Computing faithful representations for nilpotent Lie algebras

We describe three methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods in finding bounds for the smallest dimension \mu(\Lg) of a faithful \Lg-module for some nilpotent Lie algebras \Lg. In particular, we describe an infinite family of filiform nilpotent Lie algebras \Lf_n of dimension $n$ over \Q and conjecture that \mu(\Lf_n) > n+1. Experiments with our algorithms suggest that \mu(\Lf_n) is polynomial in $n$.Comment: 14 page

The Invariant Two-Parameter Function of Algebras ψ

At present, the research on invariant functions for algebras is very extended since Hrivnák and Novotný defined in 2007 the invariant functions y and j as a tool to study the Inönü–Wigner contractions (IW-contractions), previously introduced by those authors in 1953. In this paper, we introduce a new invariant two-parameter function of algebras, which we call ¯y, as a tool which makes easier the computations and allows researchers to deal with contractions of algebras. Our study of this new function is mainly focused in Malcev algebras of the type Lie, although it can also be used with any other types of algebras. The main goal of the paper is to prove, by means of this function, that the five-dimensional classical-mechanical model built upon certain types of five-dimensional Lie algebras cannot be obtained as a limit process of a quantum-mechanical model based on a fifth Heisenberg algebra. As an example of other applications of the new function obtained, its computation in the case of the Lie algebra induced by the Lorentz group SO(3, 1) is shown and some open physical problems related to contractions are also formulated.Ministerio de Ciencia e Innovación MTM2013-40455-PMinisterio de Ciencia e Innovación FQM-326 (J.N.-V.)Junta de Andalucía FQM-160 (P.P.-F.

Faithful Lie algebra modules and quotients of the universal enveloping algebra

We describe a new method to determine faithful representations of small dimension for a finite dimensional nilpotent Lie algebra. We give various applications of this method. In particular we find a new upper bound on the minimal dimension of a faithful module for the Lie algebras being counter examples to a well known conjecture of J. Milnor

A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras

We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schr\"odinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.Comment: 22 page

Naturally Graded 2-Filiform Leibniz Algebras

The Leibniz algebras appear as a generalization of the Lie algebras [8]. The classification of naturally graded p-filiform Lie algebras is known [3], [4], [5], [9]. In this work we deal with the classification of 2-filiform Leibniz algebras. The study of p-filiform Leibniz non Lie algebras is solved for p = 0 (trivial) and p = 1 [1]. In this work we get the classification of naturally graded non Lie 2-filiform Leibniz algebras

Invariants of Lie Algebras with Fixed Structure of Nilradicals

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.Comment: LaTeX2e, 19 page