Incidence algebras of simplicial complexes (Q2717072)

Incidence algebras were introduced by \textit{G.-C. Rota} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 2, 340-368 (1964; Zbl 0121.02406)]. With any locally finite partially ordered set\ \(\mathcal K\)\ its incidence algebra \(\Omega (\mathcal K)\) is associated. There is a correspondence \(\mathcal K\leftrightarrow \Omega(\mathcal K)\) and it was proved by \textit{R. P. Stanley} [Enumerative combinatorics, Vol I (1986; Zbl 0608.05001)] that a poset \(\mathcal K\) can be reconstructed from its incidence algebra \(\Omega (\mathcal K)\) up to a poset isomorphism. NEWLINENEWLINENEWLINEMeanwhile, a poset homomorphism induces no homomorphism of their incidence algebras, the author shows that if the class of posets is confined to simplicial complexes then their incidence algebras acquire the structure of differential moduli over the algebra \(\mathcal A\) of all functions on \(\mathcal K\), representing \(\Omega (\mathcal K)\) as a quotient of the universal differential envelope \(\Omega _{u}(\mathcal A)\), and the correspondence \(\mathcal K\leftrightarrow \Omega (\mathcal K)\) is a contravariant functor into the category of differential moduli over finite-dimensional semisimple commutative algebras. So simplicial complexes resemble differential manifolds from the algebraic point of view.