From ce96a6f610c1c2a6837bacd893fed164d2085a03 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Tue, 15 Oct 2019 08:13:41 -0400 Subject: [PATCH] eja: make some element tests work in trivial algebras. --- mjo/eja/eja_element.py | 49 +++++++++++++++++++++++------------------- 1 file changed, 27 insertions(+), 22 deletions(-) diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index a4af4ea..2eb2677 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -430,6 +430,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: x.inverse() == J.from_vector(x_inverse) True + Trying to invert a non-invertible element throws an error: + + sage: JordanSpinEJA(3).zero().inverse() + Traceback (most recent call last): + ... + ValueError: element is not invertible + TESTS: The identity element is its own inverse:: @@ -455,14 +462,6 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: (not x.is_invertible()) or (x.inverse().inverse() == x) True - The zero element is never invertible:: - - sage: set_random_seed() - sage: J = random_eja().zero().inverse() - Traceback (most recent call last): - ... - ValueError: element is not invertible - Proposition II.2.3 in Faraut and Korányi says that the inverse of an element is the inverse of its left-multiplication operator applied to the algebra's identity, when that inverse exists:: @@ -574,10 +573,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS: - The identity element is never nilpotent:: + The identity element is never nilpotent, except in a trivial EJA:: sage: set_random_seed() - sage: random_eja().one().is_nilpotent() + sage: J = random_eja() + sage: J.one().is_nilpotent() and not J.is_trivial() False The additive identity is always nilpotent:: @@ -621,11 +621,11 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS: The zero element should never be regular, unless the parent - algebra has dimension one:: + algebra has dimension less than or equal to one:: sage: set_random_seed() sage: J = random_eja() - sage: J.dimension() == 1 or not J.zero().is_regular() + sage: J.dimension() <= 1 or not J.zero().is_regular() True The unit element isn't regular unless the algebra happens to @@ -633,7 +633,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: J = random_eja() - sage: J.dimension() == 1 or not J.one().is_regular() + sage: J.dimension() <= 1 or not J.one().is_regular() True """ @@ -677,14 +677,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): TESTS: - The zero and unit elements are both of degree one:: + The zero and unit elements are both of degree one in nontrivial + algebras:: sage: set_random_seed() sage: J = random_eja() - sage: J.zero().degree() - 1 - sage: J.one().degree() - 1 + sage: d = J.zero().degree() + sage: (J.is_trivial() and d == 0) or d == 1 + True + sage: d = J.one().degree() + sage: (J.is_trivial() and d == 0) or d == 1 + True Our implementation agrees with the definition:: @@ -1140,13 +1143,15 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): sage: A(x^2) == A(x)*A(x) True - By definition, the subalgebra generated by the zero element is the - one-dimensional algebra generated by the identity element:: + By definition, the subalgebra generated by the zero element is + the one-dimensional algebra generated by the identity + element... unless the original algebra was trivial, in which + case the subalgebra is trivial too:: sage: set_random_seed() sage: A = random_eja().zero().subalgebra_generated_by() - sage: A.dimension() - 1 + sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1 + True """ return FiniteDimensionalEuclideanJordanElementSubalgebra(self, orthonormalize_basis) -- 2.44.2