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Journal of Lie Theory 26 (2016), No. 4, 991--1000
Copyright Heldermann Verlag 2016

Filiform Lie Algebras Without Rational Structures

Naoki Kato
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-9814, Japan

Hamrouni and Souissi gave a sufficient condition for a family of 7-dimensional filiform Lie algebras to have no rational structures and constructed an example of a 7-dimensional filiform Lie algebra without rational structures explicitly. In this paper, we give a sufficient condition for arbitrary-dimensional filiform Lie algebras to have no rational structures. Moreover, we consider a sufficient condition for products of filiform Lie algebras and arbitrary nilpotent Lie algebras to have no rational structures and give infinitely many new examples of nilpotent Lie algebras without rational structures.

Keywords: Nilpotent Lie algebras, filiform Lie algebras, rational structures.

MSC: 22E40

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