Events
DMS Algebra Seminar |
| Time: Mar 29, 2022 (02:30 PM) |
| Location: 358 Parker Hall |
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Details: Speaker: Luke Oeding
Title: A Jordan-Chevalley decomposition for tensors
Abstract: The classical Jordan canonical form of a matrix expresses a matrix in its best possible form up to similarity. A first consequence of the Jordan normal form is that every square matrix is similar to a sum of a diagonal and an upper triangular matrix. A second consequence is that all of the conjugation invariants of the matrix can be read off from the canonical form, and one obtains a complete classification of orbits for the conjugation action. Jordan decomposition generalizes nicely to adjoint representations of semisimple Lie algebras. We reinterpret the square matrix case via the special linear group action on the space of endomorphisms of a vector space. The Jordan-Chevalley decomposition expresses any element of a semisimple as a sum of a semisimple and a nilpotent element. I will explain how Vinberg and coauthors used the Jordan-Chevalley decomposition together with Dynkin’s classification of semi-simple subalgebras of semi-simple algebras to classify orbits in several special cases of tensors. I will explain our recent efforts to generalize this concept to obtain a course classification of tensors.
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