A decomposition of the Brauer-Picard group of the representation\n category of a finite group

2015/06/25 by Simon Lentner, Lentner, Simon, Jan Priel +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Finite Group Theory Research #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1506.07832

openalex publication_date 2015/06/25 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We present an approach of calculating the group of braided autoequivalences\nof the category of representations of the Drinfeld double of a finite\ndimensional Hopf algebra H and thus the Brauer-Picard group of\nH-\mod. We consider two natural subgroups and a subset as\ncandidates for generators. In this article H is the group algebra of a finite\ngroup G. As our main result we prove that any element of the Brauer-Picard\ngroup, fulfilling an additional cohomological condition, decomposes into an\nordered product of our candidates. For elementary abelian groups G our\ndecomposition reduces to the Bruhat decomposition of the Brauer-Picard group,\nwhich is in this case a Lie group over a finite field. Our results are\nmotivated by and have applications to symmetries and defects in 3d-TQFT and\ngroup extensions of fusion categories.\n

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