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Basics

Creation of algebras

See the corresponding sections on structure constant algebras, group algebras and quaternion algebras.

zero_algebra — Method
zero_algebra([T, ] K::Field) -> AbstractAssociativeAlgebra

Return the zero ring as an algebra over the field $K$.

The optional first argument determines the type of the algebra, and can be StructureConstantAlgebra (default) or MatrixAlgebra.

Examples

julia> A = zero_algebra(QQ)
Structure constant algebra of dimension 0 over QQ
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Basic properties

base_ring — Method
base_ring(A::AbstractAssociativeAlgebra) -> Field

Given a $K$-algebra $A$, return $K$.

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basis — Method
basis(A::AbstractAssociativeAlgebra) -> Vector

Given a $K$-algebra $A$ return the $K$-basis of $A$. See also coordinates to get the the coordinates of an element with respect to the bases.

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Predicates

is_zero — Method
is_zero(A::AbstractAssociativeAlgebra) -> Bool

Return whether $A$ is the zero algebra.

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is_commutative — Method
is_commutative(A::AbstractAssociativeAlgebra) -> Bool

Return whether $A$ is commutative.

Examples

julia> A = matrix_algebra(QQ, 2);

julia> is_commutative(A)
false
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is_central — Method
is_central(A::AbstractAssociativeAlgebra)

Return whether the $K$-algebra $A$ is central, that is, whether $K$ is the center of $A$.

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Creation of elements

Elements of algebras can be constructed by arithmetic with basis elements, generators or by providing coordinates.

julia> Q = quaternion_algebra(QQ, -1, -1)
Quaternion algebra
  over rational field
  defined by i^2 = -1, j^2 = -1, ij = -ji

julia> B = basis(Q);

julia> x = B[1] + B[2] + 1//3 * B[4]
1 + i + 1//3*k

julia> Q([1, 1, 0, 1//3])
1 + i + 1//3*k

Generators

gens — Method
gens(A::AbstractAssociativeAlgebra; thorough_search::Bool = false) -> Vector

Given a $K$-algebra $A$, return a subset of basis(A), which generates $A$ as an algebra over $K$.

If thorough_search is true, the number of returned generators is possibly smaller. This will in general increase the runtime. It is not guaranteed that the number of generators is minimal in any case.

The gens_with_data function computes additional data for expressing a basis as words in the generators.

Examples

julia> A = matrix_algebra(QQ, 3);

julia> gens(A; thorough_search = true)
5-element Vector{MatAlgebraElem{QQFieldElem, QQMatrix}}:
 [1 0 0; 0 0 0; 0 0 0]
 [0 0 0; 1 0 0; 0 0 0]
 [0 0 0; 0 0 0; 1 0 0]
 [0 1 0; 0 0 0; 0 0 0]
 [0 0 1; 0 0 0; 0 0 0]
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gens_with_data — Method
gens_with_data(A::AbstractAssociativeAlgebra; thorough_search::Bool = false)
                                                   -> Vector, Vector, Vector

Given a $K$-algebra $A$, return a triple $(G, B, w)$ consisting of

  • a subset $G$ of basis(A), which generates $A$ as an algebra over $K$,
  • a (new) basis $B$ and a vector w::Vector{Tuple{Int, Int}}, such that B[i] = prod(G[j]^k for (j, k) in w[i].

If thorough_search is true, the number of returned generators is possibly smaller. This will in general increase the runtime. It is not guaranteed that the number of generators is minimal in any case.

Examples

julia> A = matrix_algebra(QQ, 3);

julia> G, B, w = gens_with_data(A; thorough_search = true);

julia> B[1] == prod(G[i]^j for (i, j) in w[1])
true
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Center

center — Method
center(A::AbstractAssociativeAlgebra)
                                   -> StructureConstantAlgebra, Map

Returns the center $C$ of $A$ and the inclusion $C \to A$. Note that $C$ itself is an algebra.

Examples

julia> A = matrix_algebra(QQ, 2);

julia> C, CtoA = center(A);

julia> C
Structure constant algebra of dimension 1 over QQ
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dimension_of_center — Method
dimension_of_center(A::AbstractAssociativeAlgebra) -> Int

Given a $K$-algebra, return the $K$-dimension of the center of $A$.

Examples

julia> A = matrix_algebra(QQ, 2);

julia> dimension_of_center(A)
1
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dimension_over_center — Method
dimension_over_center(A::AbstractAssociativeAlgebra) -> Int

Given a simple $K$-algebra with center $C$, return the $C$-dimension $A$.

Examples

julia> A = matrix_algebra(QQ, 2);

julia> dimension_of_center(A)
1
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