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!
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