Ideals in PBW-algebras

Types

The OSCAR type for ideals in PBW-algebras is of parametrized form PBWAlgIdeal{D, T, S}, where D encodes the direction left, right, or two-sided, and T is the element type of the field over which the PBW-algebra is defined (the type S is added for internal use).

Constructors

left_ideal — Method

left_ideal(g::Vector{<:PBWAlgElem})

Given a vector g of elements in a PBW-algebra A, say, return the left ideal of A generated by these elements.

left_ideal(A::PBWAlgRing, g::AbstractVector)

Given a vector g of elements of A, return the left ideal of A generated by these elements.

Examples

julia> R, (x, y, z) = QQ[:x, :y, :z];

julia> L = [x*y, x*z, y*z + 1];

julia> REL = strictly_upper_triangular_matrix(L);

julia> A, (x, y, z) = pbw_algebra(R, REL, deglex(gens(R)))
(PBW-algebra over rational field in x, y, z with relations y*x = x*y, z*x = x*z, z*y = y*z + 1, PBWAlgElem{QQFieldElem, Singular.n_Q}[x, y, z])

julia> I = left_ideal(A, [x^2*y^2, x*z+y*z])
left_ideal(x^2*y^2, x*z + y*z)

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right_ideal — Method

right_ideal(g::Vector{<:PBWAlgElem})

Given a vector g of elements in a PBW-algebra A, say, return the right ideal of A generated by these elements.

right_ideal(A::PBWAlgRing, g::AbstractVector)

Given a vector g of elements of A, return the right ideal of A generated by these elements.

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two_sided_ideal — Method

two_sided_ideal(g::Vector{<:PBWAlgElem})

Given a vector g of elements in a PBW-algebra A, say, return the two-sided ideal of A generated by these elements.

two_sided_ideal(A::PBWAlgRing, g::AbstractVector)

Given a vector g of elements of A, return the two-sided ideal of A generated by these elements.

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Gröbner bases

Data Associated to Ideals

If I is an ideal of a PBW-algebra A, then

base_ring(I) refers to A,
gens(I) to the generators of I,
number_of_generators(I) / ngens(I) to the number of these generators, and
gen(I, k) as well as I[k] to the k-th such generator.

Examples

julia> D, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"])
(Weyl-algebra over rational field in variables (x, y), PBWAlgElem{QQFieldElem, Singular.n_Q}[x, y, dx, dy])

julia> I = left_ideal(D, [x, dx])
left_ideal(x, dx)

julia> base_ring(I)
Weyl-algebra over rational field in variables (x, y)

julia> gens(I)
2-element Vector{PBWAlgElem{QQFieldElem, Singular.n_Q}}:
 x
 dx

julia> number_of_generators(I)
2

julia> gen(I, 2)
dx

Operations on Ideals

Simple Ideal Operations

If I and J are PBW ideals, the following operations are available:

I + J for sums (same side),
I * J for products (for compatible left/right or two-sided ideals),
I^k for powers of two-sided ideals.

Examples:

julia> D, (x, y, dx, dy) = weyl_algebra(GF(3), [:x, :y]);

julia> I = left_ideal(D, [x^3+y^3, x*y^2])
left_ideal(x^3 + y^3, x*y^2)

julia> J = left_ideal(D, [x, y])
left_ideal(x, y)

julia> S = I + J;

julia> is_subset(I, S)
true

julia> is_subset(J, S)
true

julia> D, (x, y, dx, dy) = weyl_algebra(GF(3), [:x, :y]);

julia> I = left_ideal(D, [x^3+y^3, x*y^2])
left_ideal(x^3 + y^3, x*y^2)

julia> J = right_ideal(D, [dx^3, dy^5])
right_ideal(dx^3, dy^5)

julia> I*J
two_sided_ideal(x^3*dx^3 + y^3*dx^3, x^3*dy^5 + y^3*dy^5, x*y^2*dx^3, x*y^2*dy^5)

julia> D, (x, dx) = weyl_algebra(GF(3), [:x]);

julia> I = two_sided_ideal(D, [x^3])
two_sided_ideal(x^3)

julia> I^2
two_sided_ideal(x^6)

Intersection of Ideals

intersect — Method

intersect(I::PBWAlgIdeal{D, T, S}, Js::PBWAlgIdeal{D, T, S}...) where {D, T, S}
intersect(V::Vector{PBWAlgIdeal{D, T, S}}) where {D, T, S}

Return the intersection of two or more ideals.

Examples

julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y]);

julia> I = intersect(left_ideal(D, [x^2, x*dy, dy^2])+left_ideal(D, [dx]), left_ideal(D, [dy^2-x^3+x]))
left_ideal(-x^3 + dy^2 + x)

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intersect — Method

intersect(a::AlgAssAbsOrdIdl, b::AlgAssAbsOrdIdl) -> AlgAssAbsOrdIdl

Returns $a \cap b$.

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intersect(a::AlgAssRelOrdIdl, b::AlgAssRelOrdIdl) -> AlgAssRelOrdIdl

Returns $a \cap b$.

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intersect(I::MPolyIdeal{T}, Js::MPolyIdeal{T}...) where T
intersect(V::Vector{MPolyIdeal{T}}) where T

Return the intersection of two or more ideals.

Examples

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y])
(Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])

julia> I = ideal(R, [x, y])^2;

julia> J = ideal(R, [y^2-x^3+x]);

julia> intersect(I, J)
Ideal generated by
  x^3*y - x*y - y^3
  x^4 - x^2 - x*y^2

julia> intersect([I, J])
Ideal generated by
  x^3*y - x*y - y^3
  x^4 - x^2 - x*y^2

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intersect(a::MPolyQuoIdeal{T}, bs::MPolyQuoIdeal{T}...) where T
intersect(V::Vector{MPolyQuoIdeal{T}}) where T

Return the intersection of two or more ideals.

Examples

julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);

julia> A, _ = quo(R, ideal(R, [x^2-y^3, x-y]));

julia> a = ideal(A, [y^2])
Ideal generated by
  y^2

julia> b = ideal(A, [x])
Ideal generated by
  x

julia> intersect(a,b)
Ideal generated by
  x*y

julia> intersect([a,b])
Ideal generated by
  x*y

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intersect(I::PBWAlgIdeal{D, T, S}, Js::PBWAlgIdeal{D, T, S}...) where {D, T, S}
intersect(V::Vector{PBWAlgIdeal{D, T, S}}) where {D, T, S}

Return the intersection of two or more ideals.

Examples

julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y]);

julia> I = intersect(left_ideal(D, [x^2, x*dy, dy^2])+left_ideal(D, [dx]), left_ideal(D, [dy^2-x^3+x]))
left_ideal(-x^3 + dy^2 + x)

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intersect(M::SubquoModule{T}, N::SubquoModule{T}) where T

Given subquotients M and N such that ambient_module(M) == ambient_module(N), return the intersection of M and N regarded as submodules of the common ambient module.

Additionally, return the inclusion maps M $\cap$ N $\to$ M and M $\cap$ N $\to$ N.

Examples

julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
(Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])

julia> F = free_module(R, 1)
Free module of rank 1 over R

julia> AM = R[x;]
[x]

julia> BM = R[x^2; y^3; z^4]
[x^2]
[y^3]
[z^4]

julia> M = subquotient(F, AM, BM)
Subquotient of submodule with 1 generator
  1: x*e[1]
by submodule with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1]

julia> AN = R[y;]
[y]

julia> BN = R[x^2; y^3; z^4]
[x^2]
[y^3]
[z^4]

julia> N = subquotient(F, AN, BN)
Subquotient of submodule with 1 generator
  1: y*e[1]
by submodule with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1]

julia> intersect(M, N)
(Subquotient of submodule with 2 generators
  1: -x*y*e[1]
  2: x*z^4*e[1]
by submodule with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1], Hom: subquotient of submodule with 2 generators
  1: -x*y*e[1]
  2: x*z^4*e[1]
by submodule with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1] -> M, Hom: subquotient of submodule with 2 generators
  1: -x*y*e[1]
  2: x*z^4*e[1]
by submodule with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1] -> N)

julia> Rg, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> F = graded_free_module(Rg, 1);

julia> AM = Rg[x;];

julia> BM = Rg[x^2; y^3; z^4];

julia> M = subquotient(F, AM, BM)
Graded subquotient of graded submodule of F with 1 generator
  1: x*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1]

julia> AN = Rg[y;];

julia> BN = Rg[x^2; y^3; z^4];

julia> N = subquotient(F, AN, BN)
Graded subquotient of graded submodule of F with 1 generator
  1: y*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1]

julia> intersect(M, N)
(Graded subquotient of graded submodule of F with 2 generators
  1: -x*y*e[1]
  2: x*z^4*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1], Hom: graded subquotient of graded submodule of F with 2 generators
  1: -x*y*e[1]
  2: x*z^4*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1] -> M, Hom: graded subquotient of graded submodule of F with 2 generators
  1: -x*y*e[1]
  2: x*z^4*e[1]
by graded submodule of F with 3 generators
  1: x^2*e[1]
  2: y^3*e[1]
  3: z^4*e[1] -> N)

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Elimination

Let

\[A = K\langle x_1, \dots , x_n \mid x_jx_i = c_{ij}x_ix_j+d_{ij}, \ 1\leq i<j \leq n \rangle,\]

be a PBW-algebra. Fix a subset $\sigma\subset \{1,\dots, n\}$, write $x_\sigma$ for the set of variables $x_i$ with $i\in\sigma$, and let $A_\sigma$ be the $K$-linear subspace of $A$ which is generated by the standard monomials in $\langle x_\sigma \rangle$. Suppose there exists a global monomial ordering $>$ on $A$ which is both admissible for $A$ and an elimination ordering for $x\smallsetminus x_\sigma$. Then $A_\sigma$ is a subalgebra of $A$ with $d_{ij}\in A_\sigma$ for each pair of indices $1\leq i<j \leq n$ with $i,j\in\sigma$. In particular, $A_\sigma$ is a PBW-algebra with admissible ordering $>_\sigma$, where $>_\sigma$ is the restriction of $>$ to the set of standard monomials in $\langle x_\sigma\rangle$. Moreover, if $I\subset A$ is a nonzero (left, right, two-sided) ideal, and $\mathcal G$ is a (left, right, two-sided) Gröbner basis for $I$ with respect to $>$, then $\mathcal G\cap A_\sigma$ is a (left, right, two-sided) Gröbner basis for $I\cap A_\sigma$ with respect to $>_\sigma$. We refer to computing $I\cap A_\sigma$ as eliminating the variables in $x\smallsetminus x_\sigma$ from $I.$

Note

If the relevant $d_{ij}$ are all contained in $A_\sigma$, we also say that $A_\sigma$ is admissible for elimination.

Note

A monomial ordering which is both admissible for $A$ and an elimination ordering for $x\smallsetminus x_\sigma$ may not exist.

eliminate — Method

eliminate(I::PBWAlgIdeal, V::Vector{<:PBWAlgElem}; ordering = nothing)

Given a vector V of variables, these variables are eliminated from I. That is, return the ideal generated by all polynomials in I which only involve the remaining variables.

eliminate(I::PBWAlgIdeal, V::Vector{Int}; ordering = nothing)

Given a vector V of indices which specify variables, these variables are eliminated from I. That is, return the ideal generated by all polynomials in I which only involve the remaining variables.

Note

The return value is an ideal of the original algebra.

Note

If provided, the ordering must be an admissible elimination ordering (this is checked by the function). If not provided, finding an admissible elimination ordering may involve solving a particular linear programming problem. Here, the function is implemented so that it searches for solutions in a certain range only. If no solution is found in that range, the function will throw an error.

Examples

julia> R, (x, y, z, a) = QQ[:x, :y, :z, :a];

julia> L = [x*y-z, x*z+2*x, x*a, y*z-2*y, y*a, z*a];

julia> REL = strictly_upper_triangular_matrix(L);

julia> A, (x, y, z, a) = pbw_algebra(R, REL, deglex(gens(R)))
(PBW-algebra over rational field in x, y, z, a with relations y*x = x*y - z, z*x = x*z + 2*x, a*x = x*a, z*y = y*z - 2*y, a*y = y*a, a*z = z*a, PBWAlgElem{QQFieldElem, Singular.n_Q}[x, y, z, a])

julia> f = 4*x*y+z^2-2*z-a;

julia> I = left_ideal(A, [x^2, y^2, z^2-1, f])
left_ideal(x^2, y^2, z^2 - 1, 4*x*y + z^2 - 2*z - a)

julia> eliminate(I, [x, y, z])
left_ideal(a - 3)

julia> eliminate(I, [1, 2 ,3])
left_ideal(a - 3)

julia> try eliminate(I, [z, a]); catch e; e; end
ErrorException("no elimination is possible: subalgebra is not admissible")

julia> R, (p, q) = QQ[:p, :q];

julia> L = [p*q+q^2];

julia> REL = strictly_upper_triangular_matrix(L);

julia> A, (p, q) = pbw_algebra(R, REL, lex(gens(R)))
(PBW-algebra over rational field in p, q with relations q*p = p*q + q^2, PBWAlgElem{QQFieldElem, Singular.n_Q}[p, q])

julia> I = left_ideal(A, [p, q])
left_ideal(p, q)

julia> try eliminate(I, [q]); catch e; e; end   # in fact, no elimination ordering exists
ErrorException("could not find elimination ordering")

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Tests on Ideals

For PBW ideals, use is_zero and is_one for basic tests, is_subset for containment, and == for equality.

julia> D, (x, dx) = weyl_algebra(QQ, [:x]);

julia> I = left_ideal(D, [dx^2])
left_ideal(dx^2)

julia> J = left_ideal(D, [x*dx^4, x^3*dx^2])
left_ideal(x*dx^4, x^3*dx^2)

julia> is_zero(I)
false

julia> is_subset(I, J)
true

julia> I == J
true

julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y]);

julia> I = left_ideal(D, [x, dx])
left_ideal(x, dx)

julia> is_one(I)
true

julia> J = left_ideal(D, [y*x])
left_ideal(x*y)

julia> is_one(J)
false

julia> K = two_sided_ideal(D, [y*x])
two_sided_ideal(x*y)

julia> is_one(K)
true

julia> D3, (x, y, dx, dy) = weyl_algebra(GF(3), [:x, :y]);

julia> I3 = two_sided_ideal(D3, [x^3])
two_sided_ideal(x^3)

julia> is_one(I3)
false

ideal_membership — Method

ideal_membership(f::PBWAlgElem{T, S}, I::PBWAlgIdeal{D, T, S}) where {D, T, S}

Return true if f is contained in I, false otherwise. Alternatively, use f in I.

Examples

julia> D, (x, dx) = weyl_algebra(QQ, [:x]);

julia> I = left_ideal(D, [x*dx^4, x^3*dx^2])
left_ideal(x*dx^4, x^3*dx^2)

julia> dx^2 in I
true

julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y]);

julia> I = two_sided_ideal(D, [x, dx])
two_sided_ideal(x, dx)

julia> one(D) in I
true

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