Abstract: The main theme of the talk is the classification of gradings on simple Lie algebras over algebraically closed fields of prime characteristic.  In characteristic zero, cyclic gradings can be described in terms of finite-order automorphisms.  Following a viewpoint of Serre, this can be reformulated more intrinsically as a problem about embeddings of group schemes of roots of unity into a suitable disconnected algebraic group. I will first describe a uniform method for classifying these embeddings up to conjugacy. This approach treats inner and outer cyclic gradings within a single framework and is especially useful in positive characteristic, where one cannot in general replace the group-scheme formulation by finite-order automorphisms. I will then turn to fine gradings on the exceptional Lie algebra of type E6.  Fine gradings are governed by maximal diagonalizable subgroups of the automorphism group, so their classification reduces to the conjugacy classification of such subgroups.  For reductive algebraic groups, I will describe a characteristic-independent method for classifying maximal abelian diagonalizable subgroups.  Applying this method to E6 gives the classification of fine gradings in characteristics different from two and three.  The resulting list agrees with the known characteristic-zero classification and shows that no new fine gradings occur in these characteristics.

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Location: HH-3017

Date and Time: Wednesday, May. 13 at 01:00 PM - 02:00 PM (NDT)

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