Ten-Dimensional Levi Decomposition Lie Algebras with sl(2, R) Semi-Simple Factor

Narayana M. P. S. K. Bandara¹ ; Gerard Thompson²

¹Dept. of Mathematics, Florida A & M University, Tallahassee, U.S.A.
²Department of Mathematics & Statistics, University of Toledo, Toledo, U.S.A.

Journal of Lie Theory, Tome 34 (2024) no. 2, pp. 267-317

Résumé

\def\s{\mathfrak{s}} \def\l{\mathfrak{l}} Turkowski has classified Lie algebras that have a non-trivial Levi decomposition of dimension up to and including nine. In this work the program is continued and completes the classification of the corresponding Lie algebras in dimension ten, for which the semi-simple factor is $\s\l(2,\R)$. In the approach adopted here, one begins with a nilpotent Lie algebra {\it NR}, which will serve as the nilradical of the Levi decomposition algebra $S\rtimes N$ that is ultimately constructed. Here $N$ denotes a solvable extension of {\it NR}. Two key tools used in obtaining the classification are, the $R$-representation, that is, the action of $\s\l(2,\mathbb{R})$ as endomorphims of {\it NR} and secondly the algebra of $R$-constants, that is, the subalgebra of $N$ that commutes with the $R$-representation.

Détail

Classification : 17B05, 17B30, 17B99
Mots-clés : Semisimple factor, radical, nilradical, R-representation, R-constants Lie algebra