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arXiv · 2604.06979

A note on complex Lie Algebras isomorphic to their conjugate

Cyril Demarche
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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.

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Cyril Demarche. 2026-04-08. A note on complex Lie Algebras isomorphic to their conjugate. https://arxiv.org/abs/2604.06979

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