Date: 14 October 2016, 4:30pm
Location: Hempfield High School, Room 213
 Speaker: Vincent E. Coll, Lehigh University
   Title: Seaweed and poset algebras -- synergies and cohomology

   Abstract. Part I, Seaweeds. In my previous talk, I introduced
   seaweed algebras as subalgebras of An = sl(n) and examined certain
   newly discovered spectral invariants. We have extended the work to
   the symplectic Type-C case, Cn = sp(2n), and have strong indications
   of what is true in all other cases – these cases corresponding to the
   simple Lie algebras in the grand classification. One finds,
   curiously, that these spectral properties are not unique to
   Frobenious seaweed subalgebras of the simple Lie algebras. It seems
   that these properties are shared by certain Lie poset algebras
   recently introduced by myself and Gerstenhaber in the J. Lie Theory
   article: Cohomology of Lie semidirect products and poset algebras
   (2016).

   Part II, Posets. These Lie poset algebras are concrete matrix
   algebras and they stand at the nexus of three great cohomology
   theories: simplicial, the algebraic cohomology of Hochshild, and the
   Lie algebraic cohomology of Chevalley-Eilenberg. More explicitly, now
   classic results of Gerstenhaber and Schack elegantly established that
   simplicial cohomology is a special case of algebraic cohomology. That
   is, associated to each triangulable space, there is an associative
   matrix algebra whose algebraic cohomology is the same as the
   simplicial cohomology of the original space. These matrix algebras
   are also Lie algebras and therefore have a Lie algebra cohomology
   which controls their deformations as Lie algebras. We can establish
   that the latter cohomology is also essentially simplicial and that
   the deformation theory of these Lie poset algebras is analogous to
   that of complex analytic manifolds for which this theory is a small model.