Invariants of solvable rigid Lie algebras up to dimension 8

Abstract

The invariants of all complex solvable rigid Lie algebras up to dimension eight are computed. Moreover we show, for rank one solvable algebras, some criteria to deduce to non-existence of non-trivial invariants or the existence of fundamental sets of invariants formed by rational functions of the Casimir invariants of the associated nilradical.

A note on the classification of nine dimensional Lie algebras with nontrivial Levi decomposition

“…L 9,47 • Levi decomposition: L 9,47 = sl (2, R) − → ⊕ R g 6,15 • Describing representation: R = 2D 1 2 ⊕ 2D 0 • Structure tensor: • codim g [g, g] = 1.…”

“…Extensive work has been done on the eigenvalues of Casimir operators of classical groups and their generating functions [26,27]. For the case of non semisimple Lie algebras no general criteria for the number and structure of independent invariants exist, up to certain specific classes [1,6,9,34]. Various methods have been developed to compute the invariants of Lie algebras.…”

Non-semisimple Lie algebras with Levi factor o (3), (2, ) and their invariants

“…Another important task is to find the maximal number N (g) of functionally independent solutions of (3). For the case of the classical groups this number depends only on the dimension of a Cartan subalgebra, while for solvable Lie algebras no such general formula exists [5]. However, for a fixed algebra, this number can be described in terms of the dimension and a certain matrix associated to the commutator table.…”

“…The invariants of Lie algebras have also shown their efectiveness in the description of Hamiltonians [2], the labelling of irreducible representations or the study of coadjoint orbits [3,4]. Other important applications of invariants arise in their combination with the theories of Lie algebra contractions, deformations and rigidity [5,6,7,8]. For example, all kinematical algebras are related by a contraction procedure, which has allowed a further analysis of these algebras [7,9,10].…”

Построение Инвариантов Коприсоединенного Представления Групп Ли Методами Линейной Алгебры

“…Для неполупростых алгебр Ли до настоящего момента в общем случае задача не была решена. Основные результаты относятся к многочисленным специальным случаям (см., например, [9]). Важный результат был получен для случая разрешимых алгебр Ли произвольной размерности; в работе [10] на основе метода подвижного репера был получен алгебраический алгоритм построения инвариантов коприсоединенного представления.…”