LIE GROUP ENGINE — PhD EDITION

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LIE GROUP ENGINE — PhD EDITION

Wickerson Studios · 2026 · Powered by Claude AI

www.wickersonstudios.com

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"The universe chooses its symmetry group.

We have found it, piece by piece, in accelerators."

For PhD students in Advanced Mathematics, Theoretical

Physics, Mathematical Physics, and Computational Science.

The algebraic successor to the Advanced Tensor Engine

and Spinor Engine. Lie groups are the engine that

dictates which tensors and spinors are allowed to exist.

Every particle is a representation. Every force is a

connection on a principal bundle. Every conserved charge

is a Lie algebra generator.

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SERIES POSITION

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This is the third script in the Wickerson Studios

advanced mathematics series:

Level 1 WickersonStudios_TensorVisualiser.cs

Scalars, vectors, matrices, ML tensor shapes.

Requires: basic linear algebra.

Level 2 WickersonStudios_AdvancedTensorEngine.cs

Einstein summation, GR tensors, quantum states,

tensor decompositions, advanced ML geometry.

Requires: differential geometry, QM basics.

Level 3 WickersonStudios_SpinorEngine.cs

Clifford algebras, Weyl/Dirac/Majorana spinors,

Dirac equation, geometric algebra, topology.

Requires: Clifford algebras, QFT basics.

Level 4 WickersonStudios_LieGroupEngine.cs ← THIS FILE

Lie groups, root systems, representations,

Standard Model, GUTs, exceptional groups.

Requires: all of the above.

The series arc: Tensors (the language) → Spinors (the

square roots of tensors) → Lie Groups (the symmetry

engine that decides what exists).

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WHAT IS INCLUDED

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WickersonStudios_LieGroupEngine.cs

The Grasshopper C# Script component.

Drop into a C# Script node. Wire 12 inputs.

No NuGet packages. No DLLs. No installation.

2,543 lines. Zero external dependencies.

WickersonStudios_LieGroupEngine_README.txt

This file.

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THE 32 MODES

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── GROUP A: LIE GROUPS & ALGEBRAS (Modes 0–7) ──────────

Mode 0 — Lie Group Definition

─────────────────────────────────────────────────────────

G: smooth manifold + group, operations smooth

Renders a torus as a concrete Lie group (U(1)×U(1)),

with latitude curves as one-parameter subgroups and the

identity element marked. Param selects the example group:

0 = U(1)×U(1) — displayed torus

1 = SU(2) ≅ S³

2 = SO(3) ≅ RP³

3 = SL(2,ℝ) — hyperboloid geometry

4 = GL(n,ℝ) — n²-dimensional manifold

5 = Sp(2n,ℝ) — symplectic group

Recommended: Param 0 AnimSpeed 0.02

Mode 1 — Matrix Lie Groups

─────────────────────────────────────────────────────────

SL, O, U, Sp — matrix groups by constraint

Renders an n×n group element alongside MᵀM (should be

≈ I for orthogonal) and det(M) as a bar. Shows the

defining constraints of each classical matrix group.

Rank controls the matrix size (1–5).

Dimension table shown in Report:

GL(n,ℝ): n² SL(n,ℝ): n²−1 O(n): n(n-1)/2

U(n): n² SU(n): n²−1 Sp(2n): n(2n+1)

Recommended: Rank 3 (3×3 matrices)

Mode 2 — Lie Algebra g = T_eG

─────────────────────────────────────────────────────────

[Tᵢ, Tⱼ] = fᵢⱼₖ Tₖ (structure constants)

Renders the generators of the selected Lie algebra as

matrix components (real and imaginary parts). A pulsing

bar shows the commutator [T₁,T₂] animating. Param:

0 = su(2): 3 generators 3 = u(1): 1 generator

1 = su(3): 8 generators 4 = sl(2,ℝ): 3 generators

2 = so(3): 3 generators 5 = sp(2): 4 generators

Recommended: Param 0 or 1 AnimSpeed 0.025

Mode 3 — Exponential Map exp: g → G

─────────────────────────────────────────────────────────

exp(iθ) traces a circle on S¹ = U(1)

Animates the exponential map for U(1): a tangent vector

(the Lie algebra) on the vertical axis maps to a point

tracing the unit circle (the group S¹). The arc from

the identity to the current point is shown with length

proportional to θ.

exp(X) = I + X + X²/2! + X³/3! + ...

Recommended: AnimSpeed 0.015 (slow enough to follow)

Mode 4 — Baker-Campbell-Hausdorff Formula

─────────────────────────────────────────────────────────

log(eˣeʸ) = X+Y + ½[X,Y] + 1/12[[X,Y],Y] + ...

Shows the BCH series as a bar chart: each bar is one

order of correction. A cumulative-sum bar shows how

the total approaches the true product. Demonstrates

that group multiplication is determined by the bracket

structure alone.

Mode 5 — Adjoint Representation

─────────────────────────────────────────────────────────

ad_X(Y) = [X, Y] gauge bosons live in adjoint rep

Renders one of the three adjoint matrices for su(2) as

a 3×3 heatmap (each adjoint matrix acts on the 3D space

of the Lie algebra itself). Shows the product of two

adjoint matrices. Param selects which generator T₁, T₂,

or T₃.

The key physical fact: W±, Z⁰ are in the adjoint of

SU(2). The 8 gluons are in the adjoint of SU(3).

Recommended: Param 0 (show ad_{T₁})

Mode 6 — Killing Form B(X,Y) = Tr(ad_X ∘ ad_Y)

─────────────────────────────────────────────────────────

Negative definite ↔ compact semisimple Lie group

Renders the Killing form as a 3×3 matrix for su(2)

(Param 0, compact, all diagonal = −2) or sl(2,ℝ)

(Param 1, non-compact, mixed signs). Also shows the

su(3) case as a 4×4 block (N=4).

The Yang-Mills Lagrangian L = −(1/4g²)B(F,F) uses

the Killing form as its kinetic term.

Mode 7 — Classification

─────────────────────────────────────────────────────────

simple, semisimple, solvable, nilpotent hierarchy

Draws the full classification tree as a geometric

hierarchy. Shows the Levi decomposition g = s ⋊ r

and Cartan's complete list of simple Lie algebras:

Classical: Aₙ, Bₙ, Cₙ, Dₙ

Exceptional: G₂, F₄, E₆, E₇, E₈

These are ALL simple Lie algebras over ℂ — the

classification is complete and exhaustive.

── GROUP B: ROOT SYSTEMS (Modes 8–14) ──────────────────

Mode 8 — Cartan Subalgebra h and Weight Lattice

─────────────────────────────────────────────────────────

h = maximal abelian subalgebra Λ = weight lattice

Renders the weight lattice for Aₙ = su(n+1). Root

lattice points are shown larger, dominant weights

(positive Weyl chamber) in a different shade, and

all lattice points in a third shade. Rank controls n.

Recommended: Rank 2 (A₂ hexagonal lattice)

Mode 9 — Root System Φ

─────────────────────────────────────────────────────────

sα(β) = β − ⟨β,αᵥ⟩α (Weyl reflection)

Renders the selected root system in 2D with root

vectors from the origin. ShowWeyl adds hyperplane

walls. Param selects the system:

0 = A₁: 2 roots (line segment)

1 = A₂: 6 roots (regular hexagon) ← recommended

2 = B₂: 8 roots (two lengths, 4+4)

3 = G₂: 12 roots (handled in mode 14)

4 = A₃: 12 roots (3D projection)

5 = B₃: 18 roots (3D projection)

Recommended: Param 1 ShowWeyl TRUE AnimSpeed 0.02

Mode 10 — A-Series Roots Aₙ

─────────────────────────────────────────────────────────

Roots eᵢ − eⱼ W(Aₙ) = Sₙ₊₁

Renders A₁ through A₄ as progressively higher-

dimensional root systems. Param selects which:

0 = A₁ (2 roots) 2 = A₃ (12 roots)

1 = A₂ (6 roots) 3 = A₄ (20 roots)

The Weyl group is the symmetric group S_{n+1}.

Number of roots: n(n+1).

Mode 11 — B, C, D Series

─────────────────────────────────────────────────────────

Bₙ=so(2n+1) Cₙ=sp(2n) Dₙ=so(2n)

Shows all three classical families with their root

length ratios and accidental isomorphisms. Param:

0 = B₂ (8 roots, 2 lengths, long+short)

1 = C₂ (8 roots, B₂ dual — lengths swapped)

2 = D₃ ≅ A₃ (12 roots, 1 length)

Note D₂=A₁⊕A₁, D₃=A₃, B₁=C₁=A₁.

Mode 12 — Weyl Group W

─────────────────────────────────────────────────────────

W = reflections sα in root hyperplanes

Renders the 6 Weyl chambers of A₂ as alternating

coloured sectors. An animated point orbits through

the chambers. The Weyl group acts simply transitively

on the chamber set — one element per chamber.

ShowWeyl adds chamber boundary walls explicitly.

Recommended: ShowWeyl TRUE AnimSpeed 0.02

Mode 13 — Dynkin Diagrams

─────────────────────────────────────────────────────────

nodes = simple roots bonds = angle between roots

Renders each Dynkin diagram as a node-edge graph

in Rhino geometry. Cycle Param 0–8 to see all:

0 = Aₙ (linear chain, single bonds)

1 = Bₙ (linear + double bond at one end)

2 = Cₙ (linear + double bond, arrow reversed)

3 = Dₙ (linear + Y-fork at one end)

4 = E₆ (chain of 5 + one branch node)

5 = E₇ (chain of 6 + one branch node)

6 = E₈ (chain of 7 + one branch node)

7 = G₂ (2 nodes + triple bond)

8 = F₄ (4 nodes, single + double)

Recommended: cycle Param 0→8 slowly AnimSpeed 0.01

Mode 14 — Exceptional Roots G₂ and F₄

─────────────────────────────────────────────────────────

G₂ = Aut(𝕆) F₄ = Aut(J₃(𝕆))

Shows the exceptional root systems:

Param 0: G₂ — 12 roots (6 long + 6 short rotated 30°)

Length ratio √3. Weyl group = D₆ (order 12).

G₂ is the automorphism group of the octonions!

Param 1: F₄ — 48 roots (projected from 4D)

32 long + 16 short half-integer roots.

F₄ = automorphism of exceptional Jordan algebra.

The Freudenthal-Tits magic square connecting all five

exceptional groups via ℝ,ℂ,ℍ,𝕆 is explained in Report.

── GROUP C: REPRESENTATIONS (Modes 15–20) ──────────────

Mode 15 — Highest Weight Theory

─────────────────────────────────────────────────────────

V(λ): dominant integral weight ↔ irrep (1:1)

Renders the weight diagram of the V(Weight+1, 0)

representation of su(3) — a triangle of weight points

with step operators as arrows. The highest weight is

marked large. Weyl dimension formula is given.

Recommended: Weight 1 or 2 (dim 6 or 10)

Mode 16 — SU(2) Representations (spin-j)

─────────────────────────────────────────────────────────

dim = 2j+1 J±|j,m⟩ = √(j∓m)(j±m+1) |j,m±1⟩

Renders spin-j multiplets as weight points on a line

with raising/lowering operator arrows. Bar heights show

m-dependent Casimir contribution. Weight controls 2j:

Weight 0 = spin-0 (singlet) dim = 1

Weight 1 = spin-½ (doublet) dim = 2 ← spinor

Weight 2 = spin-1 (triplet) dim = 3

Weight 3 = spin-3/2 (quartet) dim = 4

Weight 4 = spin-2 (quintet) dim = 5

Weight 5 = spin-5/2 (sextet) dim = 6

Recommended: Weight 1→5 AnimSpeed 0.03

Mode 17 — SU(3) Representations (p,q)

─────────────────────────────────────────────────────────

dim = (p+1)(q+1)(p+q+2)/2

Renders the weight diagram of the (p,q) representation

of su(3) as a triangle/hexagon of weight points.

Param controls p (0–5), Weight controls q (0–5).

Famous representations:

(1,0) = 3 (0,1) = 3̄ (1,1) = 8 (adjoint)

(3,0) = 10 (2,0) = 6 (2,2) = 27

Recommended: Param 1, Weight 1 (dim 8, octet)

Mode 18 — Clebsch-Gordan Decomposition

─────────────────────────────────────────────────────────

V(j₁) ⊗ V(j₂) = ⊕ V(j) j from |j₁-j₂| to j₁+j₂

Shows SU(2) tensor product decomposition as stacked

weight diagrams. Param controls j₁ = Param/2,

Weight controls j₂ = Weight/2.

Examples:

Param 1, Weight 1: ½ ⊗ ½ = 1 ⊕ 0

Param 2, Weight 1: 1 ⊗ ½ = 3/2 ⊕ ½

Param 2, Weight 2: 1 ⊗ 1 = 2 ⊕ 1 ⊕ 0

Mode 19 — Young Tableaux

─────────────────────────────────────────────────────────

Schur-Weyl duality: (ℂⁿ)^⊗d = ⊕ V_λ ⊗ S^λ

Renders a Young diagram as box geometry and its

conjugate (transposed) diagram side by side.

Param controls row-1 length, Weight controls row-2

length. The hook length formula for dim(S^λ) is

given in Report.

Mode 20 — Characters and Weyl Formula

─────────────────────────────────────────────────────────

χⱼ(θ) = sin((2j+1)θ/2) / sin(θ/2)

Plots the character χⱼ(θ) of the spin-j representation

as a height curve over the angle θ ∈ [−π,π]. Param

controls j = (Param+1)/2. The Weyl character formula,

orthogonality of characters, and connection to the

Fourier series are given in Report.

Recommended: cycle Param 0→5 to see different spins

── GROUP D: PHYSICS (Modes 21–26) ──────────────────────

Mode 21 — SU(2) Physics: Spin, Isospin, Weak

─────────────────────────────────────────────────────────

j=½ doublet j=1 triplet/adjoint same algebra!

Shows the Bloch sphere (qubit state space) with an

animated state vector. Explains how the SAME Lie

algebra su(2) appears in three distinct physical

contexts — electron spin, nuclear isospin, and weak

isospin — and why this matters for the SM.

Recommended: AnimSpeed 0.025

Mode 22 — SU(3) Physics: Color, Quarks, Hadrons

─────────────────────────────────────────────────────────

3⊗3̄ = 8⊕1 confinement eightfold way

Left panel: the pseudo-scalar meson OCTET hexagon with

8 states (π, K, η). Right panel: the quark triangle

(u, d, s fundamental representation). Param selects

the physical context:

0 = meson octet display

1 = decuplet layout

2 = fundamental (quark triangle)

The prediction of Ω⁻ (strangeness −3) from filling the

decuplet multiplet is highlighted. Gell-Mann Nobel 1969.

Mode 23 — Standard Model Gauge Group

─────────────────────────────────────────────────────────

G_SM = SU(3)_c × SU(2)_L × U(1)_Y dim = 12

Renders all 12 gauge generators as geometric points:

8 gluons (outer ring, SU(3) adjoint)

3 weak bosons W₁,W₂,W₃ (middle ring, SU(2) adjoint)

1 hypercharge boson B (central point, U(1))

The Report gives the full fermion content, electroweak

mixing angle θ_W, and the 19 free parameters of the SM.

Mode 24 — Grand Unified Theories

─────────────────────────────────────────────────────────

G_SM ⊂ SU(5) ⊂ SO(10) ⊂ E₆

Renders the SM subgroup (inner ring, 12 generators)

nested inside the GUT group (outer ring, extra bosons).

Param selects the GUT:

0 = SU(5) Georgi-Glashow (rank 4, dim 24, 12 X,Y)

1 = SO(10) Fritzsch-Minkowski (rank 5, dim 45)

2 = E₆ (rank 6, dim 78, 27-dim fundamental)

4 = SO(10) with spinor 16 highlighted

(one SM generation = one irreducible rep!)

The right-handed neutrino arises automatically in

SO(10) spinor 16 — seesaw mechanism is geometric.

Mode 25 — Supersymmetry (SUSY)

─────────────────────────────────────────────────────────

{Qα, Q̄β̇} = 2σᵘαβ̇ Pᵤ graded Lie algebra

Renders boson-fermion SUSY multiplet pairs side by side

with connecting Q-operator arrows. The key anticommutator

{Q,Q̄} = 2P means energy = square of supercharge, implying

E ≥ 0 (positive energy theorem). The Haag-Łopuszański-

Sohnius theorem — that SUSY is the ONLY non-trivial

extension of Poincaré — is stated.

Mode 26 — Virasoro Algebra

─────────────────────────────────────────────────────────

[Lₙ,Lₘ] = (n−m)L_{n+m} + c/12·n(n²−1)δ

Renders the infinite tower of Virasoro modes Lₙ as

oscillations on a circle (worldsheet). Positive and

negative modes shown in different channels. Param

controls how many modes are displayed (0–5).

Central charge c = 26 (bosonic string) or c = 15

(superstring) for anomaly cancellation.

Recommended: Param 3 AnimSpeed 0.02

── GROUP E: EXCEPTIONAL & ADVANCED (Modes 27–31) ───────

Mode 27 — E₈ Root System

─────────────────────────────────────────────────────────

240 roots rank 8 dim 248 |W| = 696,729,600

Renders two classes of E₈ roots as 3D projections:

· Long-type roots: ±eᵢ±eⱼ (48 displayed, blue)

· Half-integer roots: ½Σεᵢeᵢ, even # of minus (48 displayed)

The view rotates with animation to reveal the 8D structure.

KEY FACTS:

All 240 roots have the same length (unlike B,C,F₄,G₂)

E₈ lattice = densest sphere packing in ℝ⁸ (Viazovska, 2016)

Heterotic string gauge group: E₈ × E₈ or SO(32)

Gosset polytope 4₂₁ has 240 vertices = E₈ roots

Recommended: ShowWeyl TRUE ShowRoots TRUE AnimSpeed 0.02

Mode 28 — Principal G-Bundles

─────────────────────────────────────────────────────────

Gauge theory = geometry of a connection on P(M,G)

Renders a principal bundle as a base space (spacetime M)

with circle fibres (G = U(1)) above each base point,

and animated horizontal lifts (the connection).

Gauge transformation = change of trivialisation.

Field strength F = curvature of the connection.

This mode visualises the geometric statement:

FORCES ARE CONNECTIONS ON PRINCIPAL BUNDLES.

Mode 29 — Yang-Mills Equations

─────────────────────────────────────────────────────────

Fᵤᵥ = ∂ᵤAᵥ − ∂ᵥAᵤ + [Aᵤ,Aᵥ] DᵤFᵤᵥ = Jᵥ

Renders a 2D lattice of gauge field vectors (arrows)

with the curvature Fₓᵧ shown as height bars at each

lattice point. The animated field configuration shows

how the non-abelian commutator [Aᵤ,Aᵥ] creates self-

interaction (gluons interact with each other, photons

do not). Param scales the field coupling strength.

Instanton equation F = ±⋆F and BPST solutions

are described in Report.

Recommended: Param 2 AnimSpeed 0.015

Mode 30 — Chern-Weil Theory

─────────────────────────────────────────────────────────

cₖ = [σₖ(F/2πi)] ∈ H²ᵏ(M,ℤ) (from curvature to topology)

Renders the curvature as height modulation on a sphere

(the base manifold), with the first Chern number shown

as a bar. Demonstrates that characteristic classes

are topological invariants extracted purely from the

curvature form F.

APPLICATIONS covered in Report:

· Instanton number = second Chern class c₂

· Quantum anomalies ↔ Chern-Simons forms

· TKNN invariant (QHE Hall conductance = Chern number)

· Atiyah-Singer index theorem: Ind(D) = ∫ch(E)Â(M)

Mode 31 — Kac-Moody and Affine Lie Algebras

─────────────────────────────────────────────────────────

ĝ = g⊗ℂ[t,t⁻¹] ⊕ ℂc ⊕ ℂd (affine extension)

Renders the affine Dynkin diagram of Â₁ (3-node cycle)

alongside an infinite tower of mode bars (the Fourier

modes n=−4,...,+4 of the loop algebra). Param selects

the algebra rank. Applications: 2D CFT, WZW models,

integrable systems, and the conjectured E₁₀/E₁₁

symmetries of M-theory.

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INPUTS (12)

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Mode int 0–31 Which concept to visualise.

Use the 5 group tabs to browse.

Param int 0–5 Sub-parameter. Meaning varies:

· Group A: which example algebra

· Group B: which root system

· Mode 13: which Dynkin diagram

(0=A 1=B 2=C 3=D 4=E₆ 5=E₇

— continue in same slider)

· Mode 17: p-value of (p,q) rep

· Mode 18: j₁ = Param/2

· Mode 24: which GUT group

· Mode 26: number of extra modes

Rank int 1–8 Dimension / rank of algebra.

· Mode 1: matrix size n

· Mode 8: Cartan rank n

· Most other modes: ignored

Weight int 0–5 Dominant weight / rep label.

· Mode 15: Dynkin label

· Mode 16: 2j (spin = Weight/2)

· Mode 17: q-value of (p,q) rep

· Mode 18: j₂ = Weight/2

· Mode 19: row-2 Young length

· Most other modes: ignored

CellSize float 0.2–4 Size of each 3D box in Rhino

units. 0.8–1.0 for everyday use.

2.0–3.0 for presentation/projector.

Gap float 0–1.5 Spacing between cells.

0.10–0.15 recommended.

Animate bool toggle Enables 80ms animation loop.

Requires Timer component.

AnimSpeed float 0.005–0.3 Speed per tick.

0.020–0.030 recommended.

Slow: 0.010 Fast: 0.050

Reset bool button Clears static state and rewinds

all animations.

ShowWeyl bool toggle Show Weyl chambers and reflection

hyperplanes in root system modes.

Most useful: modes 9, 12, 14.

ShowRoots bool toggle Add root lattice overlay to weight

diagrams. Most useful: modes 8, 15,

16, 17.

Precision int 1–4 Decimal places in Report text.

3 recommended. 4 for E₈/GR modes.

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OUTPUTS (7)

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RootGeo Main geometry — root system points,

weight diagram lattice, group element

matrices. Connect to Preview component.

Assign light grey / white material.

WeylGeo Weyl chambers, hyperplane walls, orbit

curves on S¹/S³. Connect to separate

Preview. Assign accent colour (amber).

AlgebraGeo Multiplication tables, Dynkin diagram

node-edge geometry, classification trees.

Connect to separate Preview. Assign

cyan or teal.

RepGeo Weight diagram of representation, CG

decomposition bars, character curves.

Connect to separate Preview. Assign

violet or purple.

BundleGeo Principal bundle fibres, Yang-Mills field

vectors, Chern class height surfaces.

Connect to separate Preview. Assign

gold or orange.

Report Full educational text for current mode.

Connect to a wide Grasshopper Panel.

Enable "Wrap Text". Use monospaced font.

Formula Clean mathematical formula (LaTeX-ready

notation). Connect to a small Panel.

RECOMMENDED MATERIAL ASSIGNMENT:

RootGeo White / light grey

WeylGeo Amber (Weyl chambers and walls)

AlgebraGeo Cyan (algebra structure)

RepGeo Violet / purple (representations)

BundleGeo Gold / orange (gauge/bundle geometry)

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LIE GROUP ENGINE — PhD EDITION

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