Roughly speaking, a separating algebra is a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this thesis, we introduce the notion of a geometric separating algebra, a more geometric notion of a separating algebra. We find two geometric formulations for the notion of separating algebra which allow us to prove, for geometric separating algebras, the results found in the literature for separating algebras, generally removing the hypothesis that the base field be algebraically closed. Using results from algebraic geometry allows us to prove that, for finite groups, when a polynomial separating algebra exists, the group is generated by reflections, and when a complete intersection separating algebra exists, the group is generated by bireflections. We also consider geometric separating algebras having a small number of generators, giving an upper bound on the number of generators required for a geometric separating algebra. We end with a discussion of two methods for obtaining new separating sets from old. Interesting, and relevant examples are presented throughout the text. Some of these examples provide answers to questions which previously appeared in print. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-08-28 14:14:04.138

First published in the Bulletin of the American Mathematical Society in Vol.68 1962, published by the American Mathematical Society

This thesis will examine Relative Local Cohomology. First we extend many well known theorems about Local Cohomology of finitely generated modules with respect to an ideal of a commutative noetherian rings so that they hold for non-finitely generated modules with respect to certain ideals of non-commutative non-noetherian rings. Then we show how similar results hold for Relative Local Cohomology. In particular we provide a relative version of the Local Duality Theorem. We then examine the links between Relative Homological Algebra and the concept of Structure Theorems and give a bound on the Castelnuovo-Mumford Regularity of rings of invariants based on the Cech Complex.

Commutative algebra finds its roots in invariant theory and the connection is drawn from a modern standpoint. The Hilbert Basis Theorem and the Nullstellenstatz were considered lemmas for classical invariant theory. The Groebner basis is a modern tool used and is implemented with the computer algebra system Mathematica. Number 14 of Hilbert\'s 23 problems is discussed along with the notion of invariance under a group action of GLn(C). Computational difficulties are also discussed in reference to Groebner bases and Invariant theory.The straitening law is presented from a Groebner basis point of view and is motivated as being a key piece of machinery in proving First Fundamental Theorem of Invariant Theory. / Master of Science

Abstract not available. / text

In 1949, Wever observed that the degree d of an invariant Lie polynomial must be a multiple of the number q of generators of the free Lie algebra. He also found that there are no invariant Lie polynomials in the following cases: q = 2, d = 4; q = 3, d = 6; d = q ≥ 3. Wever gave a formula for the number of invariants for q = 2 in the natural representation of sl(2). In 1958, Burrow extended Wevers formula to q > 1 and d = mq where m > 1. In the present thesis, we concentrate on finding invariant Lie polynomials (simply called Lie invariants) in the natural representations of sl(2) and sl(3), and in the adjoint representation of sl(2). We first review the method to construct the Hall basis of the free Lie algebra and the way to transform arbitrary Lie words into linear combinations of Hall words. To find the Lie invariants, we need to find the nullspace of an integer matrix, and for this we use the Hermite normal form. After that, we review the generalized Witt dimension formula which can be used to compute the number of primitive Lie invariants of a given degree. Secondly, we recall the result of Bremner on Lie invariants of degree ≤ 10 in the natural representation of sl(2). We extend these results to compute the Lie invariants of degree 12 and 14. This is the first original contribution in the present thesis. Thirdly, we compute the Lie invariants in the adjoint representation of sl(2) up to degree 8. This is the second original contribution in the present thesis. Fourthly, we consider the natural representation of sl(3). This is a 3-dimensional natural representation of an 8-dimensional Lie algebra. Due to the huge number of Hall words in each degree and the limitation of computer hardware, we compute the Lie invariants only up to degree 12. Finally, we discuss possible directions for extending the results. Because there are infinitely many different simple finite dimensional Lie algebras and each of them has infinitely many distinct irreducible representations, it is an open-ended problem.

In 1949, Wever observed that the degree d of an invariant Lie polynomial must be a multiple of the number q of generators of the free Lie algebra. He also found that there are no invariant Lie polynomials in the following cases: q = 2, d = 4; q = 3, d = 6; d = q ≥ 3. Wever gave a formula for the number of invariants for q = 2 in the natural representation of sl(2). In 1958, Burrow extended Wevers formula to q > 1 and d = mq where m > 1. In the present thesis, we concentrate on finding invariant Lie polynomials (simply called Lie invariants) in the natural representations of sl(2) and sl(3), and in the adjoint representation of sl(2). We first review the method to construct the Hall basis of the free Lie algebra and the way to transform arbitrary Lie words into linear combinations of Hall words. To find the Lie invariants, we need to find the nullspace of an integer matrix, and for this we use the Hermite normal form. After that, we review the generalized Witt dimension formula which can be used to compute the number of primitive Lie invariants of a given degree. Secondly, we recall the result of Bremner on Lie invariants of degree ≤ 10 in the natural representation of sl(2). We extend these results to compute the Lie invariants of degree 12 and 14. This is the first original contribution in the present thesis. Thirdly, we compute the Lie invariants in the adjoint representation of sl(2) up to degree 8. This is the second original contribution in the present thesis. Fourthly, we consider the natural representation of sl(3). This is a 3-dimensional natural representation of an 8-dimensional Lie algebra. Due to the huge number of Hall words in each degree and the limitation of computer hardware, we compute the Lie invariants only up to degree 12. Finally, we discuss possible directions for extending the results. Because there are infinitely many different simple finite dimensional Lie algebras and each of them has infinitely many distinct irreducible representations, it is an open-ended problem.

The larger area of my thesis is Algebra; more specifically, my work belongs to the following two major branches of Algebra: \emph{representation theory} and \emph{invariant theory}. In brief, the objective of representation theory is to investigate algebraic objects through their actions on vector spaces; this allows the well-developed toolkit of linear algebra to be brought to bear on complex algebraic problems. The theory has played a crucial role in nearly every subdiscipline of pure mathematics. Outside of pure mathematics, representation theory has been successfully used, for instance, in the study of symmetries of physical systems and in describing molecular structures in physical chemistry. Invariant theory is another classical algebraic theme permeating virtually all areas of pure mathematics and some areas of applied mathematics as well, notably coding theory. The theory studies actions of algebraic objects, traditionally groups and Lie algebras, on algebras, that is, vector spaces that are equipped with a multiplication. \bigskip The representation theory of (associative) algebras provides a useful setting in which to studymany aspects of the two most classical flavors of representation theory under a common umbrella: representations of groups and of Lie algebras. However, it turns out that general algebras fail to capture certain features of group representations and the same can be said for representations of Lie algebras as well. The additional structure that is needed in order to access these features is naturally provided by the important class of \emph{Hopf algebras}. Besides unifying the representation theories of groups and of Lie algebras, Hopf algebras serve a similar purpose in invariant theory, allowing for a simultaneous treatment of group actions (by automorphisms) and Lie algebras (by derivations) on algebras. More importantly, actions of Hopf algebras have the potential of capturing additional aspects of the structure of algebras they act on, uncovering features that cannot be accessed by ordinary groups or Lie algebras. \bigskip Presently, the theory of Hopf algebras is still nowhere near thelevel that has been achieved for groups and for Lie algebras over the course of the past century and earlier. This thesis aims to make a contribution to the representation and invariant theories of Hopf algebras, focusing for the most part on Hopf algebras that are not necessarily finite dimensional. Specifically, the contributions presented here can be grouped under two headings: \smallskip \noindent\qquad(i) \textbf{ Invariant Theory:} Hopf algebra actions and prime spectra, and\smallskip \noindent\qquad(ii)\textbf{ Representation Theory:} the adjoint representation of a Hopf algebra. \smallskip In the work done under the heading (i), we were able to use the action of cocommutative Hopf algebras on other algebras to "stratify" the prime spectrum of the algebra being acted upon, and then express each stratum in terms of the spectrum of a commutative domain. Additionally, we studied the transfer of properties between an ideal in the algebra being acted upon, and the largest sub-ideal of that ideal, stable under the action. We were able to achieve results for various families of acting Hopf algebras, namely \emph{cocommutative} and \emph{connected} Hopf algebras.\\The main results concerning heading (ii) concerned the subalgebra of locally finite elements of a Hopf algebra, often called the finite part of the Hopf algebra. This is a subalgebra containing the center that was used successfully to study the ring theoretic properties of group algebras, Lie algebras, and other classical structures. We prove that the finite is not only a subalgebra, but a coideal subalgebra in general, and in the case of (almost) cocommuative Hopf algebra, it is indeed a Hopf subalgebra. The results in this thesis generalize earlier theorems that were proved for the prototypical special classes of Hopf algebras: group algebras and enveloping algebras of Lie algebras. / Mathematics

The invariant theory of finite reflection groups has rich connections to geometry, topology, representation theory, and combinatorics. We consider finite reflection groups acting on vector spaces over fields of arbitrary characteristic, where many arguments of classical invariant theory break down. When the characteristic of the underlying field is positive, reflections may be nondiagonalizable. A group containing these so-called transvections has order which is divisible by the characteristic of the underlying field, so is in the modular setting. In this thesis, we examine the action on differential derivations, which include products of differential forms and derivations, and identify the structure of the set of invariants under the action of groups fixing a single hyperplane, groups with maximal transvection root spaces acting on vector spaces over prime fields, as well as special linear groups and general linear groups over finite fields.

The main object of study of this work are orbits in so-called minimal sheets in irreducible representations of semisimple groups. Let $G$ be a semisimple group. The notion of sheets goes back to Dixmier: Given a $G$-module $V$, the union of all orbits in $V$ of a fixed dimension is a locally closed subset. Its irreducible components are called sheets of $V$. We call a sheet minimal if it contains an orbit in $V$ of minimal strictly positive dimension among all orbits in $V$. In Chapter I, some notation is fixed and some basic results are proved. In Chapter II, we describe minimal sheets in simple $G$-modules, and study $G$-stable deformations of orbits in minimal sheets by means of an invariant Hilbert scheme. Invariant Hilbert Schemes have been introduced by Alexeev and Brion in 2005. These are quasi-projective schemes representing functors of families of $G$-schemes with prescribed Hilbert function. The discussion in Chapter II is closely related to the work of Jansou in the following way: Choose once and for all a highest weight vector $v_\lambda \in V(\lambda)$ for each dominant weight $\lambda \in \Lambda^+$, and let $X_\lambda = \overline{G v_\lambda} \subset V(\lambda)$ be the closure of the orbit $G v_\lambda$ of $v_\lambda$ in $V(\lambda)$. In his thesis Jansou investigates $G$-stable deformations of $X_\lambda$ in $V(\lambda)$. If $h_\lambda$ denotes the Hilbert function of $X_\lambda$, then Jansou proves that the invariant Hilbert scheme $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine space of dimension 0 or 1, depending on $G$ and $\lambda$. Furthermore, he gives a complete list of all pairs $(G,\lambda)$ such that $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine line. In the sequel, we call these weights Jansou-weights. The orbit $Gv_\lambda$ is of minimal strictly positive dimension among all $G$-orbits in $V(\lambda)$. There exist other orbit of the same dimension as $Gv_\lambda$ in $V(\lambda)$ if and only if $\lambda$ is an integral multiple of a Jansou-weight. Here, we start with a general orbit $X$ of minimal strictly positive dimension in a fixed simple $G$-module $V(\lambda)$, and we study $G$-stable deformations of $X$. In particular, we conjecture that the invariant Hilbert scheme parametrizing the $G$-stable deformations of $X$ in the closure of the sheet of $X$ is an affine space of dimension either 0 or 1. This will stand in contrast to the fact that the invariant Hilbert scheme parametrizing the $G$-stable deformations of $X$ in $V(\lambda)$ can look much more complicated. This is the content of Chapter III, in which we will focus on the group $\SL_2$, and compute some corresponding invariant Hilbert schemes. In particular, we study deformations of orbits of the form $SL_2 \cdot x^{d/2}y^{d/2}$ in the space $k[x,y]_d = V(d)$ of binary forms of degree $d$. It turns out that easiest accessible case is when $d$ is a multiple of 4, and even in this case the corresponding invariant Hilbert scheme can become very complicated. This reflects the principle that even in `simple' cases for invariant Hilbert schemes all possible sort of `bad' things (different irreducible components, non-reduced points, singularities) occur. (This `bad' behavior is also encountered in the case of the classical Grothendieck Hilbert scheme parametrizing closed subschemes of projective space with a given Hilbert polynomial.) In Chapter III Classical Invariant Theory is often used, and some computations are computer-based. Finally, in Chapter IV we turn our attention to not necessarily simple modules. In the multiplicity-free case important work has been done by Bravi and Cupit-Foutou. We translate some of their results to the case of not necessarily multiplicity-free modules. This corrects a result by Alexeev and Brion. Chapter IV is independent from the preceding chapters.