GR-Algebras: Quotients of PBW-Algebras

In analogy to the affine algebras section in the commutative algebra chapter, we describe OSCAR functionality for dealing with quotients of PBW-algebras modulo two-sided ideals.

Note

Quotients of PBW-algebras modulo two-sided ideals are also known as GR-algebras (here, GR stands for Gröbner-Ready; see Viktor Levandovskyy (2005)).

Types

GR-algebras are modelled by objects of type PBWAlgQuo{T, S} <: NCRing, their elements are objects of type PBWAlgQuoElem{T, S} <: NCRingElem. Here, T is the element type of the field over which the GR-algebra is defined (the type S is added for internal use).

Constructors

quoMethod
quo(A::PBWAlgRing, I::PBWAlgIdeal)

Given a two-sided ideal I of A, create the quotient algebra $A/I$ and return the new algebra together with the quotient map $A\rightarrow A/I$.

Examples

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

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

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, PBWAlgElem{fmpq, Singular.n_Q}[x, y, z])

julia> I = two_sided_ideal(A, [x^2, y^2, z^2])
two_sided_ideal(x^2, y^2, z^2)

julia> Q, q = quo(A, I);

julia> Q
(PBW-algebra over Rational Field in x, y, z with relations y*x = -x*y, z*x = -x*z, z*y = -y*z)/two_sided_ideal(x^2, y^2, z^2)

julia> q
Map from
PBW-algebra over Rational Field in x, y, z with relations y*x = -x*y, z*x = -x*z, z*y = -y*z to (PBW-algebra over Rational Field in x, y, z with relations y*x = -x*y, z*x = -x*z, z*y = -y*z)/two_sided_ideal(x^2, y^2, z^2) defined by a julia-function with inverse
Note

The example above, shows one way of constructing the exterior algebra on the variables x, y, z over $\mathbb Q$. For reasons of efficiency, it is recommended to use the built-in constructor exterior_algebra when working with exterior algebras in OSCAR.

source

Exterior Algebras

The $n$-th exterior algebra over a field $K$ is the quotient of the PBW-algebra

\[A=K \langle e_1,\dots, e_n \mid e_ie_j = - e_je_i \ \text { for }\ i\neq j\rangle\]

modulo the two-sided ideal

\[\langle e_1^2,\dots, e_n^2\rangle.\]

exterior_algebraMethod
exterior_algebra(K::Ring, xs::Union{AbstractVector{<:AbstractString}, 
                                AbstractVector{Symbol}, AbstractVector{Char}})

Given a field K and a vector xs of, say, $n$ Strings, Symbols, or Characters, return the $n$-th exterior algebra over K.

The generators of the returned algebra print according to the entries of xs. See the example below.

Examples

source

Data Associated to Affine GR-Algebras

Basic Data

If Q=A/I is the quotient ring of a PBW-algebra A modulo a two-sided ideal I of A, then

  • base_ring(Q) refers to A,
  • modulus(Q) to I,
  • gens(Q) to the generators of Q,
  • ngens(Q) to the number of these generators, and
  • gen(Q, i) as well as Q[i] to the i-th such generator.
Examples
julia> R, (x, y, z) = QQ["x", "y", "z"];

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

julia> REL = strictly_upper_triangular_matrix(L);

julia> A, (x, y, z) = pbw_algebra(R, REL, deglex(gens(R)));

julia> I = two_sided_ideal(A, [x^2, y^2, z^2]);

julia> Q, q = quo(A, I);

julia> base_ring(Q)
PBW-algebra over Rational Field in x, y, z with relations y*x = -x*y, z*x = -x*z, z*y = -y*z

julia> modulus(Q)
two_sided_ideal(x^2, y^2, z^2)

julia> gens(Q)
3-element Vector{PBWAlgQuoElem{fmpq, Singular.n_Q}}:
 x
 y
 z

julia> ngens(Q)
3

julia> gen(Q, 2)
y

Elements of GR-Algebras

Types

The OSCAR type for elements of quotient rings of multivariate polynomial rings PBW-algebras is of parametrized form PBWAlgQuoElem{T, S}, where T is the element type of the field over which the GR-algebra is defined (the type S is added for internal use).

Creating Elements of GR-Algebras

Elements of a GR-algebra $Q = A/I$ are created as images of elements of $A$ under the projection map or by directly coercing elements of $A$ into $Q$. The function simplify! reduces a given element with regard to the modulus $I$.

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

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

julia> REL = strictly_upper_triangular_matrix(L);

julia> A, (x, y, z) = pbw_algebra(R, REL, deglex(gens(R)));

julia> I = two_sided_ideal(A, [x^2, y^2, z^2]);

julia> Q, q = quo(A, I);

julia> f = q(y*x+z^2)
-x*y + z^2

julia> typeof(f)
PBWAlgQuoElem{fmpq, Singular.n_Q}

julia> simplify!(f);

julia> f
-x*y

julia> g = Q(y*x+x^2)
x^2 - x*y

julia> f == g
true

Data associated to Elements of GR-Algebras

Given an element f of an affine GR-algebra Q,

  • parent(f) refers to Q.

Ideals in GR-Algebras