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

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

On minimal faithful representations of a class of nilpotent lie algebras

In this work we consider 2-step nilradicals of parabolic subalgebras of the simple Lie algebra An and describe a new family of faithful nil-representations of the nilradicals na,c, a, c ∈ N. We obtain a sharp upper bound for the minimal dimension µ(na,c ) and for several pairs (a, c) we obtain µ(na,c ).Fil: Alvarez, María Alejandra. Universidad de Antofagasta; ChileFil: Rojas, Nadina Elizabeth. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba; Argentin

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.Comment: 28 pages, 3 figure

Faithful representations of minimal degree for Lie algebras with an abelian radical

In dieser Dissertation untersuchen wir die sogenannte $\mu$-Invariante von Lie Algebren. F\"ur eine endlich-dimensionale Lie Algebra $\mathfrak{g}$ ist sie die minimale Dimension eines treuen $\mathfrak{g}$-Moduls. Es ist bereits nicht-trivial zu zeigen, da\ss\ diese Invariante Werte in den nat\"urlichen Zahlen annimmt, d.h., da\ss\ jede endlich-dimensionale Lie Algebra eine endlich-dimensionale treue Darstellung besitzt. Das wurde urspr\"unglich von Ado und Iwasawa bewiesen, und ist ein fundamentales Resultat. Es hat eine lange Geschichte. In dieser Arbeit geht es um eine Verfeinerung des Ado-Iwasawa-Theorems, und zwar in folgender Hinsicht:\\[1cm] {\it Sei $\mathfrak{g}$ eine endlich-dimensionale Lie algebra. Berechne \mu(\Lg) und finde einen treuen Modul dieser Dimension. Beschreibe die Eigenschaften treuer Moduln minimaler Dimension. Berechne obere und untere Schranken f\"ur $\mu(\mathfrak{g})$ als Funktion anderer Invarianten. }\\[1cm] Im allgemeinen kann man keine explizite Formel f\"ur $\mu(\mathfrak{g})$ erwarten, insbesondere nicht f\"ur nilpotente Lie Algebren. Die Frage ist daher, ob man f\"ur reduktive bzw. halbeinfache Lie Algebren $\mu(\mathfrak{g})$ bestimmen kann. Tats\"achlich gelingt dies f\"ur den Fall da\ss\ $\mathfrak{g}$ abelsch, einfach, halbeinfach oder reduktiv ist. Der Beweis dazu ist im wesentlichen kombinatorischer Natur und verwendet klassiche Resultate der Darstellungstheorie f\"ur reduktive Lie-Algebren. Allgemeiner untersuchen wir die $\mu$-Invariante auch f\"ur Lie Algebren deren aufl\"osbares Radikal abelsch ist. Wir betrachten weitere Invarianten, die mit der $\mu$-Invariante zusammenh\"angen. Abschliessend werden dazu einige spezielle Familien von solchen Lie Algebren im Detail betrachtet

Constructing Faithful Matrix Representations of Lie Algebras

By Ado's theorem every finite dimensional Lie algebra over a field of characteristic zero has a faithful finite dimensional representation. We consider the algorithmic problem of constructing such a representation for Lie algebras given by a multiplication table. An effective version of Ado's theorem is proved. 1 Introduction When dealing with the problem of representing finite dimensional Lie algebras on computer, two presentations leap into mind: a presentation by matrices and a presentation by an array of structure constants. In the first presentation the Lie algebra is given by a finite set of matrices fA 1 ; : : : ; An g that form a basis of the Lie algebra. If A and B are two elements of the space spanned by the A i , then their Lie product is defined as [A; B] = A \Delta B \Gamma B \Delta A (where the \Delta denotes the ordinary matrix multiplication) . The second approach is more abstract. The Lie algebra is a (abstract) vector space over a field F with basis fx 1 ; : : : ; x..

Constructing Faithful Matrix Representations of Lie Algebras

By Ado's theorem every nite dimensional Lie algebra over a field of characteristic zero has a faithful finite dimensional representation. We consider the algorithmic problem of constructing such a representation for Lie algebras given by a multiplication table. An effective version of Ado's theorem is proved