aarjaneiro/planetmath huggingface dataset:
bialgebra

bialgebra

A

bialgebra

is a

vector space

that is both a unital

algebra

and a coalgebra, such that the comultiplication and counit are unital algebra homomorphisms.

A bialgebra

homomorphism

is a linear map that is both an algebra and a

coalgebra homomorphism

.

Title

bialgebra

Canonical name

Bialgebra

Date of creation

2013-03-22 13:06:05

Last modified on

2013-03-22 13:06:05

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

6

Author

mhale (572)

Entry type

Definition

Classification

msc 16W30

Related topic

QuantumDouble

Generated on Fri Feb 9 18:44:45 2018 by

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coalgebra

coalgebra

A

coalgebra

is a

vector space

A

over a field

𝕂

with a

𝕂

-

linear map

Δ

:

A

→

A

⊗

A

,
called the

comultiplication

,
and a (non-zero)

𝕂

-linear map

ε

:

A

→

𝕂

,
called the

counit

, such that

(

Δ

⊗

id

)

∘

Δ

=

(

id

⊗

Δ

)

∘

Δ

(coassociativity)

,

(

ε

⊗

id

)

∘

Δ

=

id

=

(

id

⊗

ε

)

∘

Δ

.

In of

commutative diagrams

:

Generated on Fri Feb 9 18:44:51 2018 by

LaTeXML
coalgebra homomorphism

coalgebra homomorphism

Let

(

C

,

Δ

,

ε

)

and

(

D

,

Δ

′

,

ε

′

)

be coalgebras.

Definition.

Linear map

f

:

C

→

D

is called

coalgebra homomorphism

if

Δ

′

∘

f

=

(

f

⊗

f

)

∘

Δ

and

ε

′

∘

f

=

ε

.

Examples.

1

)

Of course, if

D

is a subcoalgebra of

C

, then the

inclusion

i

:

D

→

C

is a coalgebra homomorphism. In particular, the

identity

is a coalgebra homomorphism.

2

)

If

(

C

,

Δ

,

ε

)

is a coalgebra and

I

⊆

C

is a coideal, then we have canonical coalgebra structur on

C

/

I

(please, see this entry (

http://planetmath.org/SubcoalgebrasAndCoideals

) for more details). Then the

projection

π

:

C

→

C

/

I

is a coalgebra homomorphism. Furthermore, one can show that the canonical coalgebra

structure

on

C

/

I

is a unique coalgebra structure such that

π

is a coalgebra homomorphism.

Title

coalgebra homomorphism

Canonical name

CoalgebraHomomorphism

Date of creation

2013-03-22 18:49:25

Last modified on

2013-03-22 18:49:25

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 18:44:53 2018 by

LaTeXML
coalgebra isomorphisms and isomorphic coalgebras

coalgebra isomorphisms and isomorphic coalgebras

Let

(

C

,

Δ

,

ε

)

and

(

D

,

Δ

′

,

ε

′

)

be coalgebras.

Definition

.

We will say that

coalgebra homomorphism

f

:

C

→

D

is a

coalgebra

isomorphism

, if there exists a coalgebra homomorphism

g

:

D

→

C

such that

f

∘

g

=

id

D

and

g

∘

f

=

id

C

.

Remark.

Of course every coalgebra isomorphism is a linear isomorphism, thus it is ,,

one-to-one

” and ,,onto”. One can show that the

converse

also holds, i.e. if

f

:

C

→

D

is a coalgebra homomorphism such that

f

is ,,one-to-one” and ,,onto”, then

f

is a coalgebra isomorphism.

Definition.

We will say that coalgebras

(

C

,

Δ

,

ε

)

and

(

D

,

Δ

′

,

ε

′

)

are

isomorphic

if there exists coalgebra isomorphism

f

:

C

→

D

. In this case we often write

(

C

,

Δ

,

ε

)

≃

(

D

,

Δ

′

,

ε

′

)

or simply

C

≃

D

if

structure maps

are known from the

context

.

Remarks.

Of course the

relation

,,

≃

” is an

equivalence relation

. Furthermore, (from the coalgebraic point of view) isomorphic coalgebras are the same, i.e. they share all coalgebraic

properties

.

Title

coalgebra isomorphisms and isomorphic coalgebras

Canonical

name

CoalgebraIsomorphismsAndIsomorphicCoalgebras

Date of creation

2013-03-22 18:49:28

Last modified on

2013-03-22 18:49:28

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 18:44:55 2018 by

LaTeXML
coinvariant

coinvariant

Let

V

be a comodule with a right coaction

t

:

V

→

V

⊗

A

of a coalgebra

A

.
An element

v

∈

V

is

right coinvariant

if

t

⁢

(

v

)

=

v

⊗

1

⁢

I

A

.

(1)

The set of coinvariants of

A

is a sub-comodule with the trivial coaction of

A

.
The sub-comodule of right (or left) coinvariants of

V

is sometimes denoted by

V

co

⁢

A

(or

V

co

⁢

A

).

Title

coinvariant

Canonical

name

Coinvariant

Date of creation

2013-03-22 13:57:40

Last modified on

2013-03-22 13:57:40

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

5

Author

mhale (572)

Entry type

Definition

Classification

msc 16W30

Generated on Fri Feb 9 18:44:56 2018 by

LaTeXML
comodule

comodule

Let

(

A

,

Δ

,

ε

)

be a coalgebra.
A

right

A

-comodule

is a

vector space

V

with a

linear map

t

:

V

→

V

⊗

A

, called the

right coaction

, satisfying

(

t

⊗

id

)

∘

t

=

(

id

⊗

Δ

)

∘

t

,

(

id

⊗

ε

)

∘

t

=

id

.

(1)

An

A

-comodule is also referred to as a corepresentation of

A

.

In of

commutative diagrams

:

Generated on Fri Feb 9 18:45:03 2018 by

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comodule algebra

comodule algebra

Let

H

be a

bialgebra

.
A

right

H

-

comodule algebra

is a unital

algebra

A

which is a right

H

-comodule
satisfying

t

⁢

(

a

⁢

b

)

=

t

⁢

(

a

)

⁢

t

⁢

(

b

)

=

∑

a

(

1

)

⁢

b

(

1

)

⊗

a

(

2

)

⁢

b

(

2

)

,

t

⁢

(

1

⁢

I

A

)

=

1

⁢

I

A

⊗

1

⁢

I

H

,

(1)

for all

h

∈

H

and

a

,

b

∈

A

.

There is a dual notion of a

H

-

module coalgebra

.

Example 1

Let

H

be a bialgebra.
Then

H

is itself a

H

-comodule algebra for the right

regular

coaction

t

⁢

(

h

)

=

Δ

⁢

(

h

)

.

Title

comodule algebra

Canonical

name

ComoduleAlgebra

Date of creation

2013-03-22 13:26:34

Last modified on

2013-03-22 13:26:34

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

8

Author

mhale (572)

Entry type

Definition

Classification

msc 16W30

Related topic

ModuleCoalgebra

Related topic

ModuleAlgebra

Related topic

ComoduleCoalgebra

Generated on Fri Feb 9 18:45:04 2018 by

LaTeXML
comodule coalgebra

comodule coalgebra

Let

H

be a

bialgebra

.
A

right

H

-

comodule coalgebra

is a coalgebra

A

which is a right

H

-comodule
satisfying

(

Δ

⊗

id

)

⁢

t

⁢

(

a

)

=

∑

a

(

1

)

⁢

(

1

)

⊗

a

(

2

)

⁢

(

1

)

⊗

a

(

1

)

⁢

(

2

)

⁢

a

(

2

)

⁢

(

2

)

,

(

ε

⊗

id

)

⁢

t

⁢

(

a

)

=

ε

⁢

(

a

)

⁢

1

⁢

I

H

,

(1)

for all

h

∈

H

and

a

∈

A

.

There is a dual notion of a

H

-

module algebra

.

Example 1

Let

H

be a

Hopf algebra

.
Then

H

is itself a

H

-comodule coalgebra for the

adjoint

coaction

t

⁢

(

h

)

=

h

(

2

)

⊗

S

⁢

(

h

(

1

)

)

⁢

h

(

3

)

.

Title

comodule coalgebra

Canonical name

ComoduleCoalgebra

Date of creation

2013-03-22 13:26:39

Last modified on

2013-03-22 13:26:39

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

7

Author

mhale (572)

Entry type

Definition

Classification

msc 16W30

Related topic

ModuleAlgebra

Related topic

ModuleCoalgebra

Related topic

ComoduleAlgebra

Generated on Fri Feb 9 18:45:06 2018 by

LaTeXML
example of algebras and coalgebras which cannot be turned into Hopf algebras

example of algebras and coalgebras which cannot be turned into Hopf algebras

Let

H

=

𝕄

n

⁢

(

k

)

be a matrix algebra over a field

k

with standard

multiplication

and assume that

n

>

1

. Assume that

H

can be turned into a

Hopf algebra

. In particular, there is

ε

:

H

→

k

such that

ε

is a

morphism

of

algebras

. It can be shown that matrix algebra is simple, i.e. if

I

⊆

H

is a

two-sided ideal

, then

I

=

0

or

I

=

H

. Thus we have that

ker

⁢

ε

=

0

(because

ε

⁢

(

1

)

=

1

).

Contradiction

, because

dim

k

⁢

H

>

1

=

dim

k

⁢

k

.

Now consider

H

=

𝕄

c

⁢

(

n

,

k

)

a

vector space

of all

n

×

n

matrices over

k

. We introduce coalgebra

structure

on

H

. Let

E

i

⁢

j

be a matrix in

H

with

1

in

(

i

,

j

)

place and

0

everywhere else. Of course

{

E

i

⁢

j

}

forms a basis of

H

and it is

sufficient

to define comultiplication and counit on it. Define

Δ

⁢

(

E

i

⁢

j

)

=

∑

p

=

1

n

E

i

⁢

p

⊗

E

p

⁢

j

;

ε

⁢

(

E

i

⁢

j

)

=

δ

i

⁢

j

,

where

δ

i

⁢

j

denotes

Kronecker delta

. It can be easily checked, that

(

𝕄

c

⁢

(

n

,

k

)

,

Δ

,

ε

)

is a coalgebra known as the matrix coalgebra. Also, is well known that the dual algebra

𝕄

c

⁢

(

n

,

k

)

*

is

isomorphic

to the standard matrix algebra.

Now assume that matrix coalgebra

H

=

𝕄

c

⁢

(

n

,

k

)

(where

n

>

1

) can be turned into a Hopf algebra. Since

H

is finite dimensional, then we can take dual Hopf algebra

H

*

. But the underlaying algebra structure of

H

*

is isomorphic to a matrix algebra (as we remarked earlier), which we’ve already shown to be impossible. Thus matrix coalgebra cannot be turned into a Hopf algebra.

Title

example of algebras and coalgebras which cannot be turned into Hopf algebras

Canonical

name

ExampleOfAlgebrasAndCoalgebrasWhichCannotBeTurnedIntoHopfAlgebras

Date of creation

2013-03-22 18:58:45

Last modified on

2013-03-22 18:58:45

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Example

Classification

msc 16W30

Generated on Fri Feb 9 18:45:11 2018 by

LaTeXML
Faà di Bruno’s formula

Faà di Bruno’s formula

Faà di Bruno’s

formula

is a

generalization

of the

chain rule

to

higher order derivatives

which expresses the

derivative

of a

composition

of functions as a series of

products

of derivatives:

d

n

d

⁢

x

n

⁢

f

⁢

(

g

⁢

(

x

)

)

=

∑

∑

k

=

0

n

k

⁢

m

k

=

n

n

!

m

1

!

⁢

m

2

!

⁢

m

3

!

⁢

⋯

⁢

1

!

m

1

⁢

2

!

m

2

⁢

3

!

m

3

⁢

⋯

⁢

f

(

m

1

+

⋯

+

m

n

)

⁢

(

g

⁢

(

x

)

)

⁢

∏

j

:

m

j

≠

0

(

g

(

j

)

⁢

(

x

)

)

m

j

This formula was discovered by Francesco Faà di Bruno in the 1850s and can
be proved by

induction

on the order of the derivative.

References

1

Faà di Bruno, C. F.. “Sullo sviluppo delle funzione.”

Ann. di
Scienze Matem. et Fisiche di Tortoloni

6

(1855): 479-480

2

Faà di Bruno, C. F.. “Note sur un nouvelle formule de calcul différentiel.”

Quart. J. Math.

1

(1857): 359-360

3

H. Figueroa & J. M. Gracia-Bondía, “Combinatorial Hopf Algebras in Quantum Field Theory I”

Rev. Math. Phys.

17

(2005): 881 - 975

Title

Faà di Bruno’s formula

Canonical

name

FaaDiBrunosFormula

Date of creation

2013-03-22 16:38:57

Last modified on

2013-03-22 16:38:57

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 16W30

Synonym

Faa di Bruno’s formula

Synonym

Faà di Bruno formula

Synonym

Faa di Bruno formula

Generated on Fri Feb 9 18:45:13 2018 by

LaTeXML
fundamental isomorphism theorem for coalgebras

fundamental isomorphism theorem for coalgebras

Let

(

C

,

Δ

,

ε

)

and

(

D

,

Δ

′

,

ε

′

)

be coalgebras. Recall, that if

D

0

⊆

D

is a subcoalgebra, then

(

D

0

,

Δ

|

D

0

′

,

ε

|

D

0

′

)

is a coalgebra. On the other hand, if

I

⊆

C

is a coideal, then there is a canonical coalgebra

structure

on

C

/

I

(please, see this entry (

http://planetmath.org/SubcoalgebrasAndCoideals

) for more details).

Theorem

.

If

f

:

C

→

D

is a

coalgebra homomorphism

, then

ker

⁢

(

f

)

is a coideal,

im

⁢

(

f

)

is a subcoalgebra and a

mapping

f

′

:

C

/

ker

⁢

(

f

)

→

im

⁢

(

f

)

defined by

f

′

⁢

(

c

+

ker

⁢

(

f

)

)

=

f

⁢

(

c

)

is a well defined coalgebra

isomorphism

.

Title

fundamental isomorphism theorem for coalgebras

Canonical name

FundamentalIsomorphismTheoremForCoalgebras

Date of creation

2013-03-22 18:49:30

Last modified on

2013-03-22 18:49:30

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 16W30

Generated on Fri Feb 9 18:45:15 2018 by

LaTeXML
fundamental theorem of coalgebras

fundamental theorem of coalgebras

Fundamental Theorem of Coalgebras

.

Let

(

C

,

Δ

,

ε

)

be a coalgebra over a field

k

and

x

∈

C

. Then there exists subcoalgebra

D

⊆

C

such that

x

∈

D

and

dim

k

⁢

D

<

∞

.

Proof.

Let

Δ

⁢

(

x

)

=

∑

i

b

i

⊗

c

i

.

Consider the

element

Δ

2

⁢

(

x

)

=

∑

i

Δ

⁢

(

b

i

)

⊗

c

i

=

∑

i

,

j

a

j

⊗

b

i

⁢

j

⊗

c

i

.

Note that we may assume that

(

a

j

)

are

linearly independent

and so are

(

c

i

)

. Let

D

be a

subspace

spanned by

(

b

i

⁢

j

)

. Of course

dim

k

⁢

D

<

∞

. Furthermore

x

∈

D

, because

x

=

∑

i

,

j

ε

⁢

(

a

j

)

⁢

ε

⁢

(

c

i

)

⁢

b

i

⁢

j

.

We will show that

D

is a subcoalgebra, i.e.

Δ

⁢

(

D

)

⊆

D

⊗

D

. Indeed, note that

∑

i

,

j

Δ

⁢

(

a

j

)

⊗

b

i

⁢

j

⊗

c

i

=

∑

i

,

j

a

j

⊗

Δ

⁢

(

b

i

⁢

j

)

⊗

c

i

and since

c

i

are linearly independent we obtain that

∑

j

Δ

⁢

(

a

j

)

⊗

b

i

⁢

j

=

∑

j

a

j

⊗

Δ

⁢

(

b

i

⁢

j

)

for all

i

. Thus

∑

j

a

j

⊗

Δ

⁢

(

b

i

⁢

j

)

∈

C

⊗

C

⊗

D

and since

a

j

are linearly independent, we obtain that

Δ

⁢

(

b

i

⁢

j

)

∈

C

⊗

D

for all

i

,

j

. Analogously we show that

Δ

⁢

(

b

i

⁢

j

)

∈

D

⊗

C

, thus

Δ

⁢

(

b

i

⁢

j

)

∈

C

⊗

D

∩

D

⊗

C

=

D

⊗

D

,

(please, see this entry (

http://planetmath.org/TensorProductOfSubspacesOfVectorSpaces

) for last equality) which

completes

the proof.

□

Remark.

The category of finite dimensional coalgebras is dual to the category of finite dimensional

algebras

(via

dual space

functor), so one could think that generally they are

similar

. Unfortunetly Fundamental Theorem of Coalgebras is major diffrence between algebras and coalgebras. For example consider a field

k

and its

polynomial algebra

k

⁢

[

X

]

. Then whenever

f

∈

k

⁢

[

X

]

is such that

deg

⁢

(

f

)

>

0

, then a

subalgebra generated by

f

is always infinite dimensional (if

deg

⁢

(

f

)

=

0

then subalgebra generated by

f

is

k

). This can never

occur in

coalgebras.

Title

fundamental theorem of coalgebras

Canonical name

FundamentalTheoremOfCoalgebras

Date of creation

2013-03-22 18:49:22

Last modified on

2013-03-22 18:49:22

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 18:45:18 2018 by

LaTeXML
grouplike elements

grouplike elements

Let

(

C

,

Δ

,

ε

)

be a coalgebra over a field

k

.

Definition

.

The element

g

∈

C

is called

grouplike

iff

g

≠

0

and

Δ

⁢

(

g

)

=

g

⊗

g

. The set of all

grouplike elements

in a coalgebra

C

is denoted by

G

⁢

(

C

)

.

Properties

.

0

)

The set

G

⁢

(

C

)

can be empty, but (for example) if

C

can be turned into a

bialgebra

, then

G

⁢

(

C

)

≠

∅

. In particular

Hopf algebras

always have grouplike elements.

1

)

If

g

∈

G

⁢

(

C

)

, then it follows from the counit property that

ε

⁢

(

g

)

=

1

.

2

)

It can be shown that the set

G

⁢

(

C

)

is

linearly independent

.

Title

grouplike elements

Canonical

name

GrouplikeElements

Date of creation

2013-03-22 18:58:37

Last modified on

2013-03-22 18:58:37

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Definition

Classification

msc 16W30

Generated on Fri Feb 9 18:45:20 2018 by

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grouplike elements in Hopf algebras

grouplike elements in Hopf algebras

Recall, that if

k

is a field and

G

is a group, then the

group algebra

k

⁢

G

can be turned into a

Hopf algebra

, by defining comultiplication

Δ

⁢

(

g

)

=

g

⊗

g

, counit

ε

⁢

(

g

)

=

1

and antipode

S

⁢

(

g

)

=

g

-

1

.

Now let

H

be a Hopf algebra over a field

k

, with

identity

1

, comultiplication

Δ

, counit

ε

and antipode

S

. Recall that element

g

∈

H

is called grouplike iff

g

≠

0

and

Δ

⁢

(

g

)

=

g

⊗

g

. The set of all

grouplike elements

G

⁢

(

H

)

is nonempty, because

1

∈

G

⁢

(

H

)

. Also, since comultiplication is an

algebra

morphism, then

G

⁢

(

H

)

is

multiplicative

, i.e. if

g

,

h

∈

G

⁢

(

H

)

, then

g

⁢

h

∈

G

⁢

(

H

)

. Furthermore, it can be shown that for any

g

∈

G

⁢

(

H

)

we have

S

⁢

(

g

)

∈

G

⁢

(

H

)

and

S

⁢

(

g

)

⁢

g

=

g

⁢

S

⁢

(

g

)

=

1

. Thus

G

⁢

(

H

)

is a group under

multiplication

inherited from

H

.

It is easy to see, that the

vector subspace

spanned by

G

⁢

(

H

)

is a Hopf

subalgebra

of

H

isomorphic

to

k

⁢

G

⁢

(

H

)

. It can be shown that

G

⁢

(

H

)

is always

linearly independent

, so if

H

is

finite dimensional

, then

G

⁢

(

H

)

is a

finite group

. Also, if

H

is finite dimensional, then it follows from the

Nichols-Zoeller Theorem

, that the order of

G

⁢

(

H

)

divides

dim

k

⁢

H

.

From these observations it follows that if

dim

k

⁢

H

=

p

is a

prime number

, then

G

⁢

(

H

)

is either trivial or the order of

G

⁢

(

H

)

is equal to

p

(i.e.

G

⁢

(

H

)

is cyclic of order

p

). The second case implies that

H

is isomorphic to

k

⁢

ℤ

p

and it can be shown that the first case cannot occur.

Title

grouplike elements in Hopf algebras

Canonical name

GrouplikeElementsInHopfAlgebras

Date of creation

2013-03-22 18:58:39

Last modified on

2013-03-22 18:58:39

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Definition

Classification

msc 16W30

Generated on Fri Feb 9 18:45:22 2018 by

LaTeXML
Hopf algebra

Hopf algebra

A

Hopf algebra

is a

bialgebra

A

over a field

𝕂

with a

𝕂

-linear map

S

:

A

→

A

,
called the

antipode

, such that

m

∘

(

S

⊗

id

)

∘

Δ

=

η

∘

ε

=

m

∘

(

id

⊗

S

)

∘

Δ

,

(1)

where

m

:

A

⊗

A

→

A

is the multiplication map

m

⁢

(

a

⊗

b

)

=

a

⁢

b

and

η

:

𝕂

→

A

is the unit map

η

⁢

(

k

)

=

k

⁢

1

⁢

I

.

In of a

commutative diagram

:

Generated on Fri Feb 9 18:45:29 2018 by

LaTeXML
module algebra

module algebra

Let

H

be a

bialgebra

.
A

left

H

-

module algebra

is a unital

algebra

A

which is a left

H

-module with action

h

▷

a

satisfying

h

▷

(

a

⁢

b

)

=

∑

(

h

(

1

)

▷

a

)

⁢

(

h

(

2

)

▷

b

)

,

h

▷

1

⁢

I

A

=

ε

⁢

(

h

)

⁢

1

⁢

I

A

,

(1)

for all

h

∈

H

and

a

,

b

∈

A

.

There is a dual notion of a

H

-

comodule coalgebra

.

Example 1

Let

H

be a

Hopf algebra

.
Then

H

is itself a

H

-module algebra for the

adjoint action

g

▷

h

=

∑

g

(

1

)

⁢

h

⁢

S

⁢

(

g

(

2

)

)

.

Title

module algebra

Canonical name

ModuleAlgebra

Date of creation

2013-03-22 13:26:31

Last modified on

2013-03-22 13:26:31

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

8

Author

mhale (572)

Entry type

Definition

Classification

msc 16W30

Related topic

ComoduleCoalgebra

Related topic

ModuleCoalgebra

Related topic

ComoduleAlgebra

Generated on Fri Feb 9 18:45:31 2018 by

LaTeXML
module coalgebra

module coalgebra

Let

H

be a

bialgebra

.
A

left

H

-

module coalgebra

is a coalgebra

A

which is a left

H

-module
with action

h

▷

a

satisfying

Δ

⁢

(

h

▷

a

)

=

∑

(

h

(

1

)

▷

a

(

1

)

)

⊗

(

h

(

2

)

▷

a

(

2

)

)

,

ε

⁢

(

h

▷

a

)

=

ε

⁢

(

h

)

⁢

ε

⁢

(

a

)

,

(1)

for all

h

∈

H

and

a

∈

A

.

There is a dual notion of a

H

-

comodule algebra

.

Example 1

Let

H

be a bialgebra.
Then

H

is itself a

H

-module coalgebra for the left

regular

action

g

▷

h

=

g

⁢

h

.

Title

module coalgebra

Canonical name

ModuleCoalgebra

Date of creation

2013-03-22 13:26:37

Last modified on

2013-03-22 13:26:37

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

9

Author

mhale (572)

Entry type

Definition

Classification

msc 16W30

Related topic

ComoduleAlgebra

Related topic

ModuleAlgebra

Related topic

ComoduleCoalgebra

Generated on Fri Feb 9 18:45:33 2018 by

LaTeXML
monoid bialgebra

monoid bialgebra

Let

G

be a monoid and

k

a field. Consider the

vector space

k

⁢

G

over

k

with basis

G

. More precisely,

k

G

=

{

f

:

G

→

k

|

f

(

g

)

=

0

for almost all

g

∈

G

}

.

We identify

g

∈

G

with a function

f

g

:

G

→

k

such that

f

g

⁢

(

g

)

=

1

and

f

g

⁢

(

h

)

=

0

for

h

≠

g

. Thus, every element in

k

⁢

G

is of the form

∑

g

∈

G

λ

g

⁢

g

,

for

λ

g

∈

k

. The vector space

k

⁢

G

can be turned into a

k

-

algebra

, if we define multiplication as follows:

g

⋅

h

=

g

⁢

h

,

where on the right side we have a multiplication in the monoid

G

. This definition extends linearly to entire

k

⁢

G

and defines an algebra structure on

k

⁢

G

, where

neutral element

of

G

is the

identity

in

k

⁢

G

.

Furthermore, we can turn

k

⁢

G

into a coalgebra as follows: comultiplication

Δ

:

k

⁢

G

→

k

⁢

G

⊗

k

⁢

G

is defined by

Δ

⁢

(

g

)

=

g

⊗

g

and counit

ε

:

k

⁢

G

→

k

is defined by

ε

⁢

(

g

)

=

1

. One can easily check that this defines coalgebra structure on

k

⁢

G

.

The vector space

k

⁢

G

is a

bialgebra

with with these algebra and coalgebra structures and it is called a

monoid bialgebra

.

Title

monoid bialgebra

Canonical name

MonoidBialgebra

Date of creation

2013-03-22 18:58:48

Last modified on

2013-03-22 18:58:48

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Example

Classification

msc 16W30

Generated on Fri Feb 9 18:45:35 2018 by

LaTeXML
monoid bialgebra is a Hopf algebra if and only if monoid is a group

monoid bialgebra is a Hopf algebra if and only if monoid is a group

Assume that

H

is a

Hopf algebra

with comultiplication

Δ

, counit

ε

and antipode

S

. It is well known, that if

c

∈

H

and

Δ

⁢

(

c

)

=

∑

i

=

1

n

a

i

⊗

b

i

, then

∑

i

=

1

n

S

⁢

(

a

i

)

⁢

b

i

=

ε

⁢

(

c

)

⁢

1

=

∑

i

=

1

n

a

i

⁢

S

⁢

(

b

i

)

(actualy, this condition defines the antipode), where on the left and right side we have

multiplication

in

H

.

Now let

G

be a monoid and

k

a field. It is well known that

k

⁢

G

is a

bialgebra

(please, see parent object for details), but one may ask, when

k

⁢

G

is a Hopf algebra? We will try to answer this question.

Proposition

.

A

monoid bialgebra

k

⁢

G

is a Hopf algebra if and only if

G

is a group.

Proof.

,,

⇐

” If

G

is a group, then define

S

:

k

⁢

G

→

k

⁢

G

by

S

⁢

(

g

)

=

g

-

1

. It is easy to check, that

S

is the antipode, thus

k

⁢

G

is a Hopf algebra.

,,

⇒

” Assume that

k

⁢

G

is a Hopf algebra, i.e. we have the antipode

S

:

k

⁢

G

→

k

⁢

G

. Then, for any

g

∈

G

we have

S

⁢

(

g

)

⁢

g

=

g

⁢

S

⁢

(

g

)

=

1

(because

Δ

⁢

(

g

)

=

g

⊗

g

and

ε

⁢

(

g

)

=

1

). Here

1

is the

identity

in both

G

and

k

⁢

G

. Of course

S

⁢

(

g

)

∈

k

⁢

G

, so

S

⁢

(

g

)

=

∑

h

∈

G

λ

h

⁢

h

.

Thus we have

1

=

(

∑

h

∈

G

λ

h

⁢

h

)

⁢

g

=

∑

h

∈

G

λ

h

⁢

h

⁢

g

.

Of course

G

is a basis, so this decomposition is unique. Therefore, there exists

g

′

∈

G

such that

λ

g

′

=

1

and

λ

h

′

=

0

for

h

′

≠

g

′

. We obtain, that

1

=

g

′

⁢

g

, thus

g

is

left invertible

. Since

g

was arbitrary it implies that

g

is

invertible

. Thus, we’ve shown that

G

is a group.

□

Title

monoid bialgebra is a Hopf algebra if and only if monoid is a group

Canonical name

MonoidBialgebraIsAHopfAlgebraIfAndOnlyIfMonoidIsAGroup

Date of creation

2013-03-22 18:58:51

Last modified on

2013-03-22 18:58:51

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 16W30

Generated on Fri Feb 9 18:45:37 2018 by

LaTeXML