A projective resolution for the Fomin-Kirillov algebra $\operatorname{FK}(4)$

TL;DR

This paper develops a method to construct minimal projective resolutions for quadratic algebras with a resolving datum and applies it to prove that the Fomin-Kirillov algebra FK(4) has such a datum, providing new insights into its structure.

Contribution

It introduces a general construction for projective resolutions of quadratic algebras with a resolving datum and demonstrates this for FK(4), a previously unresolved algebra.

Findings

Constructed a minimal projective resolution for FK(4)

Established that FK(4) has a resolving datum

Unified approach applicable to many quadratic algebras

Abstract

In this article we show that, given a quadratic algebra satisfying some assumptions, which we call having a resolving datum, one can construct a projective resolution of the trivial module which is obtained as iterated cones of Koszul complexes, and this projective resolution is minimal under some further assumptions. We observe that many examples of quadratic algebras studied so far have a resolving datum, and that the (minimal) projective resolutions constructed for all of them in the literature are an example of our construction. The second main result of the article is that the Fomin-Kirillov algebra $\operatorname{FK}(4)$ of index $4$ has a resolving datum.