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!
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