Super modules with basis¶
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class
sage.categories.super_modules_with_basis.
SuperModulesWithBasis
(base_category)¶ Bases:
sage.categories.super_modules.SuperModulesCategory
The category of super modules with a distinguished basis.
An \(R\)-super module with a distinguished basis is an \(R\)-super module equipped with an \(R\)-module basis whose elements are homogeneous.
EXAMPLES:
sage: C = GradedModulesWithBasis(ZZ); C Category of graded modules with basis over Integer Ring sage: sorted(C.super_categories(), key=str) [Category of filtered modules with basis over Integer Ring, Category of graded modules over Integer Ring] sage: C is ModulesWithBasis(ZZ).Graded() True
TESTS:
sage: TestSuite(C).run()
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class
ElementMethods
¶ -
even_component
()¶ Return the even component of
self
.EXAMPLES:
sage: Q = QuadraticForm(QQ, 2, [1,2,3]) sage: C.<x,y> = CliffordAlgebra(Q) sage: a = x*y + x - 3*y + 4 sage: a.even_component() x*y + 4
TESTS:
Check that this really return
A.zero()
and not a plain0
:sage: a = x + y sage: a.even_component().parent() is C True
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is_even_odd
()¶ Return
0
ifself
is an even element and1
ifself
is an odd element.EXAMPLES:
sage: Q = QuadraticForm(QQ, 2, [1,2,3]) sage: C.<x,y> = CliffordAlgebra(Q) sage: a = x + y sage: a.is_even_odd() 1 sage: a = x*y + 4 sage: a.is_even_odd() 0 sage: a = x + 4 sage: a.is_even_odd() Traceback (most recent call last): ... ValueError: element is not homogeneous sage: E.<x,y> = ExteriorAlgebra(QQ) sage: (x*y).is_even_odd() 0
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is_super_homogeneous
()¶ Return whether this element is homogeneous, in the sense of a super module (i.e., is even or odd).
EXAMPLES:
sage: Q = QuadraticForm(QQ, 2, [1,2,3]) sage: C.<x,y> = CliffordAlgebra(Q) sage: a = x + y sage: a.is_super_homogeneous() True sage: a = x*y + 4 sage: a.is_super_homogeneous() True sage: a = x*y + x - 3*y + 4 sage: a.is_super_homogeneous() False
The exterior algebra has a \(\ZZ\) grading, which induces the \(\ZZ / 2\ZZ\) grading. However the definition of homogeneous elements differs because of the different gradings:
sage: E.<x,y> = ExteriorAlgebra(QQ) sage: a = x*y + 4 sage: a.is_super_homogeneous() True sage: a.is_homogeneous() False
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odd_component
()¶ Return the odd component of
self
.EXAMPLES:
sage: Q = QuadraticForm(QQ, 2, [1,2,3]) sage: C.<x,y> = CliffordAlgebra(Q) sage: a = x*y + x - 3*y + 4 sage: a.odd_component() x - 3*y
TESTS:
Check that this really return
A.zero()
and not a plain0
:sage: a = x*y sage: a.odd_component().parent() is C True
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class
SuperModulesWithBasis.
ParentMethods
¶
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class