Conjugacy classes of nilpotent elements in complex semisimple Lie algebras are classified using the Bala-Carter theory. In this theory, nilpotent orbits in g are parametrized by the conjugacy classes of pairs (l,pl) of Levi subalgebras of g and distinguished parabolic subalgebras of [l,l]. In this thesis we present this theory and use it to give a list of representatives for nilpotent orbits in so(8) and from there we give a partition-type parametrization of them.

In 1980 Feigin and Fuchs classified the length 2 bounded representations of Vec(R), the Lie algebra of polynomial vector fields on the line, as a result of their work on the cohomology of Vec(R). This dissertation is concerned mainly with the uniserial (completely indecomposable) representations of Vec(R) with a single Casimir eigenvalue and weights bounded below. Such representations are composed of irreducible representations with semisimple Euler operator action, bounded weight space dimensions, and weights bounded below. These are known to be the tensor density modules with lowest weight λ, for any non-zero complex number λ, and the trivial module C, with Vec(R) actions π_λ and π_C, respectively. Our proofs are cohomology arguments involving the first cohomology groups of Vec(R) with values in the space of homomorphisms between two irreducible representations. These results classify the finite length uniserial extensions, with a single Casimir eigenvalue, of admissible irreducible Vec(R) representations with weights bounded below. In almost every case there is at most one uniserial representation with a given composition series. However, in the case of an odd length extension with composition series {π_1,π_C,π_1,…,π_C,π_1}, there is a one-parameter family of extensions. We also give preliminary results on uniserial representations of the Virasoro Lie algebra.

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A Hessenberg variety is a subvariety of the flag variety parametrized by two maps: a Hessenberg function on $[n]$ and a linear map on $\C^n$. We study regular nilpotent Hessenberg varieties in Lie type A by focusing on the Hessenberg function $h=(n-1,n,\ldots,n)$. We first state a formula for the $f^w_{n,1}$ which generates the local defining ideal $J_{w,h}$ for any $w\in\Ss_n$. Second, we prove that there exists a convenient monomial order so that $\lead(J_{w,h})$ is squarefree. As a consequence, we conclude that each codimension-1 regular nilpotent Hessenberg variety is locally Frobenius split (in positive characteristic). / Thesis / Master of Science (MSc)

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The classification of finite semigroups is difficult even for small orders because of their large number. Most finite semigroups are nilpotent of nilpotency rank 3. Formulae for their number up to isomorphism, and up to isomorphism and anti-isomorphism of any order are the main results in the theoretical part of this thesis. Further studies concern the classification of nilpotent semigroups by rank, leading to a full classification for large ranks. In the computational part, a method to find and enumerate multiplication tables of semigroups and subclasses is presented. The approach combines the advantages of computer algebra and constraint satisfaction, to allow for an efficient and fast search. The problem of avoiding isomorphic and anti-isomorphic semigroups is dealt with by supporting standard methods from constraint satisfaction with structural knowledge about the semigroups under consideration. The approach is adapted to various problems, and realised using the computer algebra system GAP and the constraint solver Minion. New results include the numbers of semigroups of order 9, and of monoids and bands of order 10. Up to isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups with 9 elements, 52,991,253,973,742 monoids with 10 elements, and 7,033,090 bands with 10 elements. That constraint satisfaction can also be utilised for the analysis of algebraic objects is demonstrated by determining the automorphism groups of all semigroups with 9 elements. A classification of the semigroups of orders 1 to 8 is made available as a data library in form of the GAP package Smallsemi. Beyond the semigroups themselves a large amount of precomputed properties is contained in the library. The package as well as the code used to obtain the enumeration results are available on the attached DVD.

Abstract in English / The research will analyze and compare the current research on the spectral factorization of non-singular and singular matrices. We show that a nonsingular non-scalar matrix A can be written as a product A = BC where the eigenvalues of B and C are arbitrarily prescribed subject to the condition that the product of the eigenvalues of B and C must be equal to the determinant of A. Further, B and C can be simultaneously triangularised as a lower and upper triangular matrix respectively. Singular matrices will be factorized in terms of nilpotent matrices and otherwise over an arbitrary or complex field in order to present an integrated and detailed report on the current state of research in this area. Applications related to unipotent, positive-definite, commutator, involutory and Hermitian factorization are studied for non-singular matrices, while applications related to positive-semidefinite matrices are investigated for singular matrices. We will consider the theorems found in Sourour [24] and Laffey [17] to show that a non-singular non-scalar matrix can be factorized spectrally. The same two articles will be used to show applications to unipotent, positive-definite and commutator factorization. Applications related to Hermitian factorization will be considered in [26]. Laffey [18] shows that a non-singular matrix A with det A = ±1 is a product of four involutions with certain conditions on the arbitrary field. To aid with this conclusion a thorough study is made of Hoffman [13], who shows that an invertible linear transformation T of a finite dimensional vector space over a field is a product of two involutions if and only if T is similar to T−1. Sourour shows in [24] that if A is an n × n matrix over an arbitrary field containing at least n + 2 elements and if det A = ±1, then A is the product of at most four involutions. We will review the work of Wu [29] and show that a singular matrix A of order n ≥ 2 over the complex field can be expressed as a product of two nilpotent matrices, where the rank of each of the factors is the same as A, except when A is a 2 × 2 nilpotent matrix of rank one. Nilpotent factorization of singular matrices over an arbitrary field will also be investigated. Laffey [17] shows that the result of Wu, which he established over the complex field, is also valid over an arbitrary field by making use of a special matrix factorization involving similarity to an LU factorization. His proof is based on an application of Fitting's Lemma to express, up to similarity, a singular matrix as a direct sum of a non-singular and nilpotent matrix, and then to write the non-singular component as a product of a lower and upper triangular matrix using a matrix factorization theorem of Sourour [24]. The main theorem by Sourour and Tang [26] will be investigated to highlight the necessary and sufficient conditions for a singular matrix to be written as a product of two matrices with prescribed eigenvalues. This result is used to prove applications related to positive-semidefinite matrices for singular matrices. / National Research Foundation of South Africa / Mathematical Sciences / M Sc. (Mathematics)

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O principal objetivo desta dissertação é estudar um contraexemplo para o Décimo Quarto Problema de Hilbert no caso de dimensão n = 5, que foi apresentado por Arno van den Essen ([6]) em 2006 e que é baseado em um contraexemplo de D. Daigle e G. Freudenburg ([4]). Para isso, serão estudados os conceitos fundamentais da teoria de derivações e os princípios básicos das derivações localmente nilpotentes, bem como seus respectivos corolários. Dentre esses princípios encontra-se o Princípio 13, que garante que, se B é uma k- álgebra polinomial, digamos B = k[x1; ..., xn], (onde k é um corpo de característica zero) e D é uma derivação localmente nilpotente sobre B, então seu núcleo A = ker D satisfaz A = B &cap: Frac(A). Assim encontramos o contraexemplo esperado, ao mostrar que A não é finitamente gerado sobre k. Além disso, no apêndice deste trabalho, é dada uma prova para o caso de dimensão 1 do Décimo Quarto Problema de Hilbert. / The main objective of this thesis is to study a counterexample to the Hilberts Fourteenth Problem in dimension n = 5, which was presented by Arno van den Essen ([6]) in 2006 and that is based on a counterexample of D. Daigle and G. Freudenburg ([4]). For these purpose, we study the fundamental concepts of the theory of derivations and the basic principles of locally nilpotent derivations and their corollaries. Among these principles, Principle 13 ensures that if B is a k-algebra polynomial, say B = k[x1; ..., xn], (where k is a field of characteristic zero) and D is a locally nilpotent derivation on B, then its kernel A = ker D satisfies A = B ∩ Frac(A). Once we have proved that A is not finitely generated over k, we find the expected counterexample. In addition, in the appendix of this work is given a proof for the Hilberts Fourteenth Problemin dimension n = 1.