Algebra Commons^™

A Theory Of Fundamental Strata For Twisted Formal Connections, Sorawit Viwanthananut

LSU Doctoral Dissertations

Let $G$ be a complex reductive group. A fundamental stratum for $G$ is a triple $(x,r,\beta)$ where $x$ is a point in the Bruhat-Tits building of $G$, $r$ is a nonnegative real number called depth of the stratum, and $\beta$ is a semistable functional on the Moy-Prasad filtration of $\fg$ associated to $x$ at level $r$. Fundamental strata were first introduced to classify admissible representations of a $p$-adic reductive group. More recently, Bremer and Sage have shown that fundamental strata play an important role in the geometric Langlands program and developed a theory of fundamental strata for $G$-connections. In this …

The Impact Of Loss Function Topology On Gradient Descent, Robert B. Skudnig Jr.

Theses and Dissertations

Gradient descent is a popular optimization method that utilizes a model’s prediction error to iteratively improve its parameters for a given task. The functions that measure this error can be defined to align with the user’s goals and sometimes satisfy metric or norm properties. It is common for these functions to measure over Rn, but any differentiable space allows for gradient descent to occur. There has been some research investigating the influence of topological spaces on optimization methods, but it is a limited field of study. This thesis further explores this phenomenon by applying a transformation prediction model to multiple …

The Lie Algebra Sl2(C) And Krawtchouk Polynomials, Nkosi Alexander

UNF Graduate Theses and Dissertations

The Lie algebra L = sl2(C) consists of the 2 × 2 complex matrices that have trace zero, together with the Lie bracket [y, z] = yz − zy. In this thesis we study a relationship between L and Krawtchouk polynomials. We consider a type of element in L said to be normalized semisimple. Let a, a^∗ be normalized semisimple elements that generate L. We show that a, a^∗ satisfy a pair of relations, called the Askey-Wilson relations. For a positive integer N, we consider an (N + 1)-dimensional irreducible L-module V consisting of the homogeneous polynomials in two variables …

Exact Sequences Of Inner Automorphisms Of Tensors, Peter A. Brooksbank

Faculty Journal Articles

We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner similar to those, our sequence facilitates inductive reasoning about, and calculation of the groups of symmetries of a tensor. The new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics.

Application Of Symplectic Integration On A Dynamical System, William Frazier

Electronic Theses and Dissertations

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic …

Cartan Subalgebras, Compact Roots And The Satake Diagram For Su(2, 2), Ian M. Anderson

Tutorials on... in 1 hour or less

In this worksheet we use the 15-dimensional real Lie algebra su(2, 2) to illustrate some important points regarding the general structure theory and classification of real semi-simple Lie algebras.

1. Recall that a real semi-simple Lie algebra g is called a compact Lie algebra if the Killing form is negative definite. The Lie algebra g is compact if and only if all the root vectors for any Cartan subalgebra are purely imaginary. However, if the root vectors are purely imaginary for some choice of Cartan subalgebra it is not necessarily true that the Lie algebra is compact.

2. A real …

Jordan Algebras And The Exceptional Lie Algebra F4, Ian M. Anderson

Tutorials on... in 1 hour or less

This worksheet analyzes the structure of the Jordan algebra J(3, O) and its split and exceptional versions. The algebra of derivations is related to the exceptional Lie algebra f4.

Calculation Of The Killing Form Of A Simple Lie Group, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

The Killing form of a simple Lie Algebra is determined from invariants of the extended root diagrams of the Lie algebra.

On The Matrix Equation Xa + Ax_T = 0, Stephan Ramon Garcia, Amy L. Shoemaker '14

Pomona Faculty Publications and Research

The matrix equation XA+AX^T=0, which has relevance to the study of Lie algebras, was recently studied by De Terán and Dopico (Linear Algebra Appl. 434 (2011), 44–67). They reduced the study of this equation to several special cases and produced explicit solutions in most instances. In this note we obtain an explicit solution in one of the difficult cases, for which only the dimension of the solution space and an algorithm to find a basis of this space were known previously