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SUMMARY:Sebastian Halbig (Universität Marburg)
DTSTART:20251008T090000Z
DTEND:20251008T100000Z
DTSTAMP:20260423T035713Z
UID:EQuAL/44
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EQuAL/44/">A
  non-semisimple version of the Kitaev model</a>\nby Sebastian Halbig (Univ
 ersität Marburg) as part of European Quantum Algebra Lectures (EQuAL)\n\n
 \nAbstract\nIn 1997\, Alexei Kitaev proposed a foundational model for faul
 t-tolerant quantum computation based on complex semisimple Hopf algebras. 
 Its key feature is a topologically invariant code space which is construct
 ed using combinatorial data encoded by a graph embedded into a closed orie
 nted surface. This ensures robustness against a wide range of errors. Beyo
 nd applications in quantum computing\, the model has remarkable connection
 s with combinatorics\, Hopf algebra representation theory\, homological al
 gebra\, and topological quantum field theories. In this talk\, based on jo
 int work with U.\\ Krähmer\, we present a generalisation of the Kitaev mo
 del to arbitrary finite-dimensional Hopf algebras. Two challenges prevent 
 a straightforward approach. First\, the extended Hilbert space\, a Yetter-
 -Drinfeld module whose invariant submodule is the code space\, relies on a
 n involutive antipode---a condition equivalent to the underlying Hopf alge
 bra being semisimple. Second\, topological invariance is proven using proj
 ectors assembled from (co)integrals. Since we do not have these tools at o
 ur disposal\, we follow a new approach\, inspired by homological considera
 tions. We introduce involutive Hopf bimodules\, which are related to coeff
 icients of Hopf cyclic cohomology and allow us to form appropriate\, Yette
 r–Drinfeld valued\, variants of extend Hilbert spaces. Instead of consid
 ering invariant submodules\, the analoga of the code spaces arise as biten
 sor products---combinations of cotensor and tensor products. Our proof of 
 their topological invariance relies on a notion of excision and uses actio
 ns of a group related to mapping class groups. Towards computing bitensor 
 products\, we discuss induction-restriction type identities\, which are pa
 rticularly useful for eg. small quantum groups.\n
LOCATION:https://researchseminars.org/talk/EQuAL/44/
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