### Deformation Theory Seminar

Monday, January 24, 2022 - 2:00pm

#### Alan Hylton and Vince Coll

Lehigh

Location

University of Pennsylvania

https://upenn.zoom.us

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.