8:00-12:00, 5:00-7:00
Because of a major competition during the middle of the day, I had to split my times of internship. Hence, today was another review day. I specifically reread the sections on simple, semisimple, and nilpotent algebras, and I also redid many of the exercises from those chapters. I also referenced another text that dealt with Lie algebras and representation theory, which provided another perspective on the implications of nilpotency and solvency. Now that I was able to review the basics and fundamentals of those concepts, I feel more equipped to handle the more complicated arguments involving root systems!
Thursday, May 1, 2014
Wednesday, April 23, 2014
Day 30 (4/23/14)
Time: 10:00 - 4:30
Today, I further studied the implications of root systems. First, I learned about the geometric structures that root systems provide. For example, using the inner product, one could determine the possible angles between two elements in a root system, thereby arriving at a series of vectors stemming from the origin outwards symmetrically. I showed that there were surprisingly not much variety to the angles-- for example there are less than ten possible angles total between two base vectors in the regular xy-plane!
Afterwards, the book shifted to a more graph-theoretical approach, as it began to relate root systems to "Dynkin diagrams," which is a graph with nodes and edges representing the value <a,b><b,a>. Through such a notion, we showed that to show that root systems were equivalent, it is enough to simply compare different Dynkin diagrams, of which there are only 4 kinds and 5 exceptions. Through this, we eventually were able to demonstrate that all but two of the "classical" Lie algebras are simple. While I learned a lot, I think another review day next week would help me recover some more materials that I might have missed today...We went over a lot!
In addition, today Professor Dolbin took us out to lunch, and we talked about his past life, careers, and academic choices. He told me and gave me a lot of advice about both undergraduate and graduate schools, and how he dealt with studying theoretical mathematics. While the situations that he described seem incredibly difficult and stressful, I'm nonetheless thankful for the advice, as I now know more about how to achieve my career goals. Overall today was an amazing day, as I got to bond more with Professor Dolbin!
Today, I further studied the implications of root systems. First, I learned about the geometric structures that root systems provide. For example, using the inner product, one could determine the possible angles between two elements in a root system, thereby arriving at a series of vectors stemming from the origin outwards symmetrically. I showed that there were surprisingly not much variety to the angles-- for example there are less than ten possible angles total between two base vectors in the regular xy-plane!
Afterwards, the book shifted to a more graph-theoretical approach, as it began to relate root systems to "Dynkin diagrams," which is a graph with nodes and edges representing the value <a,b><b,a>. Through such a notion, we showed that to show that root systems were equivalent, it is enough to simply compare different Dynkin diagrams, of which there are only 4 kinds and 5 exceptions. Through this, we eventually were able to demonstrate that all but two of the "classical" Lie algebras are simple. While I learned a lot, I think another review day next week would help me recover some more materials that I might have missed today...We went over a lot!
In addition, today Professor Dolbin took us out to lunch, and we talked about his past life, careers, and academic choices. He told me and gave me a lot of advice about both undergraduate and graduate schools, and how he dealt with studying theoretical mathematics. While the situations that he described seem incredibly difficult and stressful, I'm nonetheless thankful for the advice, as I now know more about how to achieve my career goals. Overall today was an amazing day, as I got to bond more with Professor Dolbin!
Thursday, April 17, 2014
Day 29 (4/16/14)
9:00 - 3:30
Today in internship, I learned about root spaces and how they could be used to determine even more information about the structure of semisimple complex Lie algebras. Root spaces are sets of "roots" or functions of weights, which as described earlier are generalizations of eigenvalues in linear algebra. Essentially, a "root" is a linear mapping "f" from the Lie subalgebra L of gl(V) to its scalar field such that there exists at least one element v in V such that for any element x in L, x(v) = f(x)v. This set of roots, the root space, is especially important because any Lie algebra over the complex numbers can be decomposed as a direct sum of weight spaces that stem from the roots.
This new notion leads to interesting results, as these root spaces eventually lead to another kind of inner products, which are operations of two vectors that output a scalar. Such a valuable tool can be used to geometrically describe specific Lie algebras in terms of symmetric vectors. After my mind was blown by the tremendously symmetric representations of these Lie algebras like sl(3,C), Professor Dolbin talked to me about how these notions could be further generalized. I saw a glimpse of such ideas when I learned about buildings and chamber systems with Dr. Abramson in school, and I was struck by the resemblances. I can't believe how cool math is and how it ties in such different branches together!
Today in internship, I learned about root spaces and how they could be used to determine even more information about the structure of semisimple complex Lie algebras. Root spaces are sets of "roots" or functions of weights, which as described earlier are generalizations of eigenvalues in linear algebra. Essentially, a "root" is a linear mapping "f" from the Lie subalgebra L of gl(V) to its scalar field such that there exists at least one element v in V such that for any element x in L, x(v) = f(x)v. This set of roots, the root space, is especially important because any Lie algebra over the complex numbers can be decomposed as a direct sum of weight spaces that stem from the roots.
This new notion leads to interesting results, as these root spaces eventually lead to another kind of inner products, which are operations of two vectors that output a scalar. Such a valuable tool can be used to geometrically describe specific Lie algebras in terms of symmetric vectors. After my mind was blown by the tremendously symmetric representations of these Lie algebras like sl(3,C), Professor Dolbin talked to me about how these notions could be further generalized. I saw a glimpse of such ideas when I learned about buildings and chamber systems with Dr. Abramson in school, and I was struck by the resemblances. I can't believe how cool math is and how it ties in such different branches together!
Tuesday, April 15, 2014
Day 28 (4/9/14)
10:00 -4:30
Today, I had to review more materials for a harder topic. I read more about linear algebra, and then worked towards some vector analysis. Because today was solely meant for review, I was not able to learn anything new in terms of Lie Algebra Theory. However, I learned to appreciate the subject of linear algebra, as I found out about the tremendous ties between the two fields of study. I can't wait to get back to Lie Algebra theory next week!
Edit: I apologize for publishing this blog entry late - I had written it on the day of the internship, but I forgot to press the submit button!
Today, I had to review more materials for a harder topic. I read more about linear algebra, and then worked towards some vector analysis. Because today was solely meant for review, I was not able to learn anything new in terms of Lie Algebra Theory. However, I learned to appreciate the subject of linear algebra, as I found out about the tremendous ties between the two fields of study. I can't wait to get back to Lie Algebra theory next week!
Edit: I apologize for publishing this blog entry late - I had written it on the day of the internship, but I forgot to press the submit button!
Wednesday, April 2, 2014
Day 27 (4/2/14)
Time: 10:30 - 5:00
After a much-needed review on previous topics like representations and modules, I continued with my journey on Lie algebra theory. Specifically, I was pleased to make much progress, as I studied both representations of Sl(2,C) and criteria for determining whether a Lie algebra is semisimple. First, through a series of very rigorous and complex exercises and lemmas, I was able to classify the irreducible modules of Sl(2,C), which again is the set of 2-by-2 matrices with complex numbers and whose diagonal elements sum to zero. This area was especially important, as irreducible modules, in one way, become a sort of building blocks for other modules.
Afterwards, after a nice lunch break, I moved on to studying the various useful conditions that are both necessary and sufficient for a Lie algebra to be semisimple. To recap, a Lie algebra is semisimple if it has no non-zero solvable ideals. While ultimately I had to consult the book multiple times, I eventually discovered two very crucial criteria, which are officially called Cartan's Criteria. The first of these criteria asserts that the complex Lie algebra L is solvable if and only if the sum of the diagonal elements of the matrix of [x,[y,-]] is equal to 0. The second and more applicable theorem states that L is semisimple if and only if the "Killing form," or the function K defined as K(x,y)=sum of diagonal elements of the matrix [x,[y,-]], is non-degenerate. While I learned a significant amount of information regarding Lie algebra theory, today's portion of my internship especially showed me how complicated and difficult mathematics can get. However, I also realized that these difficulties don't discourage me, but rather make me even more hooked into the subject!
After a much-needed review on previous topics like representations and modules, I continued with my journey on Lie algebra theory. Specifically, I was pleased to make much progress, as I studied both representations of Sl(2,C) and criteria for determining whether a Lie algebra is semisimple. First, through a series of very rigorous and complex exercises and lemmas, I was able to classify the irreducible modules of Sl(2,C), which again is the set of 2-by-2 matrices with complex numbers and whose diagonal elements sum to zero. This area was especially important, as irreducible modules, in one way, become a sort of building blocks for other modules.
Afterwards, after a nice lunch break, I moved on to studying the various useful conditions that are both necessary and sufficient for a Lie algebra to be semisimple. To recap, a Lie algebra is semisimple if it has no non-zero solvable ideals. While ultimately I had to consult the book multiple times, I eventually discovered two very crucial criteria, which are officially called Cartan's Criteria. The first of these criteria asserts that the complex Lie algebra L is solvable if and only if the sum of the diagonal elements of the matrix of [x,[y,-]] is equal to 0. The second and more applicable theorem states that L is semisimple if and only if the "Killing form," or the function K defined as K(x,y)=sum of diagonal elements of the matrix [x,[y,-]], is non-degenerate. While I learned a significant amount of information regarding Lie algebra theory, today's portion of my internship especially showed me how complicated and difficult mathematics can get. However, I also realized that these difficulties don't discourage me, but rather make me even more hooked into the subject!
Wednesday, March 26, 2014
Day 26 (3/26/14)
10:00 - 4:30
Today during internship, I encountered a subtopic that I found was extremely difficult to understand. Thus, with not having thought about Lie algebras over the last two weeks, I decided to spend today reviewing all of the materials that I have learned prior to this specific section. I revisited old exercises that I was not able to solve first, and surprisingly I was able to solve them more readily the second time. I also got a much better grasp of Lie algebras, their homomorphisms, weight spaces, and other concepts than before. Therefore, while I am partly disappointed that I was not able to make much new progress today, I am still very happy also that I was able to solidify my understanding in my studies.
Today during internship, I encountered a subtopic that I found was extremely difficult to understand. Thus, with not having thought about Lie algebras over the last two weeks, I decided to spend today reviewing all of the materials that I have learned prior to this specific section. I revisited old exercises that I was not able to solve first, and surprisingly I was able to solve them more readily the second time. I also got a much better grasp of Lie algebras, their homomorphisms, weight spaces, and other concepts than before. Therefore, while I am partly disappointed that I was not able to make much new progress today, I am still very happy also that I was able to solidify my understanding in my studies.
Friday, March 21, 2014
Day 25 (3/19/14)
Because of a sudden illness, I was not able to attend the internship today. However, I was still able to work on the exercises from the text, which was fun by itself!
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