RELATIONSHIP BETWEEN WEAK ENTWINING STRUCTURES AND WEAK CORINGS - Diponegoro University | Institutional Repository (UNDIP-IR)

! "

# $ $ % ! ! &'()&

* ++ , $

Abstract. Given a commutative ring

R

with unit,

R

-algebra

A

and

R

-coalgebra

C

.

Triple

(

A C

, ,

ψ

)

is called (weak) entwining structure if there is

R

-linear map

ψ

:

C

⊗

_R

A

→

A

⊗

_R

C

that fulfil some axioms. In the other hand, from algebra

A

and coalgebra

C

we can consider

A

⊗

_R

C

as a left

A

-module canonically such that

(

A C

, ,

ψ

)

is entwined structure if only if

A

⊗

_R

C

is a

A

-coring. In particular, we obtain that

(

A C

, ,

ψ

)

is a weak entwined structure if only if

R

A

⊗

C

is a weak

A

-coring.

Keywords : algebra, coalgebra, coring, entwining structure.

-

R

" ! - $ . / / 0 +

1 % 234

R

5 ! %

(

A

, ,

µ ι

)

R

5 ! %

(

C

, ,

∆

ε

)

-(

A C

, ,

ψ

)

% - ! 6

R

5

:

C

_R

A

A

_R

C

ψ

⊗

→

⊗

6 $ % % . / / 0 + 2(4

!

R

5 ! %

A

R

5 ! %

C

%

R

5 $

A

C

R

5 -

A

⊗

_R

C

.

" !

A

5

(

)

:

A

_R

C

_R

A

A

_R

C

,

α

⊗

⊗

→

⊗

α

(

(

a

⊗

b

)

⊗

c

)

=

a

ψ

(

c

⊗

b

)

,

A

⊗

_R

C

₍

A A

,

₎

5

% - %

A

⊗

_R

C

- + !$ . / / 0 + 274 - "

% - - + ! - + - ! $ $

(

A C

, ,

ψ

)

- !

R

A

⊗

C

A

5 ! ! " %

(

)

(

)

:

I

_A

:

A

_R

C

A

_R

C

_R

C

A

_R

C

_A

A

_R

C

,

∆

AC

I

ε

⊗

∆

CA

I

⊗

ε

1

⊗ − ⊗

1

:

I

_A

:

A

_R

C

A

.

ε

=

⊗

ε

⊗

→

1 + ! + ! % - +

! % 5 % 8 2)4 1 % 294:$ 1

-% - ! - ! % - + !

A

⊗

_R

C

.

( - ! " ! - + ! $ ! /

! % 8 . / / 0 + 234 2)4 1 % 294:$ 6

. / / 0 + 274 ! " - ! - + - ! $

7 - " % - - + - ! - + ! $

R

A

⊗

C

- + !

A

⊗

_R

C

- ! $

;9<' - ! % " $

"

C

"

F

-

∆

:

C

→

C

⊗

_F

C

ε

:

C

→

F

F

5 ! % $ ! % " " ! "

! . / / 0 + 1 % 234$ - ! " %

! - + ! 8 . / / 0 + 234 2)4 1 % 294:$

!

A

- % " ! - $

C

(

A A

,

)

(

A A

,

)

∆

:

C

→

C

⊗

_A

A

⊗

_A

C

,

₍

∀ ∈

c

C

_{) ( )}

∆

c

=

c

₁

⊗ ⊗

1

c

₂

(

A A

,

)

ε

:

C

→

A

∆

! ; "

c

∈

C

,

ε

( )

c c

_{1 2}

=

1 1

c

=

c

₁

ε

( )

c

₂

.

(

A A

,

)

C

∆

ε

.

C

A A

C

⊗

A

⊗

C

C

A A

:

C

_R

A

A

_R

C

,

ψ

⊗

→

⊗

ψ

₍

c

⊗

a

₎

=

a

_ψ

⊗

c

ψ

,

,

.

a

_ψ

∈

A c

ψ

∈

C

! ( "

1 2 R R

,

c

⊗

a

⊗

a

∈ ⊗

C

A

⊗

A

c

⊗ ∈ ⊗

a

C

_R

A

,

;$

(

a a

_{1 2}

)

c

a a c

_{1 2}

ψ ϕ

ψ ψϕ

ψ

=

($

a

_ψ

⊗

c

₁ψ

⊗

c

₂ψ

=

a c c

_{ψϕ 1}ϕ ₂ψ 3$

1

_ψ

⊗

c

ψ

= ⊗

1

c

7$

a

_ψ

ε

( )

c

ψ

=

ε

( )

c a

.

- + - ! ! - 3$;$

%

A

⊗

_R

C

%

" " ;$ - ! > ! 2<4

1 % 294$

(

A

, ,

µ ι

)

R

(

C

, ,

∆

ε

)

R

!

(

A C

, ,

ψ

)

"

R

ψ

:

C

⊗

_R

A

→

A

⊗

_R

C

,

(

a

c

)

a c

_ψ ψ

ψ

⊗

=

a

_ψ

∈

A

c

ψ

∈

C

&

( )

ab

c

a b c

ψ ϕ

ψ ψϕ

ψ

=

#

a

_ψ

(

c

ψ₁

⊗

1

)

⊗

c

ψ₂

=

a

_ψϕ

⊗

c

₁ϕ

⊗

c

₂ψ $

a

_ψ

ε

( )

c

ψ

=

ε

( )

c

ψ

1

_ψ

a

%

1

_ψ

⊗

c

ψ

=

ε

( )

c

₁ψ

1

_ψ

⊗

c

₂

.

R

5 % -

R

5 ! %

A

R

5 ! %

C

%

A

⊗

_R

C

.

- % 6 % - 8- +: - !

8- +: ! :

A

⊗

_R

C

.

1 - " ! $

!

(

A

, ,

µ ι

)

R

(

C

, ,

∆

ε

)

R

!

(

A C

, ,

ψ

)

8 :$

(

(

a

⊗

c x y

)

.

)

.

=

(

a

ψ

(

c

⊗

x

)

)

.

y

(by counital as a coalgebra).