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!
Thursday, April 17, 2014
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!
Wednesday, March 12, 2014
Day 24 (3/12/14)
10:30 - 5:00
Today I explored the specific structure of Sl2(C), the set of 2x2 matrices of complex numbers whose diagonal elements add up to 0. I learned about a very cool and intricate theorem, which used a basis of that Lie algebra to classify any irreducible module. While reading about this, I thought about how these irreducible modules act as Lego pieces, which look boring by themselves but can be fit together to yield very cool and complicated structures.
Because this theorem was very hard to conceptualize, I had to spend a lot of time on this chapter, and unfortunately I was not able to finish the exercises available. However, I was able to fully grasp this specific theorem, which the author contends has tremendous amounts of applications and possible generalizations. I can't wait to find out more!
Today I explored the specific structure of Sl2(C), the set of 2x2 matrices of complex numbers whose diagonal elements add up to 0. I learned about a very cool and intricate theorem, which used a basis of that Lie algebra to classify any irreducible module. While reading about this, I thought about how these irreducible modules act as Lego pieces, which look boring by themselves but can be fit together to yield very cool and complicated structures.
Because this theorem was very hard to conceptualize, I had to spend a lot of time on this chapter, and unfortunately I was not able to finish the exercises available. However, I was able to fully grasp this specific theorem, which the author contends has tremendous amounts of applications and possible generalizations. I can't wait to find out more!
Sunday, March 9, 2014
Day 23 (3/5/14)
10:00 - 4:30
Today, I studied the core of basic Lie algebra representation theory. Representation theory, as stated before, revolve around "homomorphisms," a kind of functions, from a Lie algebra to the set of self-transformations of a vector space. One of the most important theorems that my mentor described to me was Schur's Lemma, which requires too much background information to be described in a single blog post. However, even though it is considered to be very "elementary," it has a lot of applications in proving stronger and much more complicated theorems. Lastly, towards the end of the internship, my mentor referred again to the very general "category theory," and how even Lie algebra representations could be generalized and extended to other algebras such as groups and rings. While I still don't have a satisfying grasp on the topic, I realized that this topic is one of the most interesting ones that I have studied so far!
Today, I studied the core of basic Lie algebra representation theory. Representation theory, as stated before, revolve around "homomorphisms," a kind of functions, from a Lie algebra to the set of self-transformations of a vector space. One of the most important theorems that my mentor described to me was Schur's Lemma, which requires too much background information to be described in a single blog post. However, even though it is considered to be very "elementary," it has a lot of applications in proving stronger and much more complicated theorems. Lastly, towards the end of the internship, my mentor referred again to the very general "category theory," and how even Lie algebra representations could be generalized and extended to other algebras such as groups and rings. While I still don't have a satisfying grasp on the topic, I realized that this topic is one of the most interesting ones that I have studied so far!
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