Lie Group Networks are neural architectures designed for data that naturally resides on or is governed by continuous symmetry groups (Lie groups) — such as $SO(3)$ (3D rotations), $SE(3)$ (rigid body transformations), $SU(2)$ (quantum spin), and $GL(n)$ (general linear transformations) — operating in the Lie algebra (the linearized tangent space where group operations simplify to vector addition) and mapping to the Lie group manifold through the exponential map, enabling differentiable computation on smooth continuous symmetry structures.

What Are Lie Group Networks?

• Definition: Lie group networks process data that lives on continuous symmetry groups (Lie groups) by leveraging the Lie algebra — the tangent space at the identity element where the curved group manifold is locally linearized. The exponential map ($exp: mathfrak{g} o G$) maps from the flat algebra to the curved group, and the logarithm map ($log: G o mathfrak{g}$) maps back. Neural network operations are performed in the algebra (where standard linear operations apply) and the results are mapped back to the group when geometric quantities are needed.
• Lie Algebra Operations: In the Lie algebra, group composition (which is non-linear on the manifold) corresponds to vector addition (linear) for small transformations, and the Lie bracket $[X, Y] = XY - YX$ captures the non-commutativity of the group. Neural networks can use standard MLP operations in the algebra space, then exponentiate to obtain group elements.
• Equivariant by Design: By parameterizing transformations through the Lie algebra and constructing layers that respect the algebra's structure (equivariant linear maps between representation spaces), Lie group networks achieve equivariance to the continuous symmetry group without the discretization approximations of finite group methods.

Why Lie Group Networks Matter

• Robotics and Pose: Robot joint configurations, end-effector poses, and rigid body states are elements of $SE(3)$ — the group of 3D rotations and translations. Standard neural networks that represent poses as raw matrices or quaternions do not respect the group structure, producing interpolations and predictions that violate the geometric constraints (non-unit quaternions, non-orthogonal rotation matrices). Lie group networks operate natively on $SE(3)$, producing geometrically valid predictions by construction.
• Continuous Symmetry: Many physical symmetries are continuous — rotation by any angle, translation by any distance, scaling by any factor. Discrete group methods (4-fold rotation, 8-fold rotation) approximate these continuous symmetries with finite samples. Lie group networks handle continuous symmetries exactly through the algebraic structure.
• Quantum Mechanics: Quantum states transform under $SU(2)$ (spin) and $SU(3)$ (color charge). Lie group networks that operate on these groups can process quantum mechanical data while respecting the symmetry structure of the underlying physics, enabling equivariant quantum chemistry and particle physics applications.
• Manifold-Valued Data: When outputs must lie on a specific manifold (rotation matrices must be orthogonal, probability distributions must be non-negative and normalized), standard networks produce unconstrained outputs that require post-hoc projection. Lie group networks produce outputs that lie on the correct manifold by construction through the exponential map.

Lie Group Machinery

Lie Group Networks are continuous symmetry solvers — processing data that lives on smooth manifolds of transformations by leveraging the linearized algebra where neural network operations are natural, then mapping results back to the curved geometric space where physical meaning resides.