Elements of finite flat algebras¶
Elements of finite flat algebras.
- class dual_pairs.finite_flat_algebra_element.FiniteFlatAlgebraElement(*args: Any, **kwargs: Any)¶
An element of a finite flat algebra.
This is an abstract base class.
- inverse_of_unit()¶
TODO
- is_unit()¶
TODO
- monomial_coefficients(**kwds)¶
Return a dictionary containing the coefficients of
self.This method is required by
ModulesWithBasis.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: R.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, x^2 - 1) sage: a = A(x) sage: (3*a + 2).monomial_coefficients() {0: 2, 1: 3}
- class dual_pairs.finite_flat_algebra_element.FiniteFlatAlgebraElement_generic(*args: Any, **kwargs: Any)¶
An element of a generic finite flat algebra.
EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: A = FiniteFlatAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])]) sage: a = A.gen(1) sage: type(a) <class 'dual_pairs.finite_flat_algebra.FiniteFlatAlgebra_generic_with_category.element_class'> sage: a e1
- algebra_element()¶
Return the element of the underlying algebra corresponding to
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: A = FiniteFlatAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])]) sage: A.gen(1).algebra_element() e1
- matrix()¶
Return the matrix of multiplication by
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: A = FiniteFlatAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])]) sage: A.gen(1).matrix() [ 0 1] [-1 0]
- module_element()¶
Return the element of the underlying module corresponding to
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: A = FiniteFlatAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])]) sage: A.gen(1).module_element() (0, 1)
- class dual_pairs.finite_flat_algebra_element.FiniteFlatAlgebraElement_monogenic(*args: Any, **kwargs: Any)¶
An element of a monogenic finite flat algebra.
EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: S.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, x^4 - 16) sage: a = A(x) sage: type(a) <class 'dual_pairs.finite_flat_algebra.FiniteFlatAlgebra_monogenic_with_category.element_class'> sage: a a
- algebra_element()¶
Return the element of the underlying algebra corresponding to
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: R.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, x^3 - x - 1) sage: A(x).algebra_element() a sage: R.<x> = GF(3)[] sage: A = FiniteFlatAlgebra(GF(3), x^2 - 2) sage: A(x).algebra_element() a
- matrix()¶
Return the matrix of multiplication by
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: R.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, x^3 - x - 1) sage: A(x).matrix() [0 1 0] [0 0 1] [1 1 0]
- module_element()¶
Return the element of the underlying module corresponding to
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: R.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, x^3 - x - 1) sage: A(x).module_element() (0, 1, 0) sage: F.<c> = GF(9) sage: R.<x> = F[] sage: A = FiniteFlatAlgebra(F, x^3 - 1) sage: A(x^2).module_element() (0, 0, 1)
- class dual_pairs.finite_flat_algebra_element.FiniteFlatAlgebraElement_product(*args: Any, **kwargs: Any)¶
An element of a finite flat algebra presented as a product.
EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: S.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, [x, x, x^2 + 17]) sage: a = A([1, 2, x]) sage: type(a) <class 'dual_pairs.finite_flat_algebra.FiniteFlatAlgebra_product_with_category.element_class'> sage: a (1, 2, a2)
- algebra_element()¶
Return the element of the underlying algebra corresponding to
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: R.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, [x, x, x^2 - 5]) sage: A([1, 2, x]).algebra_element() (1, 2, a2) sage: F.<c> = GF(9) sage: R.<x> = F[] sage: A = FiniteFlatAlgebra(F, [x, x, x]) sage: A([x, 1, 2]).module_element() (0, 1, 2)
- matrix()¶
Return the matrix of multiplication by
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: R.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, [x, x, x^2 - 5]) sage: A([1, 2, x]).matrix() [1 0 0 0] [0 2 0 0] [0 0 0 1] [0 0 5 0]
- module_element()¶
Return the element of the underlying module corresponding to
self.EXAMPLES:
sage: from dual_pairs import FiniteFlatAlgebra sage: R.<x> = QQ[] sage: A = FiniteFlatAlgebra(QQ, [x, x, x^2 - 5]) sage: A([1, 2, x]).module_element() (1, 2, 0, 1) sage: R.<x> = GF(3)[] sage: A = FiniteFlatAlgebra(GF(3), [x, x^2 - 2]) sage: A([2, x]).algebra_element() (2, a1)