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.

Highlights

  • This paper is motivated first of all by the study of Wick-rotations of pseudo-Riemannian manifolds defined in [3].Given a pseudo-Riemannian manifold (M, g) of signature (p, q), it is interesting know whether it can be Wick-rotated to another space (M, g ) (w.r.t. a fixed point p ∈ M ∩ M ) of signature p + q = p + q

  • If we look at a semi-simple complex Lie group GC equipped with the left-invariant Killing form: −κ, there are natural examples of Wick-rotations to find at the identity point, because there exist real forms

  • By the theory of semi-simple Lie groups, one may always Wick-rotate a real form (G, −κ) ⊂ (GC, −κ) to a Riemannian Lie group, because of the existence of a Cartan involution of the Lie algebra g. Motivated by this example, for a general pseudo-Riemannian Lie group (G, g), an interesting question one may ask: Given a pseudo-Riemannian Lie group (G, g), when can it be Wick-rotated to a Riemannian Lie group (G, g )? Suppose (G, g) is Wick-rotated to a Riemannian Lie group (G, g ), in view of the results given in [2,3,4], the so called Wick-rotatable tensors restricted to g must be fixed by the isometry action of some Cartan involution θ ∈ O(p, q) of the metric

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Summary

Introduction

This paper is motivated first of all by the study of Wick-rotations of pseudo-Riemannian manifolds defined in [3]. Suppose (G, g) is Wick-rotated to a Riemannian Lie group (G , g ), in view of the results given in [2,3,4], the so called Wick-rotatable tensors restricted to g must be fixed by the isometry action (induced from the metric) of some (linear) Cartan involution θ ∈ O(p, q) of the metric. This could for instance be the Riemann tensor R (as mentioned above), and is related to the fact that R can be embedded into the same complex orbit as R (the Riemann tensor of (G , g ) restricted to g), i.e. O(p + q, C) · R ∋ R. A complex Lie group shall always be denoted by the symbol: GC

Real forms and left-invariant metrics
Wick-rotations of pseudo-Riemannian manifolds
Real GIT on compatible representations
The isometry action on bilinear maps into the Lie algebra
Wick-rotatable tensors of pseudo-Riemannian manifolds
An invariant of Wick-rotation of Lie groups
Conjugacy of Cartan involutions
Wick-rotating a Lorentzian signature
A remark on Wick-rotatable tensors of Lie groups
Findings
Wick-rotating an algebraic soliton

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