ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 07 Aug 2019 09:24:03 -0500central idempotent of a finite dimensional algebrahttps://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/I tried the following
A=Algebras(QQ).FiniteDimensional().WithBasis().Semisimple().Commutative()
B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0],
[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1]]), Matrix([[0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0]]), Matrix([[0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,1,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,0]]), Matrix([[0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0]]), Matrix([[0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0]]), Matrix([[0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0]]), Matrix([[0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1]])], category=A)
B.central_orthogonal_idempotents()
The above code gives me error. Could anyone tell me how to fix the code?`enter code here`Thu, 01 Aug 2019 07:38:44 -0500https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/Comment by ortollj for <p>I tried the following</p>
<pre><code>A=Algebras(QQ).FiniteDimensional().WithBasis().Semisimple().Commutative()
B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0],
[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1]]), Matrix([[0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0]]), Matrix([[0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,1,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,0]]), Matrix([[0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0]]), Matrix([[0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0]]), Matrix([[0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0]]), Matrix([[0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1]])], category=A)
B.central_orthogonal_idempotents()
</code></pre>
<p>The above code gives me error. Could anyone tell me how to fix the code?<code>enter code here</code></p>
https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47345#post-id-47345it seems my comment was irrelevant here.
I must admit that the presentation does not really matter in this question
Sorry
[maybe like that ? : ](http://img28127.imagevenue.com/img.php?image=26217_Idempotent_122_736lo.JPG)Fri, 02 Aug 2019 00:50:31 -0500https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47345#post-id-47345Comment by REKHA BISWAL for <p>I tried the following</p>
<pre><code>A=Algebras(QQ).FiniteDimensional().WithBasis().Semisimple().Commutative()
B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0],
[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1]]), Matrix([[0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0]]), Matrix([[0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,1,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,0]]), Matrix([[0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0]]), Matrix([[0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0]]), Matrix([[0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0]]), Matrix([[0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1]])], category=A)
B.central_orthogonal_idempotents()
</code></pre>
<p>The above code gives me error. Could anyone tell me how to fix the code?<code>enter code here</code></p>
https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47343#post-id-47343Thanks. I edited . I hope its readable now.Thu, 01 Aug 2019 17:51:53 -0500https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47343#post-id-47343Comment by ortollj for <p>I tried the following</p>
<pre><code>A=Algebras(QQ).FiniteDimensional().WithBasis().Semisimple().Commutative()
B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0],
[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1]]), Matrix([[0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0]]), Matrix([[0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,1,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,0]]), Matrix([[0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0]]), Matrix([[0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0]]), Matrix([[0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0]]), Matrix([[0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1]])], category=A)
B.central_orthogonal_idempotents()
</code></pre>
<p>The above code gives me error. Could anyone tell me how to fix the code?<code>enter code here</code></p>
https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47342#post-id-47342Hi Rekha
Please, edit your message, select your code and click on the 101 button, just right of the web link button. it will be more readable.Thu, 01 Aug 2019 11:02:12 -0500https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47342#post-id-47342Answer by vdelecroix for <p>I tried the following</p>
<pre><code>A=Algebras(QQ).FiniteDimensional().WithBasis().Semisimple().Commutative()
B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0],
[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1]]), Matrix([[0,1,0,0,0,0,0,0,0], [0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,1,0,0,0,0,0,0], [1,0,0,0,0,0,0,0,0], [0,1,0,0,0,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0]]), Matrix([[0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0]]), Matrix([[0,0,0,0,1,0,0,0,0], [0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,1,0,0,0,0,1,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,1,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,0]]), Matrix([[0,0,0,0,0,1,0,0,0], [0,0,0,1,0,0,0,0,0], [0,0,0,0,1,0,0,0,0], [1,0,0,0,0,0,0,0,1], [0,1,0,0,0,0,1,0,0], [0,0,1,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0]]), Matrix([[0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0]]), Matrix([[0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,2,0,0,0,2,0,0,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [2,0,0,0,2,0,0,0,1], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0]]), Matrix([[0,0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,0,0], [0,0,0,0,0,0,0,1,0], [0,0,2,2,0,0,0,0,0], [2,0,0,0,2,0,0,0,0], [0,2,0,0,0,2,0,0,0], [0,2,0,0,0,2,1,0,0], [0,0,2,2,0,0,0,1,0], [2,0,0,0,2,0,0,0,1]])], category=A)
B.central_orthogonal_idempotents()
</code></pre>
<p>The above code gives me error. Could anyone tell me how to fix the code?<code>enter code here</code></p>
https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?answer=47416#post-id-47416My analysis is that this method is very broken
- first of all the inital error is because a `__hash__` method is "missing" for `FiniteDimensionalAlgebraElement`
- next, the method `central_orthogonal_idempotents` assumes that the quasi-idempotents of a commutative algebra are defined over the base ring which is wrong in general (needs the base ring to be algebraically closed...)
- finally, even though we change QQ for QQbar in the definition of B, the subalgebras created along the way do not implement `retract`/`lift` so that the product is not working... For the example given in the documentation of `central_orthogonal_idempotents` it works by some black magic
Here is the black magic:
sage: A4 = SymmetricGroup(4).algebra(QQ)
sage: B4 = A4.submodule([A4.an_element()])
sage: B4.retract
Generic morphism:
From: Symmetric group algebra of order 4 over Rational Field
To: Free module generated by {0} over Rational Field
sage: B4.lift
Generic morphism:
From: Free module generated by {0} over Rational Field
To: Symmetric group algebra of order 4 over Rational Field
sage: C = B.submodule([B.an_element()])
sage: C.retract
...
AttributeError: 'SubmoduleWithBasis_with_category' object has no attribute 'retract'
My conclusion is that the code is not written in a way that it works for generic commutative algebra.Wed, 07 Aug 2019 04:49:34 -0500https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?answer=47416#post-id-47416Comment by vdelecroix for <p>My analysis is that this method is very broken</p>
<ul>
<li>first of all the inital error is because a <code>__hash__</code> method is "missing" for <code>FiniteDimensionalAlgebraElement</code></li>
<li>next, the method <code>central_orthogonal_idempotents</code> assumes that the quasi-idempotents of a commutative algebra are defined over the base ring which is wrong in general (needs the base ring to be algebraically closed...)</li>
<li>finally, even though we change QQ for QQbar in the definition of B, the subalgebras created along the way do not implement <code>retract</code>/<code>lift</code> so that the product is not working... For the example given in the documentation of <code>central_orthogonal_idempotents</code> it works by some black magic</li>
</ul>
<p>Here is the black magic:</p>
<pre><code>sage: A4 = SymmetricGroup(4).algebra(QQ)
sage: B4 = A4.submodule([A4.an_element()])
sage: B4.retract
Generic morphism:
From: Symmetric group algebra of order 4 over Rational Field
To: Free module generated by {0} over Rational Field
sage: B4.lift
Generic morphism:
From: Free module generated by {0} over Rational Field
To: Symmetric group algebra of order 4 over Rational Field
sage: C = B.submodule([B.an_element()])
sage: C.retract
...
AttributeError: 'SubmoduleWithBasis_with_category' object has no attribute 'retract'
</code></pre>
<p>My conclusion is that the code is not written in a way that it works for generic commutative algebra.</p>
https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47425#post-id-47425I agree that it is a bug... and more precisely at least three bugs.Wed, 07 Aug 2019 09:24:03 -0500https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47425#post-id-47425Comment by Iguananaut for <p>My analysis is that this method is very broken</p>
<ul>
<li>first of all the inital error is because a <code>__hash__</code> method is "missing" for <code>FiniteDimensionalAlgebraElement</code></li>
<li>next, the method <code>central_orthogonal_idempotents</code> assumes that the quasi-idempotents of a commutative algebra are defined over the base ring which is wrong in general (needs the base ring to be algebraically closed...)</li>
<li>finally, even though we change QQ for QQbar in the definition of B, the subalgebras created along the way do not implement <code>retract</code>/<code>lift</code> so that the product is not working... For the example given in the documentation of <code>central_orthogonal_idempotents</code> it works by some black magic</li>
</ul>
<p>Here is the black magic:</p>
<pre><code>sage: A4 = SymmetricGroup(4).algebra(QQ)
sage: B4 = A4.submodule([A4.an_element()])
sage: B4.retract
Generic morphism:
From: Symmetric group algebra of order 4 over Rational Field
To: Free module generated by {0} over Rational Field
sage: B4.lift
Generic morphism:
From: Free module generated by {0} over Rational Field
To: Symmetric group algebra of order 4 over Rational Field
sage: C = B.submodule([B.an_element()])
sage: C.retract
...
AttributeError: 'SubmoduleWithBasis_with_category' object has no attribute 'retract'
</code></pre>
<p>My conclusion is that the code is not written in a way that it works for generic commutative algebra.</p>
https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47421#post-id-47421Perhaps there is another solution? It would depend on exactly what problem the OP is trying to solve. If `central_orthogonal_idempotents` can't be solved in this case it should also say so instead of just giving a useless crash (in this case I get `TypeError: <class 'sage.sets.family.TrivialFamily_with_category'> is not hashable and does not implement _cache_key()`). So I would definitely consider that a bug.Wed, 07 Aug 2019 07:29:43 -0500https://ask.sagemath.org/question/47338/central-idempotent-of-a-finite-dimensional-algebra/?comment=47421#post-id-47421