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Deformation Theory Seminar

Monday, January 24, 2022 - 2:00pm

Alan Hylton and Vince Coll



University of Pennsylvania

see email for zoom meeting id


Seaweed algebras (``seaweeds'') are certain Lie subalgebras of the complex simple Lie algebras which are defined in terms of their attendant root systems. We review this setup and show, as a corollary to a strong cohomology result, that if $\mathfrak{s}$ is an indecomposable (not a direct sum) seaweed then $\mathfrak{s}$ is absolutely rigid.  However, decomposable seaweeds can have non-trivial cohomology and may deform; examples are given.


In characteristic $p$, the situation is more complicated since simple modular Lie algebras need not have root systems;  those that do give rise to  \textit{modular seaweeds} in a natural way.

The cohomology of indecomposable modular seaweeds with coefficients in the adjoint representation is zero, as in the characteristic zero case -- so long as the rank of the modular seaweed is sufficiently larger the rank of the seaweed.