aarjaneiro/planetmath huggingface dataset:
grounded relation

grounded relation

A

grounded relation

over a

sequence

of sets is a mathematical object consisting of two

components

. The first component is a subset of the

cartesian product

taken over the given sequence of sets, which sets are called the

domains

of the

relation

. The second component is just the cartesian product itself.

For example, if

L

is a grounded relation over a

finite sequence

of sets,

X

1

,

…

,

X

k

, then

L

has the form

L

=

(

F

⁢

(

L

)

,

G

⁢

(

L

)

)

, where

F

⁢

(

L

)

⊆

G

⁢

(

L

)

=

X

1

×

…

×

X

k

.

1

Remarks

•

In various

language

that is used,

F

⁢

(

L

)

may be called the

figure

or the

graph

of

L

, while

G

⁢

(

L

)

may be called the

ground

of

L

.

•

The default

assumption

in almost all applications is that the domains of the grounded relation are

nonempty sets

, hence departures from this assumption need to be noted explicitly.

•

In many applications all relations are considered relative to explicitly specified grounds. In these settings it is conventional to refer to grounded relations somewhat more simply as “relations”.

•

One often hears or reads the usage

`

⁢

`

⁢

L

⊆

X

1

×

…

×

X

k

⁢

"

when the speaker or writer really means

`

⁢

`

⁢

F

⁢

(

L

)

⊆

X

1

×

…

×

X

k

⁢

"

. Be charitable in your

interpretations

.

•

The

cardinality

of

G

⁢

(

L

)

is referred to as the

adicity

or the

arity

of the relation. For example, in the finite case,

L

may be described as

k

-adic

or

k

-ary

.

•

The set

dom

j

⁢

(

L

)

:=

X

j

is referred to as the

j

th

domain

of the relation.

•

In the special case where

k

=

2

, the set

X

1

is called “the domain” and the set

X

2

is called “the

codomain

” of the relation.

Title

grounded relation

Canonical name

GroundedRelation

Date of creation

2013-03-22 17:48:38

Last modified on

2013-03-22 17:48:38

Owner

Jon Awbrey (15246)

Last modified by

Jon Awbrey (15246)

Numerical id

11

Author

Jon Awbrey (15246)

Entry type

Definition

Classification

msc 08A70

Classification

msc 08A02

Classification

msc 03G15

Classification

msc 03E20

Classification

msc 03C05

Classification

msc 03B10

Related topic

Relation

Generated on Fri Feb 9 14:04:01 2018 by

LaTeXML
binary operation

binary operation

A

binary operation

on a set

X

is a

function

from the

Cartesian product

X

×

X

to

X

. A binary operation is sometimes called

internal composition

.

Rather than using function notation, it is usual to write binary operations with an

operation

symbol between

elements

, or even with no operation at all, it being understood that juxtaposed elements are to be combined using an operation that should be clear from the

context

.

Thus,

addition

of real numbers is the operation

(

x

,

y

)

↦

x

+

y

,

and

multiplication

in a groupoid is the operation

(

x

,

y

)

↦

x

⁢

y

.

Title

binary operation

Canonical

name

BinaryOperation

Date of creation

2013-03-22 13:08:12

Last modified on

2013-03-22 13:08:12

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

7

Author

mclase (549)

Entry type

Definition

Classification

msc 08A99

Synonym

internal composition

Related topic

Arity

Related topic

Operation

Generated on Fri Feb 9 14:04:10 2018 by

LaTeXML
biops

biops

Let

S

be a set and

n

∈

𝐍

. Set

𝐍

n

:=

{

i

∈

𝐍

|

i

<

n

}

. If there exists a map

⋅

:

𝐍

n

→

(

S

2

→

S

)

:

i

↦

⋅

i

where

⋅

i

:

S

2

→

S

:

(

a

,

b

)

↦

a

⋅

i

b

is a

binary operation

, then I shall say that

(

S

,

⋅

)

is an

n

-biops

. In other words, an

n

-biops is an

algebraic system

with

n

binary operations defined on it, and the

operations

are labelled

0

,

1

,

…

,

n

-

1

.

Let

(

S

,

⋅

)

be an

n

-biops. If

⋅

has the

property

p

, then I shall say that

(

S

,

⋅

)

is a

p

n

-biops.

For example if

(

S

,

⋅

)

is an

n

-biops and

⋅

is

0

-

commutative

,

0

-

associative

,

0

-alternative or

(

0

,

1

)

-

distributive

, then I shall say that

(

S

,

⋅

)

is a

0

-commutative

n

-biops,

0

-associative

n

-biops,

0

-alternative

n

-biops or

(

0

,

1

)

-distributive

n

-biops respectively.

If an

n

-biops

B

is

i

-

p

for each

i

∈

𝐍

n

then I shall say that

B

is a

p

n

-biops.

A

0

-associative

1

-biops is called a

semigroup

.
A semigroup with

identity element

is called a monoid.
A monoid with

inverses

is called a group.

A

(

0

,

1

)

-distributive

2

-biops

(

S

,

+

,

⋅

)

, such that both

(

S

,

+

)

and

(

S

,

⋅

)

are monoids, is called a rig.

A

(

0

,

1

)

-distributive

2

-biops

(

S

,

+

,

⋅

)

, such that

(

S

,

+

)

is a group and

(

S

,

⋅

)

is a monoid, is called a ring.

A rig with

0

-inverses is a ring.

A

0

-associative

2

-biops

(

S

,

⋅

,

/

)

with

0

-

identity

such that for every

{

a

,

b

}

⊂

S

we have

b

=

(

b

/

a

)

⋅

a

=

(

b

⋅

a

)

/

a

is called a group.

A

3

-biops

(

S

,

⋅

,

/

,

\

)

such that for every

{

a

,

b

}

⊂

S

we have

a

\

(

a

⋅

b

)

=

a

⋅

(

a

\

b

)

=

b

=

(

b

/

a

)

⋅

a

=

(

b

⋅

a

)

/

a

is called a

quasigroup

.

A quasigroup such that for every

{

a

,

b

}

⊂

S

we have

a

/

a

=

b

\

b

is called a loop.

A

0

-associative loop is a group.

Title

biops

Canonical

name

Biops

Date of creation

2013-03-22 14:44:49

Last modified on

2013-03-22 14:44:49

Owner

HkBst (6197)

Last modified by

HkBst (6197)

Numerical id

7

Author

HkBst (6197)

Entry type

Definition

Classification

msc 08A99

Defines

semigroup

Defines

monoid

Defines

group

Defines

rig

Defines

ring

Defines

quasigroup

Defines

loop

Generated on Fri Feb 9 14:04:12 2018 by

LaTeXML
division in group

division in group

In any group

(

G

,

⋅

)

one can introduce a

division

operation

‘‘:’’ by setting

x

:

y

=

x

⋅

y

-

1

for all

elements

x

,

y

of

G

.  On the contrary, the

group operation

and the unary

inverse

forming operation may be expressed via the division by

x

⋅

y

=

x

:

(

(

y

:

y

)

:

y

)

,

x

-

1

=

(

x

:

x

)

:

x

.

(1)

The division, which of course is not

associative

, has the

properties

1.

(

x

:

z

)

:

(

y

:

z

)

=

x

:

y

,

2.

x

:

(

y

:

y

)

=

x

,

3.

(

x

:

x

)

:

(

y

:

z

)

=

z

:

y

.

The above result may be conversed:

Theorem

.

If the operation ‘‘:’’ of the non-empty

groupoid

G

has the properties 1, 2, and 3, then

G

equipped with the ‘‘

multiplication

’’ and inverse forming by (1) is a group.

Proof.

Here we prove only the associativity of ‘‘

⋅

’’.  First we derive some auxiliary results.  Using

definitions

and the properties 1 and 2 we obtain

(

x

:

y

)

:

y

-

1

=

(

x

:

y

)

:

(

(

y

:

y

)

:

y

)

=

x

:

(

y

:

y

)

=

x

,

(

x

:

y

-

1

)

:

y

=

(

x

:

y

-

1

)

:

(

(

y

:

y

)

:

y

-

1

)

=

x

:

(

y

:

y

)

=

x

and using the property 3,

(

x

:

y

)

-

1

=

(

(

x

:

y

)

:

(

x

:

y

)

)

:

(

x

:

y

)

=

y

:

x

.

Then we get:

(

x

y

)

z

=

(

x

:

y

-

1

)

:

z

-

1

=

(

(

x

:

y

-

1

)

:

y

)

:

(

z

-

1

:

y

)

=

x

:

(

z

-

1

:

y

)

=

x

:

(

y

:

z

-

1

)

-

1

=

x

(

y

z

)

References

1

А. И. Мальцев:

Алгебраические
системы

.  Издательство
‘‘Наука’’. Москва (1970).

Title

division in group

Canonical

name

DivisionInGroup

Date of creation

2013-03-22 15:08:01

Last modified on

2013-03-22 15:08:01

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Theorem

Classification

msc 08A99

Classification

msc 20A05

Classification

msc 20-00

Related topic

Group

Related topic

Division

Related topic

Groupoid

Related topic

AlternativeDefinitionOfGroup

Defines

division groupoid

Generated on Fri Feb 9 14:04:23 2018 by

LaTeXML
filtered algebra

filtered algebra

Definition 1

.

A filtered algebra over the field

k

is an

algebra

(

A

,

⋅

)

over

k

which is endowed with a

filtration

ℱ

=

{

F

i

}

i

∈

ℕ

by

subspaces

,

compatible

with the

multiplication

in
the following sense

∀

m

,

n

∈

ℕ

,

F

m

⋅

F

n

⊂

F

n

+

m

.

A special case of filtered algebra is a

graded algebra

. In general there is
the following construction that produces a graded algebra out of a filtered
algebra.

Definition 2

.

Let

(

A

,

⋅

,

ℱ

)

be a filtered algebra then the

associated

http://planetmath.org/node/3071

graded algebra

𝒢

⁢

(

A

)

is defined as follows:

•

As a

vector space

𝒢

⁢

(

A

)

=

⊕

n

∈

ℕ

G

n

,

where,

G

0

=

F

0

,

and

⁢

∀

n

>

0

,

G

n

=

F

n

/

F

n

-

1

,

•

the multiplication is defined by

(

x

+

F

n

)

⁢

(

y

+

F

m

)

=

x

⋅

y

+

F

n

+

m

Theorem 3

.

The multiplication is well defined and endows

G

⁢

(

A

)

with the
of a graded algebra, with gradation

{

G

n

}

n

∈

N

.
Furthermore if

A

is associative then so is

G

⁢

(

A

)

.

An example of a filtered algebra is the

Clifford algebra

Cliff

⁢

(

V

,

q

)

of a vector space

V

endowed with a

quadratic form

q

. The associated
graded algebra is

⋀

V

, the

exterior algebra

of

V

.

As algebras

A

and

𝒢

⁢

(

A

)

are distinct (with the exception of the

trivial

case that

A

is graded) but as vector spaces they are

isomorphic

.

Theorem 4

.

The underlying vector spaces of

A

and

G

⁢

(

A

)

are isomorphic.

Title

filtered algebra

Canonical name

FilteredAlgebra

Date of creation

2013-03-22 13:23:55

Last modified on

2013-03-22 13:23:55

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

11

Author

Dr_Absentius (537)

Entry type

Definition

Classification

msc 08A99

Generated on Fri Feb 9 14:04:25 2018 by

LaTeXML
opposite group

opposite group

Let

G

be a group under the

operation

*

. The

opposite group

of

G

, denoted

G

op

, has the same underlying set as

G

, and its

group operation

is

*

′

defined by

g

1

*

′

g

2

=

g

2

*

g

1

.

If

G

is

abelian

, then it is equal to its opposite group. Also, every group

G

(not necessarily abelian) is

isomorphic

to its opposite group: The

isomorphism

(

http://planetmath.org/GroupIsomorphism

)

φ

:

G

→

G

op

is given by

φ

⁢

(

x

)

=

x

-

1

. More generally, any

anti-automorphism

ψ

:

G

→

G

gives rise to a corresponding isomorphism

ψ

′

:

G

→

G

op

via

ψ

′

⁢

(

g

)

=

ψ

⁢

(

g

)

, since

ψ

′

⁢

(

g

*

h

)

=

ψ

⁢

(

g

*

h

)

=

ψ

⁢

(

h

)

*

ψ

⁢

(

g

)

=

ψ

⁢

(

g

)

*

′

ψ

⁢

(

h

)

=

ψ

′

⁢

(

g

)

*

′

ψ

′

⁢

(

h

)

.

Opposite groups are useful for converting a

right action

to a left action and vice versa. For example, if

G

is a group that acts on

X

on the , then a left action of

G

op

on

X

can be defined by

g

op

⁢

x

=

x

⁢

g

.

constructions

occur in

opposite ring and opposite

category

.

Title

opposite group

Canonical name

OppositeGroup

Date of creation

2013-03-22 17:09:56

Last modified on

2013-03-22 17:09:56

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 08A99

Classification

msc 20-00

Generated on Fri Feb 9 14:04:55 2018 by

LaTeXML
quandles

quandles

Quandles

are algebraic gadgets introduced by David Joyce in

[

1

]

which can be used to define invarients of links. In the case of knots these invarients are

complete

up to equivalence, that is up to

mirror images

.

Definition 1

A

quandle

is an

algebraic structure

, specifically it is a set

Q

with two

binary operations

on it,

⊲

and

⊲

-

1

and the following axioms.

1.

q

⊲

q

=

q

⁢

∀

q

∈

Q

2.

(

q

1

⊲

q

2

)

⊲

-

1

q

2

=

(

q

1

⊲

-

1

q

2

)

⊲

q

2

⁢

∀

q

1

,

q

2

∈

Q

3.

(

q

1

⊲

q

2

)

⊲

q

3

=

(

q

1

⊲

q

3

)

⊲

(

q

2

⊲

q

3

)

⁢

∀

q

1

,

q

2

,

q

3

∈

Q

It is useful to consider

q

1

⊲

q

2

as ’

q

2

acting on

q

1

’.

Examples.

1.

Let

Q

be some group, and let

n

be some fixed integer.
Then let

g

1

⊲

g

2

=

g

2

-

n

⁢

g

1

⁢

g

2

n

,

g

1

⊲

-

1

g

2

=

g

2

n

⁢

g

1

⁢

g

2

-

n

.

2.

Let

Q

be some group. Then let

g

1

⊲

g

2

=

g

1

⊲

-

1

g

2

=

g

2

⁢

g

1

-

1

⁢

g

2

.

3.

Let

Q

be some module, and

T

some invertable

linear operator

on

Q

. Then let

m

1

⊲

m

2

=

T

⁢

(

m

1

-

m

2

)

+

m

2

,

m

1

⊲

-

1

m

2

=

T

-

1

⁢

(

m

1

-

m

2

)

+

m

2

Homomorphisms

,

isomorphisms

etc. are defined in the obvious way.
Notice that the third axiom gives us that the

operation

of a quandle

element

on the quandle given by

f

q

:

q

′

↦

q

′

⊲

q

is a homomorphism, and the second axiom ensures that this is an isomorphism.

Definition 2

The

subgroup

of the

automorphism group

of a quandle

Q

generated by the quandle operations is the

operator group

of

Q

.

References

1

D.Joyce : A Classifying

Invariant

Of Knots, The Knot
Quandle : J.P.App.Alg 23 (1982) 37-65

Title

quandles

Canonical name

Quandles

Date of creation

2013-03-22 16:42:37

Last modified on

2013-03-22 16:42:37

Owner

StevieHair (1420)

Last modified by

StevieHair (1420)

Numerical id

8

Author

StevieHair (1420)

Entry type

Definition

Classification

msc 08A99

Generated on Fri Feb 9 14:04:58 2018 by

LaTeXML
special notations in algebra

special notations in algebra

•

[

G

:

H

]

,   index (

http://planetmath.org/Coset

) of a

subgroup

H

in the group

G

•

H

◁

G

,

H

is a

normal subgroup

of

G

•

G

/

H

,

quotient group

of

G

with respect to the normal subgroup

H

•

R

/

I

or

R

-

I

,

quotient ring

or difference ring of

R

with respect to the ideal

I

•

Z

⁡

(

G

)

,   center of the group

G

•

R

⁢

[

G

]

,

group ring

of the group

G

over the ring

R

•

R

⁢

[

α

]

,

ring adjunction

•

K

⁢

(

α

)

,

field adjunction

•

S

-

1

⁢

R

,

localization

of the ring

R

at

S

•

R

𝔭

,   localization of the ring

R

at the complement of the

prime ideal

𝔭

•

(

R

,

𝔪

)

,

local ring

R

with its

maximal ideal

𝔪

•

M

R

,   a left

R

-module (

http://planetmath.org/Module

)

M

•

M

R

,   a right

R

-module (

http://planetmath.org/Module

)

M

•

K

/

k

,

field extension

where

K

is an

extension field

of

k

•

[

K

:

k

]

,   degree (

http://planetmath.org/ExtensionField

) of the field extension

K

/

k

•

K

¯

,

algebraic closure

of a field

K

•

R

⁢

α

1

+

…

+

R

⁢

α

n

or

⟨

α

1

,

…

,

α

n

⟩

,

left ideal

of the ring

R

generated by (

http://planetmath.org/IdealGeneratedByASet

)

α

1

, …,

α

n

•

A

1

≅

A

2

,

isomorphism

(

http://planetmath.org/Isomorphism

) between the systems (

http://planetmath.org/AlgebraicSystem

)

A

1

and

A

2

Title

special notations in algebra

Canonical name

SpecialNotationsInAlgebra

Date of creation

2013-03-22 15:11:50

Last modified on

2013-03-22 15:11:50

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

22

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 08A99

Synonym

notations in algebra

Generated on Fri Feb 9 14:05:00 2018 by

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algebraic categories without free objects

algebraic categories without free objects

An

initial object

is always a free object. So in the

context

of

algebraic
systems

with a

trivial

object, such as groups, or modules, there is always
at least one free object. However, we usually dismiss this example as it
does not lead to any useful results such as the

existence

of

presentations

.

However, there are many ways in which a cateogry of

algebraic

objects can
fail to include non-trivial free objects.

1

Restriction to finite sets

The

restriction

of a

category

which naturally includes

infinite

objects
can often be restricted to just the finite objects and in so doing often
remove all non-trivial free objects.

•

The category of

finite groups

has only the trivial free object. Indeed,
even the rank 1

free group

, the integers

ℤ

is already infinite.

•

Similarly, finite modules of in a module category over an infinite
ring are never free. For examples use the rings

ℤ

,

ℤ

m

⁢

[

x

]

,
etc.

However, this is not always the case. For example, if we consider finite

ℤ

p

-modules (

vector spaces

) each of these are free.

2

Homomorphism restrictions

In the category of rings with 1 it is often beneficial to force all

ring
homomorphisms

to be unital. However, this restriction can prevent
the construction of free objects.

Suppose

F

is a free ring in the category of rings with positive characteristic. Then we ask, what is the

characteristic

of

F

?

If it is

m

>

0

then we choose another ring

R

of a different characteristic,
a characteristic relatively prime to

m

, and then there can be no
unital

homomorphism

from

F

to

R

. So

F

must have characteristic 0.
In contrast to the above examples we have not excluded infinite objects
in this restriction. This example is even more powerful than those
above as it also exclude the existance of an initial object, so indeed
NO free objects exist in this category.

If we return to the full category of

unital rings

we observe
every ring is a

ℤ

-

algebra

we can use the

free associative
algebras

ℤ

⁢

⟨

X

⟩

does exist here.

Title

algebraic categories without free objects

Canonical name

AlgebraicCategoriesWithoutFreeObjects

Date of creation

2013-03-22 16:51:25

Last modified on

2013-03-22 16:51:25

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

4

Author

Algeboy (12884)

Entry type

Example

Classification

msc 08B20

Generated on Fri Feb 9 14:05:25 2018 by

LaTeXML
free algebra

free algebra

Let

𝒦

be a class of

algebraic systems

(of the same type

τ

). Consider an

algebra

A

∈

𝒦

generated by (

http://planetmath.org/SubalgebraOfAnAlgebraicSystem

) a set

X

=

{

x

i

}

indexed by

i

∈

I

.

A

is said to be a

free algebra

over

𝒦

, with

free generating set

X

, if for any algebra

B

∈

𝒦

with any subset

{

y

i

∣

i

∈

I

}

⊆

B

, there is a

homomorphism

ϕ

:

A

→

B

such that

ϕ

⁢

(

x

i

)

=

y

i

.

If we define

f

:

I

→

A

to be

f

⁢

(

i

)

=

x

i

and

g

:

I

→

B

to be

g

⁢

(

i

)

=

y

i

, then freeness of

A

means the

existence

of

ϕ

:

A

→

B

such that

ϕ

∘

f

=

g

.

Note that

ϕ

above is necessarily unique, since

{

x

i

}

generates

A

. For any

n

-ary

polynomial

p

over

A

, any

z

1

,

…

,

z

n

∈

{

x

i

∣

i

∈

I

}

,

ϕ

⁢

(

p

⁢

(

z

1

,

…

,

z

n

)

)

=

p

⁢

(

ϕ

⁢

(

z

1

)

,

…

,

ϕ

⁢

(

z

n

)

)

.

For example, any

free group

is a free algebra in the class of groups. In general, however, free algebras do not always exist in an arbitrary

class of algebras

.

Remarks

.

•

A

is free over itself (meaning

𝒦

consists of

A

only) iff

A

is free over some

equational class

.

•

If

𝒦

is an equational class, then free algebras exist in

𝒦

.

•

Any

term algebra

of a given

structure

τ

over some set

X

of

variables

is a free algebra with free generating set

X

.

Title

free algebra

Canonical name

FreeAlgebra

Date of creation

2013-03-22 16:51:05

Last modified on

2013-03-22 16:51:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08B20

Synonym

free algebraic system

Related topic

TermAlgebra

Defines

free generating set

Generated on Fri Feb 9 14:05:34 2018 by

LaTeXML
free commutative algebra

free commutative algebra

Fix a

commutative

unital ring

K

and a set

X

. Then a commutative

associative

K

-

algebra

F

is said to be

free on

X

if there exists an injection

ι

:

X

→

F

such that for all functions

f

:

X

→

A

where

A

is a commutative

K

-algebra determine a unique algebra homomorphism

f

^

:

F

→

A

such that

ι

⁢

f

^

=

f

. This is an example of a

universal mapping property

for commutative

associative algebras

and in

categorical

settings is often explained with the following

commutative diagram

:

\xymatrix

⁢

&

⁢

X

⁢

\ar

⁢

[

l

⁢

d

]

ι

⁢

\ar

⁢

[

r

⁢

d

]

f

⁢

&

⁢

F

⁢

\ar

⁢

[

r

⁢

r

]

f

^

⁢

&

⁢

&

⁢

A

.

To

construct

a free commutative associative algebra we observe that commutative
associative algebras are a

subcategory

of associative algebras and thus we
can make use of

free associative algebras

in the construction and proof.

Theorem 1

.

Given a set

X

, and a commutative unital ring

K

, the free commutative associative

K

-algebra on

X

is the

polynomial ring

K

⁢

[

X

]

.

Proof.

Let

A

be any commutative associative

K

-algebra and

f

:

X

→

A

.
Recall

K

⁢

⟨

X

⟩

is the free associative

K

-algebra on

X

and
so by the universal mapping property of this free object there exists a
map

f

^

:

K

⁢

⟨

X

⟩

→

A

such that

ι

K

⁢

⟨

X

⟩

⁢

f

^

=

f

.

We also have a map

p

:

K

⁢

⟨

X

⟩

→

K

⁢

[

X

]

which effectively
maps words over

X

to words over

X

. Only in

K

⁢

[

X

]

the
indeterminants commute. Since

A

is commutative,

f

^

factors through

p

, in the sense that there exists a map

f

~

:

K

⁢

[

X

]

→

A

such
that

p

⁢

f

~

=

f

^

. Thus

f

~

is the desired map which proves

K

⁢

[

X

]

is free in the

category

of commutative
associative algebras.
∎

Title

free commutative algebra

Canonical name

FreeCommutativeAlgebra

Date of creation

2013-03-22 16:51:22

Last modified on

2013-03-22 16:51:22

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

5

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 08B20

Related topic

PolynomialRing

Defines

free commutative algebra

Generated on Fri Feb 9 14:05:39 2018 by

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free Lie algebra

free Lie algebra

Fix a set

X

and a commuative

unital ring

K

. A free

K

-

Lie algebra

𝔏

on

X

is any Lie algebra together with an injection

ι

:

X

→

𝔏

such that for any

K

-Lie algebra

𝔤

and function

f

:

X

→

𝔤

implies
the existance of a unique Lie algebra

homomorphism

f

^

:

𝔏

→

𝔤

where

ι

⁢

f

^

=

f

. This

universal mapping property

is commonly expressed
as a

commutative diagram

:

\xymatrix

⁢

&

⁢

X

⁢

\ar

⁢

[

l

⁢

d

]

ι

⁢

\ar

⁢

[

r

⁢

d

]

f

⁢

&

⁢

𝔏

⁢

\ar

⁢

[

r

⁢

r

]

f

^

⁢

&

⁢

&

⁢

𝔤

.

To

construct

a

free Lie algebra

is generally and indirect process.
We begin with any

free associative algebra

K

⁢

⟨

X

⟩

on

X

,
which can be constructed as the tensor algebra over a free

K

-module
with basis

X

. Then

K

⁢

⟨

X

⟩

-

is a

K

-Lie algebra with the
standard

commutator bracket

[

a

,

b

]

=

a

⁢

b

-

b

⁢

a

for

a

,

b

∈

K

⁢

⟨

X

⟩

.

Now define

𝔉

⁢

𝔏

K

⁢

⟨

X

⟩

as the

Lie subalgebra

of

K

⁢

⟨

X

⟩

-

generated by

X

.

Theorem 1

(Witt).

[

1

, Thm V.7]

F

⁢

L

K

⁢

⟨

X

⟩

is a free Lie algebra on

X

and its

universal
enveloping algebra

is

K

⁢

⟨

X

⟩

.

It is generally not true that

𝔉

⁢

𝔏

K

⁢

⟨

X

⟩

=

K

⁢

⟨

X

⟩

-

. For example, if

x

∈

X

then

x

2

∈

K

⁢

⟨

X

⟩

but

x

2

is not in

𝔉

⁢

𝔏

K

⁢

⟨

X

⟩

.

References

1

Nathan Jacobson

Lie Algebras

, Interscience Publishers, New York, 1962.

Title

free Lie algebra

Canonical name

FreeLieAlgebra

Date of creation

2013-03-22 16:51:11

Last modified on

2013-03-22 16:51:11

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

5

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 08B20

Related topic

LieAlgebra

Related topic

UniversalEnvelopingAlgebra

Related topic

PoincareBirkhoffWittTheorem

Defines

free Lie algebra

Generated on Fri Feb 9 14:05:41 2018 by

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