

1

. Therefore declaring

ı

^

=

-

1

can be greatly misleading. Such a conflict does not arise for polynomials with real roots since

ℝ

is a central subalgebra.

Title

The Hamiltonian ring is not a complex algebra

Canonical name

TheHamiltonianRingIsNotAComplexAlgebra

Date of creation

2013-03-22 16:01:57

Last modified on

2013-03-22 16:01:57

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

10

Author

Algeboy (12884)

Entry type

Result

Classification

msc 16W99

Generated on Fri Feb 9 18:45:58 2018 by

LaTeXML
near-ring

near-ring

Definitions

A

near-ring

is a set (

http://planetmath.org/Set

)

N

together with two

binary operations

, denoted

+

:

N

×

N

→

N

and

⋅

:

N

×

N

→

N

, such that

1.

(

a

+

b

)

+

c

=

a

+

(

b

+

c

)

and

(

a

⋅

b

)

⋅

c

=

a

⋅

(

b

⋅

c

)

for all

a

,

b

,

c

∈

N

(

associativity

of both

operations

)

2.

There exists an

element

0

∈

N

such that

a

+

0

=

0

+

a

=

a

for all

a

∈

N

(

additive identity

)

3.

For all

a

∈

N

, there exists

b

∈

N

such that

a

+

b

=

b

+

a

=

0

(

additive inverse

)

4.

(

a

+

b

)

⋅

c

=

(

a

⋅

c

)

+

(

b

⋅

c

)

for all

a

,

b

,

c

∈

N

(

right distributive law

)

Note that the axioms of a near-ring differ from those of a ring in that they do not require

addition

to be

commutative

(

http://planetmath.org/Commutative

), and only require distributivity on one side.

A

near-field

is a near-ring

N

such that

(

N

∖

{

0

}

,

⋅

)

is a group.

Notes

Every element

a

in a near-ring has a unique additive inverse, denoted

-

a

.

We say

N

has an

identity element

if there exists an element

1

∈

N

such that

a

⋅

1

=

1

⋅

a

=

a

for all

a

∈

N

.
We say

N

is

distributive

if

a

⋅

(

b

+

c

)

=

(

a

⋅

b

)

+

(

a

⋅

c

)

holds for all

a

,

b

,

c

∈

N

.
We say

N

is

commutative

if

a

⋅

b

=

b

⋅

a

for all

a

,

b

∈

N

.

Every commutative near-ring is distributive.
Every distributive near-ring with an identity element is a

unital ring

(see the attached proof (

http://planetmath.org/ConditionOnANearRingToBeARing

)).

Example

A natural example of a near-ring is the following. Let

(

G

,

+

)

be a group (not necessarily

abelian

(

http://planetmath.org/AbelianGroup2

)), and let

M

be the set of all

functions

from

G

to

G

. For two functions

f

and

g

in

M

define

f

+

g

∈

M

by

(

f

+

g

)

⁢

(

x

)

=

f

⁢

(

x

)

+

g

⁢

(

x

)

for all

x

∈

G

. Then

(

M

,

+

,

∘

)

is a near-ring with

identity

, where

∘

denotes composition of functions.

References

1

Günter Pilz,

Near-Rings

,
North-Holland, 1983.

Title

near-ring

Canonical name

Nearring

Date of creation

2013-03-22 13:25:12

Last modified on

2013-03-22 13:25:12

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

22

Author

yark (2760)

Entry type

Definition

Classification

msc 16Y30

Synonym

near ring

Synonym

nearring

Related topic

Ring

Defines

commutative near-ring

Defines

commutative near ring

Defines

commutative nearring

Defines

distributative near-ring

Defines

distributative near ring

Defines

distributative nearring

Defines

near field

Defines

nearfield

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idempotent semiring

idempotent semiring

A semiring

S

is called an

idempotent semiring

, or

i-semiring

for short, if,

addition

+

is an

idempotent

binary operation

:

a

+

a

=

a

,

for all

⁢

a

∈

S

.

Some properties of an i-semiring

S

.

1.

If we define a

binary relation

≤

on

S

by

a

≤

b

iff

a

+

b

=

b

then

≤

becomes a

partial order

on

S

. Indeed, for

a

+

a

=

a

implies

a

≤

a

; if

a

≤

b

and

b

≤

a

, then

b

=

a

+

b

=

a

; and finally, if

a

≤

b

and

b

≤

c

, then

a

+

c

=

a

+

(

b

+

c

)

=

(

a

+

b

)

+

c

=

b

+

c

=

c

so

a

≤

c

.

2.

0

≤

a

for any

a

∈

S

, because

0

+

a

=

a

.

3.

Define

a

∨

b

as the

supremum

of

a

and

b

(with respect to

≤

). Then

a

∨

b

exists and

a

∨

b

=

a

+

b

.

To see this, we have

a

+

(

a

+

b

)

=

(

a

+

a

)

+

b

=

a

+

b

, so

a

≤

a

+

b

. Similarly

b

≤

a

+

b

. If

a

≤

c

and

b

≤

c

, then

(

a

+

b

)

+

c

=

a

+

(

b

+

c

)

=

a

+

c

=

c

. So

a

+

b

≤

c

.

4.

Collecting all the information above, we see that

(

S

,

+

)

is an

upper semilattice

with

+

as the join

operation

on

S

and

0

the bottom

element

.

5.

Additon and

multiplication

respect partial ordering: suppose

a

≤

b

, then for any

c

∈

S

,

(

c

+

a

)

+

(

c

+

b

)

=

(

c

+

c

)

+

(

a

+

b

)

=

c

+

b

, hence

c

+

a

≤

c

+

b

; also,

c

⁢

b

=

c

⁢

(

a

+

b

)

=

c

⁢

a

+

c

⁢

b

implies

c

⁢

a

≤

c

⁢

b

.

Remark

.

S

in general is not a

lattice

, and

1

is not the top element of

S

.

The main example of an i-semiring is a

Kleene algebra

used in the theory of computations.

Title

idempotent semiring

Canonical name

IdempotentSemiring

Date of creation

2013-03-22 15:52:12

Last modified on

2013-03-22 15:52:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16Y60

Synonym

i-semiring

Synonym

dioid

Generated on Fri Feb 9 18:46:02 2018 by

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semifield

semifield

There are different definitions of

semifield

.  We give three such which are not

equivalent

(

http://planetmath.org/

Biconditional

).

Let

K

be a set with two

binary operations

“

+

” and “

⋅

”.

•

Semifield

(

K

,

+

,

⋅

)

is a

semiring

where all non-zero

elements

have a

multiplicative inverse

.

•

Semifield  is the

algebraic system

(

K

,

+

,

⋅

)

,  where

(

K

,

+

)

is a group (

identity

:=

0

),  the

multiplication

“

⋅

”

distributes over

the

addition

“

+

”,

K

the multiplicative identity

:=

1

and all

equations

a

⁢

x

=

b

and

y

⁢

a

=

b

with

a

≠

0

have solutions

x

,

y

in

K

.

•

Semifield

(

K

,

+

,

⋅

)

satisfies

all

postulates

of field except the

associativity

of the multiplication “

⋅

”.

Title

semifield

Canonical name

Semifield

Date of creation

2013-03-22 15:45:46

Last modified on

2013-03-22 15:45:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16Y60

Classification

msc 12K10

Related topic

NonAssociativeAlgebra

Generated on Fri Feb 9 18:46:04 2018 by

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semiring

semiring

\PMlinkescapephrase

right annihilator

A

semiring

is a set

A

with two

operations

,

+

and

⋅

, such that

0

∈

A

makes

(

A

,

+

)

into a

commutative monoid

,

1

∈

A

makes

(

A

,

⋅

)

into a monoid, the operation

⋅

distributes (

http://planetmath.org/

Distributivity

)
over

+

, and for any

a

∈

A

,

0

⋅

a

=

a

⋅

0

=

0

. Usually,

a

⋅

b

is instead written

a

⁢

b

.

A ring

(

R

,

+

,

⋅

)

, can be described as a semiring for which

(

R

,

+

)

is
required to be a group. Thus every ring is a semiring.
The

natural numbers

ℕ

form a semiring, but not a ring, with the usual

multiplication

and

addition

.

Every semiring

A

has a

quasiorder

⪯

given by

a

⪯

b

if and only if there exists some

c

∈

A

such that

a

+

c

=

b

. Any

element

a

∈

A

with an

additive inverse

is smaller than
any other element. Thus if

A

has a nonzero element

a

with an additive
inverse, then the elements

-

a

,

0

,

a

form a cycle with respect to

⪯

.
If

+

is an

idempotent

(

http://planetmath.org/Idempotency

) operation,
then

⪯

is a

partial order

.
Addition and (left and right) multiplication are

order-preserving

operators

(

http://planetmath.org/Poset

).

Title

semiring

Canonical name

Semiring

Date of creation

2013-03-22 12:27:46

Last modified on

2013-03-22 12:27:46

Owner

mps (409)

Last modified by

mps (409)

Numerical id

11

Author

mps (409)

Entry type

Definition

Classification

msc 16Y60

Related topic

Ring

Related topic

KleeneAlgebra

Generated on Fri Feb 9 18:46:06 2018 by

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anticommutative

anticommutative

A

binary operation

“

⋆

” is said to be

anticommutative

if it satisfies the

identity

y

⋆

x

=

-

(

x

⋆

y

)

,

(1)

where the minus denotes the element in the

algebra

in question.  This implies that

x

⋆

x

=

-

(

x

⋆

x

)

,  i.e.

x

⋆

x

must be the

neutral element

of the addition of the algebra:

x

⋆

x

=

𝟎

.

(2)

Using the

distributivity

of “

⋆

” over “

+

” we see that the indentity (2) also implies (1):

𝟎

=

(

x

+

y

)

⋆

(

x

+

y

)

=

x

⋆

x

+

x

⋆

y

+

y

⋆

x

+

y

⋆

y

=

x

⋆

y

+

y

⋆

x

A well known example of anticommutative operations is the vector product in the algebra

(

ℝ

3

,

+

,

×

)

,  satisfying

b

→

×

a

→

=

-

(

a

→

×

b

→

)

,

a

→

×

a

→

=

0

→

.

Also we know that the subtraction of numbers obeys identities

b

-

a

=

-

(

a

-

b

)

,

a

-

a

=

0

.

An important anticommutative operation is the

Lie bracket

.

Title

anticommutative

Canonical name

Anticommutative

Date of creation

2014-02-04 7:50:58

Last modified on

2014-02-04 7:50:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 17A01

Synonym

anticommutative operation

Synonym

anticommutativity

Related topic

Supercommutative

Related topic

AlternativeAlgebra

Related topic

Subcommutative

Generated on Fri Feb 9 18:47:29 2018 by

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associator

associator

Let

A

be a

non-associative algebra

over a field. The

associator

of

A

, denoted by

[

,

,

]

, is a trilinear (

http://planetmath.org/multilinear

) map from

A

×

A

×

A

to

A

given by:

[

a

,

b

,

c

]

=

(

a

⁢

b

)

⁢

c

-

a

⁢

(

b

⁢

c

)

.

Just as the

commutator

measures how close an

algebra

is to being

commutative

, the associator measures how close it is to being

associative

.

[

,

,

]

=

0

identically iff

A

is associative.

References

1

R. D. Schafer,

An Introduction on

Nonassociative Algebras

, Dover, New York (1995).

Title

associator

Canonical name

Associator

Date of creation

2013-03-22 14:43:21

Last modified on

2013-03-22 14:43:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 17A01

Related topic

AlternativeAlgebra

Related topic

PowerAssociativeAlgebra

Related topic

FlexibleAlgebra

Related topic

Commutator

Defines

anti-associative

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