Struct Root 

pub struct Root { /* private fields */ }

A root in the dual space of a Cartan subalgebra.

Mathematically: α ∈ 𝔥* represented as a vector in ℝⁿ (rank n). For SU(n+1) (type Aₙ), roots are differences of standard basis vectors: eᵢ - eⱼ.

Example

use lie_groups::Root;

// SU(3) root: e₁ - e₂
let alpha = Root::new(vec![1.0, -1.0, 0.0]);
assert_eq!(alpha.rank(), 3);
assert!((alpha.norm_squared() - 2.0).abs() < 1e-10);

Implementations

impl Root

pub fn new(coords: Vec<f64>) -> Self

Create a new root from coordinates.

pub fn coords(&self) -> &[f64]

Returns the coordinate vector of this root.

pub fn rank(&self) -> usize

Rank of the Lie algebra (dimension of Cartan subalgebra).

pub fn inner_product(&self, other: &Root) -> f64

Inner product ⟨α, β⟩ (standard Euclidean).

pub fn norm_squared(&self) -> f64

Squared norm ⟨α, α⟩.

pub fn reflect(&self, beta: &Root) -> Root

Weyl reflection s_α(β) = β - 2⟨β,α⟩/⟨α,α⟩ · α.

pub fn cartan_integer(&self, beta: &Root) -> i32

Cartan integer ⟨β, α^∨⟩ = 2⟨β,α⟩/⟨α,α⟩.

pub fn is_positive(&self) -> bool

Check if this root is positive (first nonzero coordinate is positive).

pub fn negate(&self) -> Root

Negate this root.

Trait Implementations

impl Clone for Root

fn clone(&self) -> Root

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for Root

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Display for Root

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl PartialEq for Root

fn eq(&self, other: &Root) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl StructuralPartialEq for Root