Quaternion Algebras

Quaternion algebras can be created over any field: the rationals, number fields, finite fields, …

The main constructor for the quaternion algebra over F, such that \(i^2 = a\) and \(j^2 = b\) is QuaternionAlgebraStructure::new(F, a, b).

use algebraeon::structures::Rational;
use algebraeon::rings::quaternion_algebra::QuaternionAlgebraStructure;
use algebraeon::rings::structure::*;
use algebraeon::structures::{EqSignature, MetaType};

let h = QuaternionAlgebraStructure::new(
    Rational::structure(),
    -Rational::ONE,
    -Rational::TWO,
);

let i = h.i();
let j = h.j();
let k = h.mul(&i, &j);

// ij = k and ji = -k
assert!(h.equal(&k, &h.k()));
assert!(h.equal(&h.mul(&j, &i), &h.neg(&k)));

Elements can also be written as strings and read with parse_quaternion. The factors of a product are multiplied in the order they are written, so "i*j" and "j*i" give different elements.

use algebraeon::structures::Rational;
use algebraeon::rings::parsing::parse_quaternion;
use algebraeon::rings::quaternion_algebra::QuaternionAlgebraStructure;
use algebraeon::rings::structure::*;
use algebraeon::structures::{EqSignature, MetaType};

let h = QuaternionAlgebraStructure::new(
    Rational::structure(),
    -Rational::ONE,
    -Rational::TWO,
);

let q = parse_quaternion("2 + 3i + 5j - 2k", &h).unwrap();
assert!(h.equal(&q, &h.from_components(
    Rational::from(2),
    Rational::from(3),
    Rational::from(5),
    Rational::from(-2),
)));

assert!(h.equal(&parse_quaternion("i*j", &h).unwrap(), &h.k()));
assert!(h.equal(&parse_quaternion("j*i", &h).unwrap(), &h.neg(&h.k())));