Time: 9:30 - 4
Today, as usual, I continued with my studies in Lie algebras. We call two Lie algebras isomorphic if we can find some mapping, or function, that sends each element from one Lie algebra to a unique element in the other, and vice versa. First, I studied the various classifications of Lie algebras, up to isomorphisms. For instance, I discovered that there can only be one Lie algebra with dimension 1, and up to two with dimension 2. However, because the cases for dimension 3 were too difficult, I resorted to the solutions from the textbook. Surprisingly, even though the jump from 2 to 3 is seemingly insignificant, there are infinitely many different Lie algebras with dimension 3!
Afterwards, I began studying the notion of "solvable" Lie algebras. I had previously seen this similar notion when I studied Galois theory, which utilized solvable groups. Both topics were very similar-- they both involve "chains" of sub-algebras that end with the trivial Lie algebra and carry special properties. While I did not get to study this topic extensively yet, I was amazed at how much group theory and Lie algebra theory overlapped. I remembered the discussion I had with Professor Dolbin regarding "category theory" and the notion of studying these different abstract algebraic structures as generally as possible. It's really cool seeing how these various topics-- group theory and Lie algebra theory in today's case-- come together and overlap.
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