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

quo — Method

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 $A\to A/I$.

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{QQFieldElem, 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

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

modulo the two-sided ideal

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

exterior_algebra — Function

exterior_algebra(K::Field, numVars::Int)
exterior_algebra(K::Field, listOfVarNames::AbstractVector{<:VarName})

The first form returns an exterior algebra with coefficient field K and numVars variables: numVars must be positive, and the variables are called e1, e2, ....

The second form returns an exterior algebra with coefficient field K, and variables named as specified in listOfVarNames (which must be non-empty).

NOTE: Creating an exterior_algebra with many variables will create an object occupying a lot of memory (probably cubic in numVars).

Examples

julia> ExtAlg, (e1,e2)  =  exterior_algebra(QQ, 2);

julia> e2*e1
-e1*e2

julia> (e1+e2)^2  # result is automatically reduced!
0

julia> ExtAlg, (x,y)  =  exterior_algebra(QQ, ["x","y"]);

julia> y*x
-x*y

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{QQFieldElem, 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{QQFieldElem, 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.