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Rigidity of Microsphere Heaps Abstract ˆ RAHMOELLER, MARGARET LYNN . On Demazure Crystals for the Quantum Affine AlgebraUq s l (n) . (Under the direction of Dr. Kailash Misra.) Lie algebras and their representations have been an important area of mathematical research due to their relevance in several areas of mathematics and physics. It is known that the representation of a Lie algebra can be studied by considering the representations of its universal enveloping al- gebra, which is an associative algebra. In 1968, Victor Kac and Robert Moody defined a class of infinite dimensional Lie algebras called affine Lie algebras. An affine Lie algebra can be viewed as the universal central extension of the Lie algebra of polynomial maps from the unity circle to ˆ a finite dimensional simple Lie algebra. In this thesis, we consider the affine Lie algebra s l (n) associated with the finite dimensional simple Lie algebra s l (n) of n n trace zero matrices over the field of complex numbers. We consider certain representations of× the quantum affine Lie algebra ˆ ˆ ˆ Uq (s l (n)), which is a q -deformation of the universal enveloping algebra U (s l (n)) of s l (n) intro- duced by Michio Jimbo and Vladimir Drinfeld in 1985. In 1988, it was shown by George Lusztig that ˆ ˆ the q -deformation of a representation ofUq (s l (n)) parallels the representation of s l (n) for generic q . It is known that for any dominant integral weight λ there is a unique (up to isomorphism) irreducible ˆ Uq (s l (n))- module V (λ). It was shown by Kashiwara that this irreducible module V (λ) admits a crystal base (L(λ), B (λ)). The set B (λ) is called a crystal, and it provides a nice combinatorial tool to study the combinatorial properties of the representation space V (λ). ˆ Let W be the Weyl group of the affine Lie algebra s l (n). For w W we consider the subspace Vw (λ) 2 + ˆ generated by the extremal weight vector vw λ and the positive half Uq (s l (n)) of the quantum affine ˆ S algebra Uq (s l (n)). We note that w W Vw (λ) = V (λ). These subspaces Vw (λ) are called Demazure 2 modules and are used for inductive arguments. In 1993, Kashiwara showed that Vw (λ) admits a crystal base and that a certain subset Bw (λ) of the crystal B(λ) is the crystal for Vw (λ). Kang, Kashi- wara, Misra, Miwa, Nakashima and Nakayashiki gave the path realizations of affine crystals as a semi-infinite tensor product of some finite crystals called perfect crystals in 1991. In this thesis, we use this path realization and study the combinatorial properties of certain De- ˆ mazure crystals Bw (λ) for the quantum affine algebra Uq (s l (n)). In particular, it is shown that the ˆ intersection and union of a certain family of Demazure crystals for Uq (s l (n)) can be realized as tensor products of finitely many perfect crystals. © Copyright 2015 by Margaret Lynn Rahmoeller All Rights Reserved ˆ On Demazure Crystals for the Quantum Affine Algebra Uq s l (n) by Margaret Lynn Rahmoeller A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Mathematics Raleigh, North Carolina 2015 APPROVED BY: Dr. Ernest Stitzinger Dr. Thomas Lada Dr. Donald Martin Dr. Kailash Misra Chair of Advisory Committee Dedication I dedicate this to my Mom. She is my biggest role model, and has supported me wholeheartedly throughout my entire life. Thank you, Mom, for all of your encouragement when life threw me curveballs, your patience when I was struggling, your comfort when I was sad or upset, and your unconditional love. ii Biography Margaret Rahmoeller was born in June of 1988 in the wonderful Webster Groves, a suburb of St. Louis, Missouri. Nicknamed Maggie, she was the third child of Ken and Diane Rahmoeller, their first being Katie and second being Brad. Not too long after, Maggie was the proud older sister of Karen and Brenda. Maggie grew up playing outside, climbing trees, collecting bugs and slugs for her teachers, and being an all-around adorable child. As a child, she dreamed of being an astronaut or an actress. Little did she know that she was destined to be a teacher. In sixth grade, she began playing the oboe, much to the horror of her family. A unique woodwind instrument, the oboe is typically called the “duck" instrument. However, after starting lessons and practicing very hard, she began to sound more like an oboist. She already had a love for music, since she had taken piano lessons for several years; however, playing the oboe increased her appreciation for classical music. At the end of high school, she had a very important decision to make- what to study in college. She chose to study mathematics and music, since she loved them both dearly. She spent an amazing four years studying both at McKendree University, located in Lebanon, Illinois. Along the way, she also picked up a computer science minor. Although most of her time was filled with her studies and tutoring, she managed to spend quality time with stupendous new friends, performing with the St. Louis Symphony Youth Orchestra as principal oboist, and helping out with math club. At the end of her time at good ‘ole McK, she had another important decision to make- what to do after graduation. Maggie spent a summer doing an internship in Maryland before heading off to graduate school to study mathematics at North Carolina State University in Raleigh, North Carolina. During her first semester as a teaching assistant for a Calculus III class, she had the opportunity to teach one lecture. She fell in love with teaching that day. By participating in pedagogical programs such as Certificate of Accomplishment in Teaching and Preparing the Professoriate, she learned as much as she could about different aspects of teaching. She also performed as principal oboist with both the Raleigh Civic Symphony and the Raleigh Civic Chamber Orchestra, ensembles at NCSU consisting of both students and community members. In her fourth year at NCSU, she joined the Raleigh Civic Symphony Association, a nonprofit orga- nization, which works in partnership with the music department to promote the orchestras at NCSU. iii But most of her time was spent studying mathematics. She was introduced to Lie algebras in her third year, and quickly became intrigued. She decided to work with Dr. Kailash Misra in the field of Lie algebra representation theory. After several years of hard work, she will be defending her thesis in front of a live audience in the summer of 2015. Then, in August of 2015, she will begin a position at Roanoke College in Salem, Virginia as a Visiting Assistant Professor. She is super excited for the next chapter of her life to begin! iv Acknowledgements Graduate school is hard work. The friends you make and the mentors you find, along with your family, are what helped to push me through the experience. I would like to first thank my advisor, Dr. Kailash Misra, whose impressive knowledge and innate ability to explain mathematics helped me through the toughest years of my graduate school career. I honestly could not have had a better advisor. Dr. Misra is an incredibly kind-hearted man and a superb teacher and mentor. Dr. Misra, thank you for all of those "epsilon" talks, all of the opportunities for travel and networking, and all of the wisdom you shared with me. I’d also like to thank my family. Thank you Mom for being there for all my tearful phone calls, for encouraging me to stay strong and push through the hard times, and for celebrating all of the wonderful times. Dad, thank you for all of the fun trips in which we toured all of North Carolina, for all the advice you shared with me, and for all the help with the ridiculous electronics in my life. I can’t wait for all of our future trips in other states. I’d like to thank my older sister Katie for all of the care packages she sent me, and my brother Brad for all of his fun stories and great humor, which kept me smiling and laughing even during the rough times. I’d like to thank my sister Karen for all of the wonderful phone conversations we had and for the paper chain she made me for the final countdown. And I’d like to thank my sister Brenda for all of the adorable dog pictures she sent me, which always cracked me up. I would also like to thank my Grandpa for his words of wisdom and encouragement. I’ve always looked up to him, both literally and figuratively, and have enjoyed all of the times we have shared. Thank you to Mr. and Mrs. Nuelle, who I have always considered grandparents, for the many memorable lunches and for many of my wonderful childhood memories. Mr. Nuelle, thank you for pushing me to finish my degree in five years and for all the math documentaries we watched together. And I of course want to thank one Mr. Russ for all of the fun times in orchestra at State and the delightful duets we played. You are an incredibly talented bassoonist and a wonderful friend. I had other mentors at NCSU who I’d also like to thank. Professor Stitz, you have always been im- mensely helpful, whether by having an answer to all of my questions, or helping me make important decisions, or by always having a smile and words of encouragement.
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