Invariants of Lie Algebras with Fixed Structure of Nilradicals

TL;DR

This paper introduces an algebraic algorithm utilizing Cartan's moving frames to compute invariants of Lie algebras with fixed nilradical structures, demonstrating its effectiveness on solvable Lie algebras of various dimensions.

Contribution

It develops a new algorithm for calculating invariants of Lie algebras with specific nilradical structures, extending previous low-dimensional applications to higher dimensions.

Findings

Successfully computes invariants for families of solvable Lie algebras.

Extends the applicability of the algorithm to higher-dimensional algebras.

Provides invariants for most solvable Lie algebras of certain dimensions.

Abstract

An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], which deals with low-dimensional Lie algebras, here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable Lie algebras of general dimension $n<\infty$ restricted only by a required structure of the nilradical. Specifically, invariants are calculated here for families of real/complex solvable Lie algebras. These families contain, with only a few exceptions, all the solvable Lie algebras of specific dimensions, for whom the invariants are found in the literature.