dhl-mm 0.1.2

Up to 913x faster Lie algebra products for all five exceptional algebras with differentiable PyTorch equivariant layers

DHL-MM: Dynamic Hodge-Lie Matrix Multiplication

Fast Lie algebra multiplication for all five exceptional algebras (G₂, F₄, E₆, E₇, E₈) using sparse structure constants. Includes adjoint-equivariant PyTorch layers, PyTorch Geometric integration, and Lie-algebra-valued quantum simulation.

pip install dhl-mm

$$z_k ;=; \Pi^{h^2}\varphi ;\frac{1}{2} !!!\sum{(i,,j,,k),\in, \mathcal{F}(\mathfrak{g})} !!! x_i ; y_j ; f_{ij}^{;k} ;, \qquad 3 \notin \lbrace \deg C_p(\mathfrak{g}) \rbrace ;;\forall; \mathfrak{g} \in \lbrace G_2,, F_4,, E_6,, E_7,, E_8 \rbrace$$

Left: the computation — sparse gather-multiply-scatter over nonzero structure constants $\mathcal{F}(\mathfrak{g})$, with Z[φ] lattice projection $\Pi^{h^2}_\varphi$ for error control. Right: why it works — no exceptional algebra has a degree-3 Casimir invariant, so the symmetric $d$-tensor vanishes and antisymmetric constants capture the full product.

Quick Start

import dhl_mm

# Load any exceptional algebra — cached, <25ms
e8 = dhl_mm.algebra("E8")
g2 = dhl_mm.algebra("G2")

# Sparse Lie bracket (913× fewer ops than dense for E8)
import numpy as np
x, y = np.random.randn(248), np.random.randn(248)
z = e8.bracket(x, y)

Differentiable PyTorch bracket

from equivariant import SparseLieBracket
import torch

bracket = SparseLieBracket.from_algebra("E8")  # or "G2", "F4", "E6", "E7"
x = torch.randn(248, requires_grad=True)
y = torch.randn(248)
z = bracket(x, y)       # full autograd support
z.sum().backward()       # gradients flow through sparse scatter-add

Adjoint-equivariant neural network

from equivariant import ExceptionalEGNN

model = ExceptionalEGNN(
    in_dim=14, hidden_dim=32, out_dim=1,
    n_layers=3, algebra_name="G2", equivariant=True
)
nodes = torch.randn(10, 14)
edge_index = torch.tensor([[0,1,2,3],[1,2,3,0]], dtype=torch.long)
prediction = model(nodes, edge_index)

PyTorch Geometric

from dhl_mm.pyg import LieBracketConv  # requires torch_geometric

conv = LieBracketConv("E8", equivariant=True)
out = conv(node_features, edge_index)   # equivariant message passing

Quantum simulation

from dhl_mm import E8SpinLattice

lattice = E8SpinLattice(n_sites=8, algebra_name="E8")
state = lattice.random_initial_state()
trajectory = lattice.evolve(state, dt=0.001, steps=500, order=2)
# 8 sites × 500 steps in ~4 seconds on CPU

Quantum Simulation Results

Lie-algebra-valued lattice dynamics under adjoint commutator flow. The sparse bracket makes it feasible to simulate E₈-valued spin chains on CPU.

Killing norm conservation — flat line proves the integrator preserves algebraic structure. Drift ~3×10⁻⁶ over 500 steps.

Correlation spreading — Killing inner product between sites shows information propagation across the lattice.

G₂ vs E₈ — same lattice geometry, different algebras. G₂ (14-dim) drifts ~10⁻⁹, E₈ (248-dim) drifts ~10⁻⁶. Both well-conserved.

Compression Table

Algebra	Dim	Roots	Nonzero f	Full n³	Compression	Jacobi Error
G₂	14	12	120	2,744	22.9×	~1e-14
F₄	52	48	1,196	140,608	117.6×	~1e-16
E₆	78	72	2,208	474,552	215×	~1e-13
E₇	133	126	5,544	2,352,637	424×	~1e-13
E₈	248	240	16,694	15,252,992	913×	~1e-16

All algebras verified to machine epsilon. No approximations.

Why It Works

None of the exceptional Lie algebras have a cubic Casimir invariant:

Algebra	Casimir Degrees
G₂	2, 6
F₄	2, 6, 8, 12
E₆	2, 5, 6, 8, 9, 12
E₇	2, 6, 8, 10, 12, 14, 18
E₈	2, 8, 12, 14, 18, 20, 24, 30

No degree 3 means the symmetric structure constants d_{ij}^k vanish identically for all five. The full product T_i·T_j projected onto the Lie algebra equals [T_i, T_j]/2 exactly. The entire product is determined by the antisymmetric structure constants alone.

How It Works

1. Sparse Structure Constant Engine

Instead of multiplying n×n matrices, the product of two Lie algebra elements X = Σ xᵢTᵢ, Y = Σ yⱼTⱼ is:

(XY)_k = (1/2) Σ_{(i,j,k) ∈ f} xᵢ · yⱼ · f_{ij}^k

where f_{ij}^k are precomputed sparse entries. This is a gather-multiply-scatter operation. Structure constants are precomputed and shipped as .npz files — first algebra load takes <25ms.

2. Z[φ] Exact Arithmetic

Coefficients stored as integer pairs (a, b) representing a + bφ where φ is the golden ratio. Multiplication uses φ² = φ + 1. No floating-point accumulation errors for algebraic inputs.

3. Defect Equation Monitor

h²(t) = h²_Λ - (κ/3) · ρ_defect(t)

Tracks accumulated deviation from the Z[φ] lattice. When h²(t) drops below threshold, coefficients are projected back to the nearest lattice point. Self-correcting computation.

4. Adjoint-Equivariant Neural Network Layers

• AdjointLinearLayer — by Schur's lemma, the only linear map commuting with the adjoint action is a scalar multiple of the identity. Single learnable parameter.
• AdjointBilinearLayer — the two independent adjoint-invariant bilinear operations: antisymmetric bracket + symmetric Killing form scalar.
• EquivariantLieConvLayer — message passing with bracket-based nonlinearity x + α·[agg, W(agg)], following the Lie Neurons pattern (Lin et al. 2024). Adjoint equivariance verified to ~1e-15 for all 5 algebras.
• LieBracketConv — PyTorch Geometric MessagePassing layer, drop-in compatible with any PyG pipeline.

5. Quantum Simulation

• LieHamiltonian — sparse commutator [H, ρ] for Lie-algebra-valued time evolution.
• EquivariantTrotterSuzuki — first and second-order integrators with Killing norm drift tracking.
• E8SpinLattice — nearest-neighbor coupled evolution on a 1D lattice of algebra-valued sites. 8-site E₈ lattice evolves in ~4s on CPU with Killing norm drift ~3×10⁻⁶.

6. Optional C Extension

A pybind11 C++ sparse kernel with OpenMP batched support is included (dhl_mm/csrc/). Falls back to NumPy if not compiled. Build with:

pip install pybind11 && python setup.py build_ext --inplace

Project Structure

dhl_mm/                 Core library (pip install dhl-mm)
  __init__.py             algebra() factory with cached loading
  exceptional_engine.py   ExceptionalAlgebra class for all 5 algebras
  roots.py                Root system builders (G2, F4, E6, E7, E8)
  structure.py            Structure constant computation (Chevalley basis)
  casimir.py              Casimir degree analysis + d-tensor verification
  quantum.py              Trotter-Suzuki evolution + E8 spin lattice
  e8.py                   E8 root system + Frenkel-Kac cocycle
  engine.py               DHLMM class (original E8 engine)
  zphi.py                 Exact Z[phi] arithmetic
  defect.py               Friedmann-style h²(t) error tracker
  pyg.py                  PyTorch Geometric LieBracketConv layer
  csparse.py              C extension wrapper with numpy fallback
  csrc/                   pybind11 C++ sparse kernel (optional)
  data/                   Precomputed .npz structure constants (5 algebras)

equivariant/            PyTorch equivariant neural network layers
  sparse_kernel.py        SparseLieBracket, SparseKillingForm (differentiable)
  layers.py               LieConvLayer, EquivariantLieConvLayer, AdjointLinearLayer
  model.py                ExceptionalEGNN architecture
  benchmark.py            Multi-algebra benchmark suite

exceptional/            Backward-compatible re-exports (delegates to dhl_mm/)

examples/               Demo scripts with output plots
  quantum_sim_demo.py     E8 spin lattice simulation + G2 vs E8 comparison

notebooks/              Interactive demos (Colab-ready)
  e8_in_5_minutes.ipynb   All 5 algebras, benchmarks, equivariance
  equivariant_gnn_demo.ipynb  Train an equivariant GNN on synthetic data

tests/                  Test suites
  test_quantum.py         Quantum simulation: conservation, Trotter accuracy

benchmarks/             Performance benchmarks
  sparse_kernel_bench.py  C extension vs numpy vs dense, all algebras

scripts/                Build utilities
  precompute.py           Generate .npz structure constant caches

Running Tests

python exceptional/tests/test_all.py              # All 5 algebras: Jacobi, antisymmetry, Killing
python equivariant/tests/test_equivariance.py      # Equivariance, gradients, consistency
python tests/test_quantum.py                       # Quantum sim: conservation, Trotter accuracy
python equivariant/benchmark.py                    # Sparse vs dense timing, all algebras
python benchmarks/sparse_kernel_bench.py           # C extension vs numpy vs dense
python examples/quantum_sim_demo.py                # E8 spin lattice demo (generates plots)

Applications

• Quantum simulation (Hamiltonian commutators, Trotter-Suzuki decomposition)
• Lattice gauge theory (exceptional gauge group computations)
• Lie group integration (Runge-Kutta-Munthe-Kaas methods)
• Equivariant neural networks (molecular property prediction, physics-informed ML)
• Post-quantum cryptography (E₈ lattice operations)
• Geometric algebra / Clifford algebra acceleration

See APPLICATIONS_ROADMAP.md for detailed directions.

Requirements

• Python ≥3.9
• NumPy ≥1.20
• PyTorch ≥2.0 (for equivariant layers, optional)
• SciPy ≥1.10 (for equivariance tests, optional)
• torch_geometric (for LieBracketConv, optional)
• matplotlib (for quantum simulation plots, optional)

License

MIT