Classification of nilpotent Lie algebras of nilpotency class 3 having a derived subalgebra of dimension three
Saboura Yousefi, Azam Kaheni, Farangis Johari
TL;DR
The paper addresses the classification of nilpotent Lie algebras of nilpotency class $3$ with derived subalgebra dimension $3$ and with $Z(L)=L^3\cong A(2)$, providing a complete description in arbitrary dimension and an explicit seven-dimensional listing. It builds on the Skjelbred–Sund construction and invariants such as $t(L)$, $s(L)$, and the newly introduced $t(L^{2})$ to organize the classification and relate it to existing seven-dimensional classifications. The main contributions include a structural decomposition framework and a concrete set of four non-isomorphic seven-dimensional stem algebras, $L_{1},L_{2},L_{3},L_{4}$, with explicit presentations. This advances understanding of how the derived subalgebra and center constrain the architecture of class-3 nilpotent Lie algebras and provides a basis for further exploration of central extensions and Schur multipliers in low dimensions.
Abstract
In this paper, we investigate nilpotent Lie algebras $ L $ of nilpotency class $3 $ and provide a complete classification of those satisfying $ \dim L^2 = 3 $ and $Z(L) = L^3 \cong A(2). $ Furthermore, we explicitly characterize the structure of such Lie algebras in the case when $ \dim L = 7. $