nLab periodic triangulated category

Definition

A triangulated category is $m$-periodic for integer $m\geq 2$, if the $m$-th power of the suspension (translation) functor

$$\Sigma: X\mapsto X[1]$$

is isomorphic to the identity functor.

Examples

Triangulated categories of matrix factorizations are 2-periodic.

Literature

• Shunya Saito, Tilting objects in periodic triangulated categories, Math. Z. 304, 47 (2023) doi arXiv:2011.14096

• Tobias Dyckerhoff, Mikhail Kapranov, Triangulated surfaces in triangulated categories, J. Eur. Math. Soc. 20:6, 1473–1524 (2018) doi

• J. Sheng, X. Chen, F. Xu, On two Hall algebra approaches to odd periodic triangulated categories, Sci. China Math. 61, 317–324 (2018) doi

On a global realization of Kac-Moody Lie algebras as a Hall type Lie algebras arising from certain 2-periodic triangulated categories:

• Fan Xu, Xueqing Chen, Some remarks on triangulated categories and Kac–Moody algebras, J. Alg. 388 (2013) 22-42 doi