Constructing a complex Lie algebra isomorphic to its complex conjugate but not definable over reals

Mikhail Borovoi, Willem A. de Graaf, Robert M. Guralnick

arXiv:2607.19513v1 Primary: math.RA Secondary: math.RT

Abstract

Following an idea of Demarche and using computer calculations of the authors and ideas of LLM Claude Fable, we construct an explicit 10-dimensional complex two-step nilpotent Lie algebra that is isomorphic to its complex conjugate but cannot be defined over the field of real numbers R.

Disclosure

“basis algorithm, we could compute the set of involutions in GW : no involutions. However, a Gröbner basis computation of the whole finite group GW or even of the set of elements of order 3 in GW , never ends. Instead, we use an idea of LLM Claude Fable. Let H = GW denote the stabilizer of W in G = GL(6, C). Then H naturally acts on W . We identify 2 V ∗ with the space Matskew V 6 of skew-symmetric 6×6 matrices. Namel”
PDF page 7

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-23T19:03:46.280420Z

Structural counts

Pages 9 pdf

Theorems 2 source

Lemmas 11 source

Propositions 4 source

Corollaries 5 source

Definitions 1 source

Displayed equations 67 source

Bibliography entries 7 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file A-note-260721-arXiv.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.