Hall algebras for odd periodic triangulated categories

TL;DR

This paper introduces a new class of Hall algebras for odd periodic triangulated categories, extending previous work to broader settings with specific finiteness conditions and a $t$-periodic translation functor.

Contribution

It generalizes the construction of Hall algebras to odd periodic triangulated categories, broadening the scope of algebraic structures associated with such categories.

Findings

Defined Hall algebras for odd $t>1$ in triangulated categories.

Extended previous results to new classes of categories.

Provided a framework for further algebraic and categorical research.

Abstract

We define the Hall algebra associated to any triangulated category under some finiteness conditions with the $t$-periodic translation functor $T$ for odd $t>1.$ This generalizes the results in \cite{Toen2005} and \cite{XX2006}.