Wick-rotations of pseudo-Riemannian Lie groups

Abstract

We study Wick-rotations of left-invariant metrics on Lie groups, using results from real GIT (Helleland and Hervik, 2018; Helleland and Hervik, 2019). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group (a Lie group with a left-invariant metric) can be Wick-rotated to a Riemannian Lie group. We define a Cartan involution of a general Lie algebra, and prove a general version of É. Cartan’s result, namely the existence and conjugacy of Cartan involutions.