dual of a coalgebra is an algebra, the

dual of a coalgebra is an algebra, the

Let

R

be a

commutative ring

with unity. Suppose we have a coassociative coalgebra

(

C

,

Δ

)

and an associative algebra

A

, both over

R

. Since

C

and

A

are both

R

-modules, it follows that

Hom

R

⁢

(

C

,

A

)

is also an

R

-module. But in fact we can give it the structure of an associative

R

-

algebra

. To do this, we use the

convolution product

. Namely, given morphisms

f

and

g

in

Hom

R

⁢

(

C

,

A

)

, we define their product

f

⁢

g

by

(

f

⁢

g

)

⁢

(

x

)

=

∑

x

f

⁢

(

x

(

1

)

)

⋅

g

⁢

(

x

(

2

)

)

,

where we use the Sweedler notation

Δ

⁢

(

x

)

=

∑

x

x

(

1

)

⊗

x

(

2

)

for the comultiplication

Δ

. To see that the convolution product is associative, suppose

f

,

g

, and

h

are in

Hom

R

⁢

(

C

,

A

)

. By applying the coassociativity of

Δ

, we may write

(

(

f

⁢

g

)

⁢

h

)

⁢

(

x

)

=

∑

x

(

f

⁢

(

x

(

1

)

)

⋅

g

⁢

(

x

(

2

)

)

)

⋅

h

⁢

(

x

(

3

)

)

and

(

f

⁢

(

g

⁢

h

)

)

⁢

(

x

)

=

∑

x

f

⁢

(

x

(

1

)

)

⋅

(

g

⁢

(

x

(

2

)

)

)

⋅

h

⁢

(

x

(

3

)

)

.

Since

A

has an associative product, it follows that

(

f

⁢

g

)

⁢

h

=

f

⁢

(

g

⁢

h

)

.

In the foregoing, we have not assumed that

C

is counitary or that

A

is

unitary

. If

C

is counitary with counit

ε

:

C

→

R

and

A

is unitary with identity

1

:

R

→

A

, then their

composition

1

∘

ε

:

C

→

A

is the identity for the convolution product.

Example

.

Let

C

be a coassociative coalgebra over

R

. Then

R

itself is an associative

R

-algebra. The algebra

Hom

R

⁢

(

C

,

R

)

is called the

algebra dual to the coalgebra

C

.

We have seen that any coalgebra dualizes to give an algebra. One might expect that a

similar

construction could be performed on

Hom

R

⁢

(

A

,

R

)

to give a coalgebra dual to

A

. However, this is not the case. Thus coalgebras (based on “factoring”) are more fundamental than algebras (based on “multiplying”).

(The proof will be provided at a later stage).

Remark on Al/gebraic Duality

–

Mirror or tangled ‘duality’ of algebras and ‘gebras’:

An interesting twist to duality was provided in Fauser’s publications on al/gebras
where mirror or tangled ‘duality’ has been defined for Grassman-Hopf al/gebras. Thus,
an algebra not only has the usual reversed arrow dual coalgebra but a mirror (or tangled)
gebra which is quite distinct from the coalgebra.

Note:

The dual of a

quantum group

is a

Hopf algebra

.

References

1

W. Nichols and M. Sweedler,

Hopf algebras and

combinatorics

, in

Proceedings of the conference on

umbral calculus

and Hopf algebras

, ed. R. Morris, AMS, 1982.

2

B. Fauser:

A treatise on quantum

Clifford Algebras

. Konstanz,
Habilitationsschrift.

arXiv.math.QA/0202059 (2002).

3

B. Fauser: Grade Free product Formulae from Grassman–Hopf Gebras.
Ch. 18 in R. Ablamowicz, Ed.,

Clifford Algebras: Applications to Mathematics, Physics and Engineering

, Birkhäuser: Boston, Basel and Berlin, (2004).

4

J. M. G. Fell.: The

Dual Spaces

of C*–Algebras.,

Transactions of the American
Mathematical Society

,

94

: 365–403 (1960).

Title

dual of a coalgebra is an algebra, the

Canonical name

DualOfACoalgebraIsAnAlgebraThe

Date of creation

2013-03-22 16:34:20

Last modified on

2013-03-22 16:34:20

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Derivation

Classification

msc 16W30

Related topic

GrassmanHopfAlgebrasAndTheirDualCoAlgebras

Related topic

DualityInMathematics

Related topic

QuantumGroups

Defines

dualities of algebraic structures

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