Central invariants and Frobenius–Schur indicators for semisimple quasi-Hopf algebras

By Geoffrey Mason, Siu-Hung Ng. 2003

72 citations recorded by the source corpus.

Abstract In this paper, we obtain a canonical central element νH for each semi-simple quasi-Hopf algebra H over any field k and prove that νH is invariant under gauge transformations. We show that if k is algebraically closed of characteristic zero then for any irreducible representation of H which affords the character χ, χ(ν H ) takes only the values 0, 1 or −1, moreover if H is a Hopf algebra or a twisted quantum double of a finite group then χ(νH) is the corresponding Frobenius–Schur indicator. We also prove an analog of a theorem of Larson–Radford for split semi-simple quasi-Hopf algebras over any field k. Using this result, we establish the relationship between the antipode S, the values of χ(νH), and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dc8cc2ae8e75cf50e4e20f81ffddc2d87e03e393

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Citations in the corpus

On the Frobenius–Schur indicators for quasi-Hopf algebras (2003, background): “It can be reformulated as in [11] to be a property of bilinear forms on V : An isomorphism f : V → V ∗ is the same as a nondegenerate bilinear form σ : V ⊗V → k, by the correspondence σ(v, w) = f(v)(w).”
Frobenius–Schur indicators and exponents of spherical categories (2005, methodology): “Direct computation shows that ord(u) = 4. By [ MN05 ], the trace-element g of D(H) is given by g = X”
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups (2006, methodology): “s of Dω(G) where Gis either extra-special group or elementary abelian, and ωis any 3-cocycle. Frobenius-Schur indicators, their higher analogs, and Frobenius-Schur exponents have been investigated in [MN2], [NS1], [NS2], [NS3] in the general context of semisimple quasi-Hopf algebras and pivotal fusi”