Invariants of Lie algebras

TL;DR

This thesis introduces new invariants for Lie algebras that help describe their nilpotent structures and facilitate classification, with applications to contractions and potential links to Jordan algebras.

Contribution

It develops novel invariant characteristics for Lie algebras, extending their descriptive power and applicability to higher dimensions and related algebraic structures.

Findings

Invariants effectively describe nilpotent parametric continua.

Invariants are useful for classifying lower-dimensional Lie algebras.

A contraction criterion involving invariants is established.

Abstract

In this thesis new objects to the existing set of invariants of Lie algebras are added. These invariant characteristics are capable of describing the nilpotent parametric continuum of Lie algebras. The properties of these invariants, in view of possible alternative classifications of Lie algebras, are formulated and their behaviour on known lower--dimensional Lie algebras investigated. It is demonstrated that these invariants, in view of their application on graded contractions of sl(3,C), are also effective in higher dimensions. A necessary contraction criterion involving these invariants is derived and applied to lower--dimensional cases. Possible application of these invariant characteristics to Jordan algebras is investigated.