A note on complex Lie Algebras isomorphic to their conjugate
Algebraic Geometry
2026-04-09 v1 Number Theory
Representation Theory
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 -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.
Cite
@article{arxiv.2604.06979,
title = {A note on complex Lie Algebras isomorphic to their conjugate},
author = {Cyril Demarche},
journal= {arXiv preprint arXiv:2604.06979},
year = {2026}
}
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6 pages