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Exploring and Classifying the Nilpotent Ideals in a Borel Subalgebra
Applicants:
Nicholas Lohr
68809 Leet Road
Niles, MI 49120
(574) 286-7986 nicklohr7@gmail.com
Nicole Kroeger
255 Hurley Hall University of Notre Dame
Notre Dame, IN 46556
(574) 631-9585 nkroeger@nd.edu
Sponsors:
Ken Andrzejewski
1311 South Logan
Mishawaka, IN 46544
(574) 259-5257
Douglas Sisk
1311 South Logan
Mishawaka, IN 46544
(574) 259-5257 kandrzejewski@marianhs.org dsisk@marianhs.org
Dates:
Start: 6 October 2013 End: N/A
Cost: $24.50
Hypothesis: The nilpotent ideals in a Borel subalgebra can be explored and classified.

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Abstract
The purpose of this experiment is to find a classification for the nilpotent ideals in a Borel Subalgebra. A nilpotent ideal of a Lie algebra L is a subspace I of L such that the Lie bracket is in the element of I for all x in the element of L and y in the element of
I , and if there is a natural number k such that the product of any k elements in I is zero, then I is nilpotent (Erdmann 4). A Borel subalgebra is a type of solvable subalgebra that utilizes these nilpotent ideals, and this project will attempt to simplify the process of determining these algebraic structures. Lie algebra has many applications in subjects including Quantum Mechanics, Quantum Physics, Particle Physics, and Differential
Geometry. In order to simplify these complex subjects, it is vital to have a solid foundation in mathematics, like Lie algebra. Overall, the in-depth analysis of Lie algebra is done in order to further understand complex sciences.

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Introduction
Many subjects in science utilize and apply the language of mathematics in order to further explain their respective field. As the complexity of concepts in science increase, the knowledge and application of mathematics needed increases as well.
Therefore, in order to understand science, it is necessary to comprehend and apply higher level math. The type of math I am exploring is Lie algebra. Lie algebra is a mixture between Linear algebra and Abstract algebra, and it is a study of algebraic structures called Lie algebras. This project encompasses certain algebraic structures called nilpotent ideals found in Lie algebras. A nilpotent ideal of a Lie algebra L is a subspace I of L such that the Lie bracket is in the element of I for all x in the element of
L and y in the element of I , and if there is a natural number k such that the product of any k elements in I is zero, then I is nilpotent (Erdmann 4). The attempted classification will further be applied to other areas of science, including Quantum Mechanics,
Quantum Physics, Particle Physics, Differential Geometry, and General Relativity.
Instead of pursuing a project involving the classical scientific method, I decided to deviate from the norm and perform mathematical research. As a high school student, I am very interested in analysis and proof.
I have proven Archimedes’
Quadrature of the
Parabola by means of integration and use of Heron’s formula, and that the limit as n increases without bound of the quotient of n and the n th root of n factorial is the transcendental number e . Being very analytically-minded, I have decided to switch over to more algebraic studies, which lead me to Lie algebra. This project is a challenge in the several prerequisites for Lie algebra in Linear and Abstract algebra. The nilpotent ideals of a Borel subalgebra can be classified and explored.

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Methods
As a mathematical research project, the procedures are not clearly identified step-wise. This project mostly consists of independently studying Lie algebra from an undergraduate book by Karin Erdmann and Mark Wildon called Introduction to Lie
Algebra . The chapters prevalent to my topic will be studied. If there are any Linear or
Abstract algebra terms that are unfamiliar, I will use my other sources from my annotated bibliography to comprehend the concepts. I will also converse with my mentor if any terms are persistently puzzling. Once I understand Borel subalgebras and nilpotent ideals, I will use my background knowledge on functions, relations, analysis, proofs by induction, and proofs by contradiction in order to classify the nilpotent ideals.
It is anticipated that a proof by induction will be used because nilpotent ideals use natural numbers in their definition. After I find or not find a solution, I will talk with my mentor for help and guidance on my next steps.

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Significance
This project’s significance lies in the knowledge that individuals gleam from it. As the applicant, my mathematical knowledge will be broadened in the algebraic methods that I can apply to future mathematical and scientific studies. Several scientists in the fields of Quantum Mechanics, Quantum Physics, Particle Physics, Differential
Geometry, and General Relativity will more easily understand nilpotent ideals in Borel subalgebras and how to identify them in order to apply them to their respective field.
This project may affect how Lie algebras are taught at universities, and it may influence the teaching methods for nilpotent ideals and Borel subalgebras. Overall, this project is for the good of others, making complex mathematics more intuitive and applicable.

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Budget
Total Expenses Item Cost Total
Writing Utensils $10.00 $20.00
Paper $10.00
Requested Expenses Item
Gasoline
Cost Total
$24.50 $24.50
The requested expenses for this project are for transportation. In order to do mathematical research, I have to travel to Notre Dame from Marian High School, and money is needed for fuel. I drive a 2000 Subaru Legacy with a combined 21 miles per gallon. Gas costs about $3.50 per gallon, and it takes about 7 miles to drive from Marian
High School to Notre Dame. I will meet with my mentor around 21 times. Using dimensional analysis for these values, it will cost around $24.50, hence the value in the table. The writing utensils and paper are provided by Notre Dame, and this project will cost about $10.00 for each category, totaling around $20.00 that is already paid for.

Literature Citations:
Imani, Paniz. Introduction To Lie Algebras.
Web. 15 October 2013.
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