morphisms of path algebras induced from morphisms of quivers

morphisms of path algebras induced from morphisms of quivers

Let

Q

=

(

Q

0

,

Q

1

,

s

,

t

)

,

Q

′

=

(

Q

0

′

,

Q

1

′

,

s

′

,

t

′

)

be quivers and let

F

:

Q

→

Q

′

be a

morphism

of quivers.

Proposition

1.

If

w

=

(

α

1

,

…

,

α

n

)

is a path in

Q

, then

F

⁢

(

w

)

=

(

F

1

⁢

(

α

1

)

,

…

,

F

1

⁢

(

α

n

)

)

is a path in

Q

′

.

Proof.

Indeed, for any

i

=

1

,

…

,

n

-

1

we calculate

t

′

⁢

(

F

1

⁢

(

α

i

)

)

=

F

0

⁢

(

t

⁢

(

α

i

)

)

=

F

0

⁢

(

s

⁢

(

α

i

+

1

)

)

=

t

′

⁢

(

F

1

⁢

(

α

i

+

1

)

)

,

which

completes

the proof.

□

Proposition 2.

Let

w

,

u

be paths in

Q

. If

w

is

compatible

(

http://planetmath.org/PathAlgebraOfAQuiver

) with

u

then

F

⁢

(

w

)

is compatible (

http://planetmath.org/PathAlgebraOfAQuiver

) with

F

⁢

(

u

)

. The

inverse

implication

holds if and only if

F

0

is an

injective function

.

Proof.

Assume that we have the following

presentations

:

w

=

(

w

1

,

…

,

w

n

)

;

u

=

(

u

1

,

…

,

u

n

)

.

If

t

⁢

(

w

n

)

=

s

⁢

(

u

1

)

, then

t

′

⁢

(

F

1

⁢

(

w

n

)

)

=

F

0

⁢

(

t

⁢

(

w

n

)

)

=

F

0

⁢

(

s

⁢

(

u

1

)

)

=

s

′

⁢

(

F

1

⁢

(

u

1

)

)

which shows the first part of the thesis.

For the second part note, that if

F

0

is injective, then the above equalities can be reversed to obtain that

t

⁢

(

w

n

)

=

s

⁢

(

u

1

)

.

On the other hand assume that

F

0

is not injective, i.e.

F

0

⁢

(

a

)

=

F

0

⁢

(

b

)

for some distinct vertices

a

,

b

∈

Q

0

. Then for

stationary

paths

e

a

and

e

b

we have that

t

′

⁢

(

F

1

⁢

(

e

a

)

)

=

F

0

⁢

(

t

⁢

(

e

a

)

)

=

F

0

⁢

(

a

)

=

F

0

⁢

(

b

)

=

F

0

⁢

(

s

⁢

(

e

b

)

)

=

s

′

⁢

(

F

1

⁢

(

e

b

)

)

so paths

(

F

1

⁢

(

e

a

)

)

and

(

F

1

⁢

(

e

b

)

)

are compatible (

http://planetmath.org/PathAlgebraOfAQuiver

), although

(

e

a

)

,

(

e

b

)

are not.

□

Definition.

Let

k

be a field. The linear map

F

¯

:

k

⁢

Q

→

k

⁢

Q

′

defined on a basis of

k

⁢

Q

by

F

¯

⁢

(

w

)

=

F

⁢

(

w

)

is said to be

induced

from

F

.

Proposition 3.

The linear map

F

¯

:

k

⁢

Q

→

k

⁢

Q

′

induced from

F

:

Q

→

Q

′

is a

homomorphism

of

algebras

if and only if

F

0

is injective.

Proof.

Indeed, we will show that

F

¯

preservers

multiplication

of compatible paths (

http://planetmath.org/PathAlgebraOfAQuiver

). If

w

=

(

w

1

,

…

,

w

n

)

;

u

=

(

u

1

,

…

,

u

m

)

are compatible paths (

http://planetmath.org/PathAlgebraOfAQuiver

) in

Q

, then

F

¯

⁢

(

w

⋅

u

)

=

F

¯

⁢

(

(

w

1

,

…

,

w

n

,

u

1

,

…

,

u

m

)

)

=

(

F

1

⁢

(

w

1

)

,

…

,

F

1

⁢

(

w

n

)

,

F

1

⁢

(

u

1

)

,

…

⁢

F

1

⁢

(

u

m

)

)

=

F

¯

⁢

(

w

)

⋅

F

¯

⁢

(

u

)

,

which completes this part.

Now assume that

w

,

u

are paths, which are not compatible (

http://planetmath.org/PathAlgebraOfAQuiver

). If

F

0

is injective, then by proposition 2

F

⁢

(

w

)

and

F

⁢

(

u

)

are also not compatible (

http://planetmath.org/PathAlgebraOfAQuiver

) and thus

F

¯

⁢

(

w

⋅

u

)

=

F

¯

⁢

(

0

)

=

0

=

F

¯

⁢

(

w

)

⋅

F

¯

⁢

(

u

)

.

On the other hand, if

F

0

is not injective, then there are paths

w

,

u

which are not compatible (

http://planetmath.org/PathAlgebraOfAQuiver

), but

F

⁢

(

w

)

,

F

⁢

(

u

)

are. Assume, that

F

¯

is a homomorphism of algebras. Then

0

=

F

¯

⁢

(

0

)

=

F

¯

⁢

(

w

⋅

u

)

=

F

¯

⁢

(

w

)

⋅

F

¯

⁢

(

u

)

≠

0

because of the compatibility (

http://planetmath.org/PathAlgebraOfAQuiver

). The

contradiction

shows that

F

¯

is not a homomorphism of algebras. This completes the proof.

□

Title

morphisms of path algebras induced from morphisms of quivers

Canonical name

MorphismsOfPathAlgebrasInducedFromMorphismsOfQuivers

Date of creation

2013-03-22 19:17:03

Last modified on

2013-03-22 19:17:03

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

6

Author

joking (16130)

Entry type

Definition

Classification

msc 14L24

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