Calabi-Yau (CY) triangulated categories are those satisfying a
useful and important
duality, characterised by a number called the CY dimension. Much
has been carried out on understanding positive CY triangulated
categories, especially in the
context of cluster-tilting theory. Even though CY dimension is
considered to be a positive (or fractional) number, there are
examples of CY triangulated categories where this”dimension” or
parameter is negative,
for example, stable module categories of self-injective algebras.
Therefore, negative CY
triangulated categories constitute a class of categories that
further systematic study. In this talk, we will give a brief
survey regarding what is so far known about the structure of
CY triangulated categories, and we will focus on an example given by
triangulated categories generated by spherical objects.