# Invariant functions and contractions of algebras.

- Jose María Escobar Rica

- Juan Núñez-Valdés Director
- Pedro Pérez Fernández Director

Defence university: Universidad de Sevilla

Year of defence: 2019

- Francisco Barranco Paulano Chair
- Desamparados Fernández Ternero Secretary
- Ángel Francisco Tenorio Villalón Committee member
- Manuel Ceballos González Committee member
- Isabel María Ortiz Rodríguez Committee member

Type: Thesis

## Abstract

This manuscript deals with invariant functions and contractions of certain types of algebras, as Lie, Heisenberg, Malcev or Kinematical algebras, for instance. The most cited algebras in the manuscript are filiform Lie algebras. The motivation for dealing with these algebras is due that they are the most structured algebras within the nilpotent Lie algebras, which allows us to use and study them easier than other Lie algebras. Filiform Lie algebras were introduced by Vergne, in the late 60's of the past century. She showed that, within the variety of nilpotent Lie multiplications on a fixed vector space, the non-filiform ones can be relegated to small-dimensional components. In any case, as it has been just mentioned, other different types of algebras, as Heisenberg, Malcev or Kinematical algebras, for instance, are also dealt with in this manuscript. The main novel aspects and results obtained in this research which are shown in this manuscript are the following. - The author has computed the values of the invariant functions ψ and φ for the filiform Lie algebras of dimensions 3, 4 and 5, for the Heisenberg algebra of dimension 3 and for other different types of algebras, all of them in the lower dimension (Chapter 3). - As a relevant aspect of the research, we have introduced in the manuscript three new invariant functions for algebras, the one-parameter invariant function υ_g the two-parameter invariant function ψ̄_g and the two-parameter invariant function φ ̄_g. We have obtained the main properties, some results and several applications of such functions to the study of contractions of algebras. These three new functions have allowed to give a step forward towards the knowledge of this topic and to make easier the computations needed for it (Chapter 4). - With respect to the study of the proper contractions of filiform Lie algebras of lower dimensions, we have obtained two main results (Chapter 5). The first one is that there exists a proper contraction from the filiform Lie algebra f_3 both to the algebra 3g_1 and the algebra g_32, whereas it exists no proper contraction either between f_3 and g_31, or f_3 and g_33. The second result is that there is no proper contraction from a filiform Lie algebra to a Heisenberg algebra. It implies that filiform Lie algebras cannot appear as a classical limit from the contraction of a quantum mechanical model built upon a Heisenberg algebra because in that case there would be a contraction from the Heisenberg algebra to the filiform Lie algebra of the same dimension. -Finally, as application of the introduced new invariant function υ_g, Kinematical Lie algebras are widely dealt with in this manuscript (Chapter 6). The main result obtained by the author on this subject has been the computation of the values of such a function for the eight kinematical Lie algebras studied by Tolar (see Chapter 6). It also suppose a step forward in the knowledge of the study of contractions of algebras. Therefore, according to the previous paragraphs, the structure of the manuscript is the following In Chapter 1, we expose a brief historical evolution on invariant functions and contractions of certain types of algebras, with the objective of framing the problem under study and offering a historical overview of the evolution that these topics have followed so far. Chapter 2 consists of those already known basic concepts and results on different types of algebras, gradings, invariant functions of them and contractions that we use throughout the manuscript. In Chapter 3 we compute the invariant functions ψ and φ, introduced by Hrivnák and Novotny, in the particular case of model filiform Lie algebras of lower dimensions, particularly dimensions 3 and 4. Our intention is to deal with those of greater dimensions in a similar way in future work. In Chapter 4 we firstly introduce a new two-parameter invariant function ψ̄_g and compute its value for different types of algebras: Malcev, Lie and other algebras. Secondly, two other invariant functions for algebras are also introduced: the one-parameter invariant function υ_g and the two-parameter invariant function φ̄_g. Chapter 5 is devoted to the study of contractions of algebras, particularly proper contractions of algebras. We show several examples of proper contractions between different types of algebras. Kinematical algebras are dealt with in Chapter 6. We consider the kinematical Lie algebras of four-dimensional spacetime, introduced by Tolar, and compute the one-parameter invariant function υ introduced in the previous chapter for the eight kinematical Lie algebras given by Tolar. Finally, a last chapter devoted to pose and analyze some open problems coming from this research has been also included.