# A General Framework for Equivariant Neural Networks on Reductive Lie Groups

arXiv: https://arxiv.org/abs/2306.00091 · HTML: https://mlsanity.com/paper/2306.00091/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 16 references that are in the map.
- First posted on arXiv 31 May 2023.

## Builds on (most cited first, 12 of 12)

- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Particle Transformer for Jet Tagging](https://mlsanity.com/paper/2202.03772/) (2022)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [Tensor-reduced atomic density representations](https://mlsanity.com/paper/2210.01705/) (2022)
- [BIP: Boost Invariant Polynomials for Efficient Jet Tagging](https://mlsanity.com/paper/2207.08272/) (2022)

## Built on by (most cited first, 16 of 16)

- [Lie Group Decompositions for Equivariant Neural Networks](https://mlsanity.com/paper/2310.11366/) (2023)
- [Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks](https://mlsanity.com/paper/2409.11772/) (2024)
- [A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks](https://mlsanity.com/paper/2410.04692/) (2024)
- [Molecular Representations in Implicit Functional Space via Hyper-Networks](https://mlsanity.com/paper/2601.22327/) (2026)
- [mathbbZ2× mathbbZ2 Equivariant Quantum Neural Networks: Benchmarking against Classical Neural Networks](https://mlsanity.com/paper/2311.18744/) (2023)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [Equivariant Matrix Function Neural Networks](https://mlsanity.com/paper/2310.10434/) (2023)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Thickness-aware E(3)-Equivariant 3D Mesh Neural Networks](https://mlsanity.com/paper/2505.21572/) (2025)
- [SEAL - A Symmetry EncourAging Loss for High Energy Physics](https://mlsanity.com/paper/2511.01982/) (2025)
- [How unconstrained machine-learning models learn physical symmetries](https://mlsanity.com/paper/2603.24638/) (2026)
- [A Geometric Insight into Equivariant Message Passing Neural Networks on Riemannian Manifolds](https://mlsanity.com/paper/2310.10448/) (2023)
- [The Generalized Skew Spectrum of Graphs](https://mlsanity.com/paper/2505.23609/) (2025)
- [High-Order Deep Meta-Learning with Category-Theoretic Interpretation](https://mlsanity.com/paper/2507.02634/) (2025)
- [bispectrum: Selective G-Bispectra Made Practical](https://mlsanity.com/paper/2605.07270/) (2026)
- [E3DGS: Unified Geometric-Photometric Equivariance for 3D Gaussian Splatting via Color-as-Geometry Embedding](https://mlsanity.com/paper/2607.15536/) (2026)

## Abstract (arXiv)

Reductive Lie Groups, such as the orthogonal groups, the Lorentz group, or the unitary groups, play essential roles across scientific fields as diverse as high energy physics, quantum mechanics, quantum chromodynamics, molecular dynamics, computer vision, and imaging. In this paper, we present a general Equivariant Neural Network architecture capable of respecting the symmetries of the finite-dimensional representations of any reductive Lie Group G. Our approach generalizes the successful ACE and MACE architectures for atomistic point clouds to any data equivariant to a reductive Lie group action. We also introduce the lie-nn software library, which provides all the necessary tools to develop and implement such general G-equivariant neural networks. It implements routines for the reduction of generic tensor products of representations into irreducible representations, making it easy to apply our architecture to a wide range of problems and groups. The generality and performance of our approach are demonstrated by applying it to the tasks of top quark decay tagging (Lorentz group) and shape recognition (orthogonal group).