Equivariant mappings and invariant sets on Minkowski space
Miram Manoel, Leandro Nery de Oliveira
TL;DR
The paper addresses the problem of understanding invariant functions and equivariant mappings for Lorentz-subgroup actions on Minkowski space $\mathbb{R}^{n+1}_1$, adapting invariant-theoretic methods from Euclidean settings to the pseudo-Riemannian context with inner product $\langle\cdot,\cdot\rangle$ of signature $(n,1)$. It introduces a framework where invariant rings $\mathcal{I}(\Gamma)$ and equivariant mappings $\mathcal{M}(\Gamma)$ are constructed, using the gradient relation $J\nabla f$ for invariants and a diagonal-product correspondence $f(x,y)=\langle g(x),y\rangle$ to generate equivariants, and provides a concrete algorithm for groups generated by involutions via Reynolds operators. An illustrative example computes generators for $\mathcal{I}(\Gamma)$ and a basis for $\mathcal{M}(\Gamma)$ in a nontrivial Lorentz-subgroup, demonstrating the method's practical effectiveness. The paper further develops invariant-subspace theory in low dimensions, showing when invariant complements exist and classifying fixed-point subspaces in $\mathbb{R}^{2}_1$ and $\mathbb{R}^{3}_1$, with implications for symmetry-based reduction in relativistic dynamical systems. Overall, these results yield concrete tools for symmetry-detected structure in Lorentzian settings and provide a foundation for systematic invariant and equivariant analysis on Minkowski space.
Abstract
In this paper we introduce the systematic study of invariant functions and equivariant mappings defined on Minkowski space under the action of the Lorentz group. We adapt some known results from the orthogonal group acting on the Euclidean space to the Lorentz group acting on the Minkowski space. In addition, an algorithm is given to compute generators of the ring of functions that are invariant under an important class of Lorentz subgroups, namely when these are generated by involutions, which is also useful to compute equivariants. Furthermore, general results on invariant subspaces of the Minkowski space are presented, with a characterization of invariant lines and planes in the two lowest dimensions.