Time: 8:30 - 3:30
Today, I learned about two very important theorems in Lie algebra theory-- Engel's Theorem and Lie's Theorem. Engel's Theorem gives a powerful criterion for determining when a vector space has a basis that represents a corresponding Lie algebra by a strictly upper triangular matrix, which only has nonzero elements above the main diagonal of the matrix. On the other hand, Lie's Theorem finds a similar criterion for the Lie algebra being an upper triangular matrix, which has nonzero elements on or above the main diagonal. Because these theorems were very complex and difficult to prove, I had to spend a lot of time on them, although eventually I was able to grasp them. Nonetheless, I was not able to study much else, as I spent almost all of the remaining time on exercises provided by the text. I read that these structures and maps have physical applications, which I can't wait to find out!
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