Ambidextrous objects and trace functions for nonsemisimple categories

Abstract

We provide a necessary and sufficient condition for a simple object in a pivotal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck k"> <mml:semantics> <mml:mi mathvariant="double-struck">k</mml:mi> <mml:annotation encoding="application/x-tex">\Bbbk</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 comma p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(1,p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> minimal model in conformal field theory, and quantum groups at a root of unity.