Homologies of monomial operads and algebras

TL;DR

This paper investigates the homology of monomial operads and algebras, providing a simple description of homology in terms of Dyck paths and exploring conditions for homological dichotomy.

Contribution

It introduces a straightforward method to compute homology of monomial algebras and operads, including the development of combinatorial tools for quadratic operads.

Findings

Homology of subcomplexes is at most one-dimensional.

Dichotomy in homology holds when monomial relations form an order.

Methods for computing homology in truncated binary operads are demonstrated.

Abstract

We consider the bar complex of a monomial non-unital associative algebra $A=k \langle X \rangle / (w_1,...,w_t)$. It splits as a direct sum of complexes $B_w$, defined for any fixed monomial $w=x_1...x_n \in A$. We give a simple argument, showing that the homology of this subcomplex is at most one-dimensional, and describe the place where the nontrivial homology appears. It has a very simple expression in terms of the length of the generalized Dyck path associated to a given monomial in $w \in A$. The operadic analogue of the question about dichotomy in homology is considered. It is shown that dichotomy holds in case when monomial tree-relations form an order. Examples are given showing that in general dichotomy and homological purity does not hold. For quadratic operads, the combinatorial tool for calculating homology in terms of relation graphs is developed. Example of using these methods to compute homology in truncated binary operads is given.