Comultiplicative map on projective resolution for a family of algebras one of which is cluster-tilted type D4

TL;DR

This paper constructs a comultiplicative map on projective resolutions for a family of algebras, including a cluster-tilted algebra of type D4, and explores the Hochschild cohomology ring structure using a star product.

Contribution

It introduces a novel comultiplicative map and a star product to analyze the Hochschild cohomology of a specific algebra family, including cluster-tilted type D4.

Findings

Explicit comultiplicative map for the algebra family.

Description of Hochschild cohomology ring structure.

Application to cluster-tilted algebra of type D4.

Abstract

We construct a comultiplicative map on the projective bimodule resolution for a family of algebras one of which is cluster-tilted of type D4. The comultiplicative map is presented in terms of idempotents associated with vertices of the quiver and a chain homotopy map is used to describe the multiplicative structure on the Hochschild cohomology ring of the first member of the family. We define a star product used to describe the cup product structure on the Hochschild cohomology ring for all family members.