JuliaCon 2025

Lie groups are a tool to work with groups, namely they are a manifold equipped with a smooth group operation. In practice these appear when dealing with rotations on a Euclidean space, the set of translations and rotations, or so-called symplectic matrices in Hamiltonian systems.
A main challenge is that these spaces are nonlinear. For example, adding two rotation matrices does usually not yield a rotation matrix. It also requires some additional tools to for example talk about what a “direction” is, for example in optimisation, when looking for a descent direction in order to find a minimizer of a function defined on a Lie group.

In this talk we introduce the package LieGroups.jl to perform numerical operations on and with Lie groups, for example in state estimation in robotics.
We first give a gentle introduction into Lie groups and Lie algebras, we discuss the main features of the package, which are

• an interface to define Lie groups based on manifolds defined in Manifolds.jl
• an interface for Lie algebras, especially employing its vector space structure
• the main tools to define and work with Lie groups
• a library of Lie groups and an overview of features available