Lie Algebras Attached to Simplicial Complexes

Incidence algebras of simplicial complexes

With any locally finite partially ordered set K its incidence algebra Ω(K) is associated. We shall consider algebras over fields with characteristic zero. In this case there is a correspondence K ↔ Ω(K) such that the poset K can be reconstructed from its incidence algebra up to an isomorphism — due to Stanley theorem. In the meantime, a monotone mapping between two posets in general induces no ...

On Isomorphism of Simplicial Complexes and Their Related Algebras

Noncompact homogeneous Einstein manifolds attached to graded Lie algebras

In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the nilradical is of two-step. New examples of Einstein solvmanifolds with three-st...

Combinatorial Hopf Algebras of Simplicial Complexes

We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their f -vectors. We al...

on isomorphism of simplicial complexes and their related algebras