Finite-Dimensional Algebras¶
-
sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra.
FiniteDimensionalAlgebra
¶ Create a finite-dimensional \(k\)-algebra from a multiplication table.
INPUT:
k
– a fieldtable
– a list of matricesnames
– (default:'e'
) string; names for the basis elementsassume_associative
– (default:False
) boolean; ifTrue
, then the category is set tocategory.Associative()
and methods requiring associativity assume thiscategory
– (default:MagmaticAlgebras(k).FiniteDimensional().WithBasis()
) the category to which this algebra belongs
The list
table
must have the following form: there exists a finite-dimensional \(k\)-algebra of degree \(n\) with basis \((e_1, \ldots, e_n)\) such that the \(i\)-th element oftable
is the matrix of right multiplication by \(e_i\) with respect to the basis \((e_1, \ldots, e_n)\).EXAMPLES:
sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])]) sage: A Finite-dimensional algebra of degree 2 over Finite Field of size 3 sage: TestSuite(A).run() sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]), Matrix([[0,1,0], [0,0,0], [0,0,0]]), Matrix([[0,0,0], [0,0,0], [0,0,1]])]) sage: B Finite-dimensional algebra of degree 3 over Rational Field