Wick-rotations of pseudo-Riemannian Lie groups

TL;DR

This paper investigates the conditions under which pseudo-Riemannian Lie groups can be Wick-rotated to Riemannian Lie groups, introducing invariants and extending Cartan's involution results using real GIT techniques.

Contribution

It introduces an invariant for Wick-rotation of Lie groups and characterizes when such groups can be Wick-rotated to Riemannian counterparts, extending Cartan's involution theory.

Findings

An invariant for Wick-rotation of Lie groups is established.

Criteria for Wick-rotation to Riemannian Lie groups are described.

A generalization of Cartan's involution existence and conjugacy is proved.

Abstract

We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (\cite{1}, \cite{2}, \cite{3}). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group can be Wick-rotated to a Riemannian Lie group. We also prove a general version (for general Lie algebras) of $\acute{E}$. Cartan's result, namely the existence and conjugacy of Cartan involutions.