A note on complex Lie Algebras isomorphic to their conjugate

TL;DR

This paper investigates whether complex Lie algebras isomorphic to their conjugate are defined over the reals, providing a counterexample and analyzing the descent obstructions using Brauer groups.

Contribution

It constructs a 10-dimensional nilpotent complex Lie algebra counterexample and computes the generic descent obstruction via Brauer groups.

Findings

Counterexample disproves the conjecture that all such Lie algebras are real-defined.

Provides a method to compute the descent obstruction using Brauer groups.

Enhances understanding of the relationship between complex and real Lie algebras.

Abstract

A real Lie algebra defines by extension of scalars a complex Lie algebra that is isomorphic to its Galois conjugate. In this paper, we are interested in the converse property: given a complex Lie algebra that is isomorphic to its conjugate, is it defined over the real numbers? We prove the existence of a $10$-dimensional nilpotent complex Lie algebra for which the answer is negative, disproving a recent conjecture by Der\'e. In addition, we compute the generic obstruction to this descent problem in terms of Brauer groups.