#19473: FiniteDimensionalAlgebra.is_unitary is not sufficient
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Reporter: darij | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.10
Component: algebra | Keywords: finite-dimensional algebra,
Merged in: | linear algebra
Reviewers: | Authors:
Work issues: | Report Upstream: N/A
Commit: | Branch:
Stopgaps: | Dependencies:
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Here is the semigroup algebra (over `QQ`) of the semigroup with two
elements whose product is given by `x*y == y`:
{{{
sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0],[1,0]]),
Matrix([[0,1],[0,1]])])
sage: for x in A.basis():
for y in A.basis():
print x*y == y
....:
True
True
True
True
}}}
Sage claims that it is unitary:
{{{
sage: A.is_unitary()
True
}}}
based on the following wrong concept:
{{{
.. NOTE::
If a finite-dimensional algebra over a field admits a left
identity,
then this is the unique left identity, and it is also a
right identity.
}}}
On a slightly related note, the `table` method on
`FiniteDimensionalAlgebra` returns mutable matrices, and mutating them
corrupts the algebra.
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Ticket URL: <http://trac.sagemath.org/ticket/19473>
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