Product of $e in A/J(A)$ with an element of $J(A)$ [GAP]

-2

I have defined a matrix algebra over the rationals, calculated its radical, and took the semi-simple coefficient using h:=NaturalHomomorphismByIdeal and $ImagesSource(h)$. Afterwards, I calculated the idempotents of the semisimple coefficient using 'CentralIdempotentsOfAlgebra'. I also found the generators of the radical $J(A)$, as well as its square $J(A)^2$.

Now, I am trying to use the information to calculate the quiver of the original algebra. For that, I need to calculate the product of the idempotents $e_i$ and $e_j$ with the Jacobson radical, with arrows existing from $e_i$ to $e_j$ iff $q:=e_iJ(A)e_j/e_iJ(A)^2e_jneq 0$. And the number of arrows is the dimension of the vector space $q$. I cannot seem to calculate the dimension using the standard $Dimension$ function, or a basis using the standard $Basis$ function, since I have $e_i$ and $e_j$ as elements of the semi-simple coefficient, given as a linear combination of the generators $[v.1,cdots,v.6]$, while the generators of $J(A)$ are matrices.

A second problem is that $IsAlgebraWithOne(sscoef)$ returns $False$, while $One(sscoef)$ returns $v.1$, and $IsAlgebra(sscoef)$ returns $True$.

What am I missing?

Here is the code:

Constructing a non-commutative polynomial algebra

LoadPackage("GBNP");

A:=FreeAssociativeAlgebraWithOne(Rationals,"a","b","c");;

a:=A.a;;b:=A.b;;c:=A.c;;z:=Zero(A);;e:=One(A);;

relation1:=a^2-e;;relation2:=b^2-e;;relation3:=c^2-e;;relation4:=ab+bc+ca+e;; relation5:=ac+cb+ba+e;;

relationlist:=GP2NPList([relation1,relation2,relation3,relation4,relation5]);;

grobnerbasis:=SGrobner(relationlist);;

qabasis:=BaseQA(grobnerbasis,3,0);;

qamatrices:=MatricesQA(3,qabasis,grobnerbasis);;

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

matrixalgebra:=AlgebraWithOne(Rationals, qamatrices);;

jacobsonradical:=RadicalOfAlgebra(matrixalgebra);;

h:=NaturalHomomorphismByIdeal(matrixalgebra,jacobsonradical);;

sscoef:=ImagesSource(h);;

idempotents:=CentralIdempotentsOfAlgebra(sscoef);;

gensofsscoef:=GeneratorsOfAlgebra(sscoef);;

Caluclating the generators' respective commutators

commutatorlist:=;;

for i in [1..Length(gensofsscoef)] do

for j in [i..Length(gensofsscoef)] do

Add(commutatorlist, gensofsscoef[i]*gensofsscoef[j] - gensofsscoef[j]*gensofsscoef[i]);

od;

od;

commutatorlist;

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

gensofjacobsonradical:=GeneratorsOfAlgebra(jacobsonradical);;

idemstimesjacobson:=;

for e_i in [1..Length(idempotents)] do

Add(idemstimesjacobson,);

for e_j in [1..Length(idempotents)] do

Add(idemstimesjacobson[e_i],);

for genofjacobson in gensofjacobsonradical do

Add(idemstimesjacobson[e_i][e_j], idempotents[e_i]genofjacobsonidempotents[e_j]);
od;

od;

od;

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

jacobsonradicalsquared:=;

for genofjacobson1 in gensofjacobsonradical do

for genofjacobson2 in gensofjacobsonradical do

Add(jacobsonradicalsquared, genofjacobson1*genofjacobson2);

od;

od;

idemstimesjacobsonsquared:=;

for e_i in [1..Length(idempotents)] do

Add(idemstimesjacobsonsquared,);

for e_j in [1..Length(idempotents)] do

Add(idemstimesjacobsonsquared[e_i],);

for genofjacobson in jacobsonradicalsquared do

Add(idemstimesjacobsonsquared[e_i][e_j], idempotents[e_i]genofjacobsonidempotents[e_j]);

od;

od;

od;

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

vertices:=;

for e_i in [1..Length(idempotents)] do

Add(vertices, );

for e_j in [1..Length(idempotents)] do

numerator:=Ideal(matrixalgebra,idemstimesjacobson[e_i][e_j]);

denominator:=Ideal(matrixalgebra, idemstimesjacobsonsquared[e_i][e_j]);

if Dimension(numerator/denominator) > 0 then

Add(vertices[e_i], [e_i, e_j, Dimension(numerator/denominator)]);

fi;

od;

od;

vertices;

The end-result should be the quiver of $A$. However, the $Dimension$ function returns the following error:

Error, usage: TwoSidedIdeal( , [, "basis"] ) at /proc/cygdrive/C/gap-4.9.3/lib/ideal.gi:42 called from
Ideal( matrixalgebra, idemstimesjacobson[e_i][e_j] ) at stdin:152 called from
( )
called from read-eval loop at stdin:158

edited Dec 24 '18 at 14:51

asked Dec 23 '18 at 21:53

UnexpectedExpectation

769

$begingroup$
Could you please provide more details? One should be able to reproduce your setup to see what happens. "I cannot seem to calculate ..." does not give enough details - it's necessary to now what's exactly the problem is.
$endgroup$
– Alexander Konovalov
Dec 23 '18 at 23:09

$begingroup$
Also, do you mean "semi-simple coefficient" or "semi-simple component"?
$endgroup$
– Alexander Konovalov
Dec 23 '18 at 23:13

$begingroup$
I mean the semi-simple coefficient, $A/J(A)$, where $A$ is a right-artinian algebra, and $J(A)$ its jacobson radical. I have added comments and the entire code to the post. Hope that clarifies the question.
$endgroup$
– UnexpectedExpectation
Dec 24 '18 at 14:21

$begingroup$
Thanks - please also indent all GAP input and output by 4 spaces, this is needed to display it properly. Also, use indentation for loop bodies and if statements - this makes the code more readable and easier to understand.
$endgroup$
– Alexander Konovalov
Dec 24 '18 at 22:56

1

$begingroup$
Your error is not in Dimension but (as the error message tells you) earlier when you create the ideal that you call numerator. You might want to step through the commands one by one and check whether you actually create the objects you want.
$endgroup$
– ahulpke
Dec 25 '18 at 9:10

|
show 1 more comment

-2

A second problem is that $IsAlgebraWithOne(sscoef)$ returns $False$, while $One(sscoef)$ returns $v.1$, and $IsAlgebra(sscoef)$ returns $True$.

What am I missing?

Here is the code:

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

The end-result should be the quiver of $A$. However, the $Dimension$ function returns the following error:

Error, usage: TwoSidedIdeal( , [, "basis"] ) at /proc/cygdrive/C/gap-4.9.3/lib/ideal.gi:42 called from
Ideal( matrixalgebra, idemstimesjacobson[e_i][e_j] ) at stdin:152 called from
( )
called from read-eval loop at stdin:158

edited Dec 24 '18 at 14:51

asked Dec 23 '18 at 21:53

UnexpectedExpectation

769

$begingroup$
Could you please provide more details? One should be able to reproduce your setup to see what happens. "I cannot seem to calculate ..." does not give enough details - it's necessary to now what's exactly the problem is.
$endgroup$
– Alexander Konovalov
Dec 23 '18 at 23:09

$begingroup$
Also, do you mean "semi-simple coefficient" or "semi-simple component"?
$endgroup$
– Alexander Konovalov
Dec 23 '18 at 23:13

$begingroup$
I mean the semi-simple coefficient, $A/J(A)$, where $A$ is a right-artinian algebra, and $J(A)$ its jacobson radical. I have added comments and the entire code to the post. Hope that clarifies the question.
$endgroup$
– UnexpectedExpectation
Dec 24 '18 at 14:21

$begingroup$
Thanks - please also indent all GAP input and output by 4 spaces, this is needed to display it properly. Also, use indentation for loop bodies and if statements - this makes the code more readable and easier to understand.
$endgroup$
– Alexander Konovalov
Dec 24 '18 at 22:56

1

$begingroup$
Your error is not in Dimension but (as the error message tells you) earlier when you create the ideal that you call numerator. You might want to step through the commands one by one and check whether you actually create the objects you want.
$endgroup$
– ahulpke
Dec 25 '18 at 9:10

|
show 1 more comment

-2

A second problem is that $IsAlgebraWithOne(sscoef)$ returns $False$, while $One(sscoef)$ returns $v.1$, and $IsAlgebra(sscoef)$ returns $True$.

What am I missing?

Here is the code:

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

The end-result should be the quiver of $A$. However, the $Dimension$ function returns the following error:

Error, usage: TwoSidedIdeal( , [, "basis"] ) at /proc/cygdrive/C/gap-4.9.3/lib/ideal.gi:42 called from
Ideal( matrixalgebra, idemstimesjacobson[e_i][e_j] ) at stdin:152 called from
( )
called from read-eval loop at stdin:158

edited Dec 24 '18 at 14:51

asked Dec 23 '18 at 21:53

UnexpectedExpectation

769

A second problem is that $IsAlgebraWithOne(sscoef)$ returns $False$, while $One(sscoef)$ returns $v.1$, and $IsAlgebra(sscoef)$ returns $True$.

What am I missing?

Here is the code:

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

The end-result should be the quiver of $A$. However, the $Dimension$ function returns the following error:

Error, usage: TwoSidedIdeal( , [, "basis"] ) at /proc/cygdrive/C/gap-4.9.3/lib/ideal.gi:42 called from
Ideal( matrixalgebra, idemstimesjacobson[e_i][e_j] ) at stdin:152 called from
( )
called from read-eval loop at stdin:158

linear-algebra abstract-algebra gap

edited Dec 24 '18 at 14:51

asked Dec 23 '18 at 21:53

UnexpectedExpectation

769

edited Dec 24 '18 at 14:51

asked Dec 23 '18 at 21:53

UnexpectedExpectation

769

edited Dec 24 '18 at 14:51

asked Dec 23 '18 at 21:53

UnexpectedExpectation

769

asked Dec 23 '18 at 21:53

UnexpectedExpectation

769

asked Dec 23 '18 at 21:53

UnexpectedExpectation

769

$begingroup$
Could you please provide more details? One should be able to reproduce your setup to see what happens. "I cannot seem to calculate ..." does not give enough details - it's necessary to now what's exactly the problem is.
$endgroup$
– Alexander Konovalov
Dec 23 '18 at 23:09

$begingroup$
Also, do you mean "semi-simple coefficient" or "semi-simple component"?
$endgroup$
– Alexander Konovalov
Dec 23 '18 at 23:13

$begingroup$
I mean the semi-simple coefficient, $A/J(A)$, where $A$ is a right-artinian algebra, and $J(A)$ its jacobson radical. I have added comments and the entire code to the post. Hope that clarifies the question.
$endgroup$
– UnexpectedExpectation
Dec 24 '18 at 14:21

$begingroup$
Thanks - please also indent all GAP input and output by 4 spaces, this is needed to display it properly. Also, use indentation for loop bodies and if statements - this makes the code more readable and easier to understand.
$endgroup$
– Alexander Konovalov
Dec 24 '18 at 22:56

1

$begingroup$
Your error is not in Dimension but (as the error message tells you) earlier when you create the ideal that you call numerator. You might want to step through the commands one by one and check whether you actually create the objects you want.
$endgroup$
– ahulpke
Dec 25 '18 at 9:10

|
show 1 more comment

$begingroup$
Could you please provide more details? One should be able to reproduce your setup to see what happens. "I cannot seem to calculate ..." does not give enough details - it's necessary to now what's exactly the problem is.
$endgroup$
– Alexander Konovalov
Dec 23 '18 at 23:09

$begingroup$
Also, do you mean "semi-simple coefficient" or "semi-simple component"?
$endgroup$
– Alexander Konovalov
Dec 23 '18 at 23:13

$begingroup$
I mean the semi-simple coefficient, $A/J(A)$, where $A$ is a right-artinian algebra, and $J(A)$ its jacobson radical. I have added comments and the entire code to the post. Hope that clarifies the question.
$endgroup$
– UnexpectedExpectation
Dec 24 '18 at 14:21

$begingroup$
Thanks - please also indent all GAP input and output by 4 spaces, this is needed to display it properly. Also, use indentation for loop bodies and if statements - this makes the code more readable and easier to understand.
$endgroup$
– Alexander Konovalov
Dec 24 '18 at 22:56

1

$begingroup$
Your error is not in Dimension but (as the error message tells you) earlier when you create the ideal that you call numerator. You might want to step through the commands one by one and check whether you actually create the objects you want.
$endgroup$
– ahulpke
Dec 25 '18 at 9:10

Could you please provide more details? One should be able to reproduce your setup to see what happens. "I cannot seem to calculate ..." does not give enough details - it's necessary to now what's exactly the problem is.

– Alexander Konovalov
Dec 23 '18 at 23:09

Also, do you mean "semi-simple coefficient" or "semi-simple component"?

– Alexander Konovalov
Dec 23 '18 at 23:13

I mean the semi-simple coefficient, $A/J(A)$, where $A$ is a right-artinian algebra, and $J(A)$ its jacobson radical. I have added comments and the entire code to the post. Hope that clarifies the question.

– UnexpectedExpectation
Dec 24 '18 at 14:21

Thanks - please also indent all GAP input and output by 4 spaces, this is needed to display it properly. Also, use indentation for loop bodies and if statements - this makes the code more readable and easier to understand.

– Alexander Konovalov
Dec 24 '18 at 22:56

Your error is not in Dimension but (as the error message tells you) earlier when you create the ideal that you call numerator. You might want to step through the commands one by one and check whether you actually create the objects you want.

– ahulpke
Dec 25 '18 at 9:10

|
show 1 more comment

1 Answer
1

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oldest

votes

As for the second question, IsAlgebraWithOne is not a test function, but (as the manual tells) a category. As your example shows, other algebras also contain an One element.

But for example the AlgebraWithOne generated by the matrix $left(begin{array}{cc}0&1\0&0end{array}right)$ is two-dimensional (and contains the identity matrix), while the Algebra generated by it is one-dimensional (and contains a one that is not the identity matrix).

answered Dec 25 '18 at 9:17

ahulpke

7,1671026

add a comment |

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1 Answer
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As for the second question, IsAlgebraWithOne is not a test function, but (as the manual tells) a category. As your example shows, other algebras also contain an One element.

answered Dec 25 '18 at 9:17

ahulpke

7,1671026

add a comment |

As for the second question, IsAlgebraWithOne is not a test function, but (as the manual tells) a category. As your example shows, other algebras also contain an One element.

answered Dec 25 '18 at 9:17

ahulpke

7,1671026

add a comment |

As for the second question, IsAlgebraWithOne is not a test function, but (as the manual tells) a category. As your example shows, other algebras also contain an One element.

answered Dec 25 '18 at 9:17

ahulpke

7,1671026

As for the second question, IsAlgebraWithOne is not a test function, but (as the manual tells) a category. As your example shows, other algebras also contain an One element.

answered Dec 25 '18 at 9:17

ahulpke

7,1671026

answered Dec 25 '18 at 9:17

ahulpke

7,1671026

answered Dec 25 '18 at 9:17

ahulpke

7,1671026

answered Dec 25 '18 at 9:17

ahulpke

7,1671026

add a comment |

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Krdytkyu

Product of $e in A/J(A)$ with an element of $J(A)$ [GAP]

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

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Product of $e in A/J(A)$ with an element of $J(A)$ [GAP]

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

Constructing a non-commutative polynomial algebra

Transform it into a matrix algebra, so that we can use the standard GAP-functions to calculate the radical.

Caluclating the generators' respective commutators

We have that it is commutative.Now we calculate the arrows. e_iJ(A)e_j/e_iJ(A)^2e_j

Here, we construct a Length(idempotents)xLength(gensofjacobsonradical)xLength(idempotents) array.

With this, we have e_iJ(A)e_j and e_iJ(A)^2e_j. Need to save the dimension of the quotient as the number of vertices.

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Mont Emei

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