modules over bound quiver algebra and bound quiver representations

modules over bound quiver algebra and bound quiver representations

Let

(

Q

,

I

)

be a bound quiver over a fixed field

k

. Denote by

Mod

⁢

A

(resp.

mod

⁢

A

) the

category

of all (resp. all finite-dimensional) (right) modules over

algebra

A

and by

REP

Q

,

I

(resp.

rep

Q

,

I

) the category of all (resp. all finite-dimensional, (see this entry (

http://planetmath.org/QuiverRepresentationsAndRepresentationMorphisms

) for details) bound

representations

.

We will also allow

I

=

0

(which is an admissible ideal only if lengths of paths in

Q

are

bounded

, in particular when

Q

is finite and acyclic). In this case bound representations are simply representations.

Theorem

.

If

Q

is a connected and finite quiver,

I

and admissible ideal in

k

⁢

Q

and

A

=

k

⁢

Q

/

I

, then there exists a

k

-

equivalence of categories

F

:

Mod

⁢

A

→

REP

Q

,

I

which restricts to the equivalence of categories

F

′

:

mod

⁢

A

→

rep

Q

,

I

.

Sketch of the proof.

We will only define

functor

F

and its

quasi-inverse

G

. For proof that

F

is actually an equivalence please see

[

1

, Theorem 1.6]

(this not difficult, but rather technical proof).

Let

e

a

be a

stationary

path in

a

∈

Q

0

and put

ϵ

a

=

e

a

+

I

∈

A

. Now if

M

is a module in

Mod

⁢

A

, then define a representation

F

⁢

(

M

)

=

(

M

a

,

M

α

)

by putting

M

a

=

M

⁢

ϵ

a

(

M

is a

right module

over

A

). Now for an arrow

α

∈

Q

1

define

M

α

:

M

s

⁢

(

α

)

→

M

t

⁢

(

α

)

by putting

M

α

⁢

(

x

)

=

x

⁢

α

¯

, where

α

¯

=

α

+

I

∈

A

. It can be shown (see

[

1

]

) that

F

⁢

(

M

)

is a bound representation.

On module

morphisms

F

acts as follows. If

f

:

M

→

M

′

is a module morphism, then define

F

⁢

(

f

)

=

(

f

a

)

a

∈

Q

0

where

f

a

:

M

a

→

M

a

′

is a

restriction

, i.e.

f

a

⁢

(

x

)

=

f

⁢

(

x

)

. It can be shown that

f

a

is well-defined (i.e.

f

a

⁢

(

x

)

∈

M

a

′

) and in this manner

F

is a functor.

The

inverse

functor is defined on objects as follows: for a representation

(

M

a

,

M

α

)

put

G

⁢

(

M

)

=

⊕

a

∈

Q

0

M

a

.

Now we will define right

k

⁢

Q

-module

structure

on

G

⁢

(

M

)

. For a stationary path

e

a

in

a

∈

Q

0

and for

x

=

(

x

a

)

∈

G

⁢

(

M

)

put

x

⋅

e

a

=

x

a

.

Now for a path

w

=

(

a

1

,

…

,

a

n

)

from

a

to

b

in

k

⁢

Q

we consider the evaluation map (see this entry (

http://planetmath.org/RepresentationsOfABoundQuiver

) for details)

f

w

:

M

a

→

M

b

and we put

(

x

⋅

w

)

c

=

δ

b

⁢

c

⁢

f

w

⁢

(

x

a

)

,

where

δ

b

⁢

c

denotes the Kronecker delta. It can be shown that

G

⁢

(

M

)

is a

k

⁢

Q

-module with the property that

G

⁢

(

M

)

⁢

I

=

0

. In particular

G

⁢

(

M

)

is a

k

⁢

Q

/

I

-module.

Now, if

f

=

(

f

a

)

:

M

→

M

′

is a

morphism of representations

then we define

G

⁢

(

f

)

=

⊕

a

∈

Q

0

f

a

:

G

⁢

(

M

)

→

G

⁢

(

M

)

.

It can be shown that

G

⁢

(

f

)

is indeed an

A

-

homomorphism

and that

G

is a functor.

Also, it follows easily from definitions that both

F

and

G

take finite-dimensional objects to finite-dimensional.

It remains to show that these two functors are quasi-inverse. For the proof please see

[

1

, Theorem 1.6]

.

□

Corollary.

If

Q

is a finite, connected and acyclic quiver, then there exists an equivalence of categories

Mod

⁢

k

⁢

Q

≃

REP

Q

which restricts to the equivalence of categories

mod

⁢

k

⁢

Q

≃

rep

Q

.

Proof.

Since

Q

is finite and acyclic, then the

zero ideal

I

=

0

is admissible (because lengths of paths are bounded, so

R

Q

m

=

0

for some

m

⩾

1

, where

R

Q

denotes the arrow ideal). The thesis follows from the theorem.

□

References

1

I. Assem, D. Simson, A. SkowroÃÆski,

Elements of the

Representation Theory

of

Associative Algebras

, vol 1.

, Cambridge University Press 2006, 2007

Title

modules over bound quiver algebra and bound quiver representations

Canonical name

ModulesOverBoundQuiverAlgebraAndBoundQuiverRepresentations

Date of creation

2013-03-22 19:17:34

Last modified on

2013-03-22 19:17:34

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Theorem

Classification

msc 14L24

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