Ambidextrous objects and trace functions for nonsemisimple categories

Understand

We provide a necessary and sufficient condition for a simple object in a pivotal k-category to be ambidextrous.

• In turn, these objects imply the existence of nontrivial trace functions in the category.
• These functions play an important role in low-dimensional topology as well as in studying the category itself.
• In particular, we prove they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the $(1,p)$ minimal model in conformal field theory, and quantum groups at a root of unity.