# Stable equivalences of Morita type for $\Phi$-Beilinson-Green algebras

## TL;DR

This paper introduces a method to generate new stable equivalences of Morita type between algebras using $ ext{Phi}$-Beilinson-Green algebras, expanding understanding of algebra equivalences and their invariants.

## Contribution

It presents a novel approach to construct stable Morita type equivalences via $ ext{Phi}$-Beilinson-Green algebras, generalizing previous results and exploring graded algebra cases.

## Key findings

- Constructed new stable equivalences of Morita type for $ ext{Phi}$-Beilinson-Green algebras.
- Provided examples of derived equivalent algebras not stably equivalent of Morita type.
- Established a link between graded stable equivalences and Beilinson-Green algebras.

## Abstract

In this paper, we present a method to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between finite dimensional algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible set $\Phi$ of natural numbers and any generator $X$ of the $A$-module category, the $\Phi$-Beilinson-Green algebras $\scr G^{\Phi}_A(X)$ and $\scr G^{\Phi}_B(N\otimes_AX)$ are stably equivalent of Morita type. In particular, if $\Phi=\{0\}$, we get a known result in literature. As another consequence, we construct an infinite family of derived equivalent algebras of the same dimension and of the same dominant dimension such that they are pairwise not stably equivalent of Morita type. Finally, we will prove that, if there is a graded stable equivalence of Morita type between graded algebras, then we can get a stable equivalence of Morita type between Beilinson-Green algebras associated with graded algebras