Incidence, or poset, algebras can be given a Lie structure by taking the commutator product. Curiously, these "Lie poset algebras" have only recently been introduced into the literature. Here, we initiate the study of the index of Lie poset subalgebras of $A_n=\mathfrak{sl}(n+1)$ by providing closed-form index formulas. Index-zero Lie algebras are called Frobenius and are of interest to those working in deformation theory. Using our new index formulas, we characterize Frobenius type-A Lie poset algebras corresponding to posets of restricted height. This classification follows from a combinatorial recipe for the construction of all posets corresponding to these algebras. The topical theory of the "spectrum" of these Frobenius type-A Lie poset algebras is also investigated -- with some very promising initial results. We conclude by showing how this theory can be extended to the other classical types.

### CAGE: Philadelphia Area Combinatorics and Alg. Geometry Seminar

Thursday, October 3, 2019 - 3:00pm

#### Nicholas Mayers

Lehigh University