1996/01/20 by Thomas Kerler, Kerler, Thomas · 1 citation
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.q-alg/9601021
openalex publication_date 1996/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the relations between the invariants \τRT, \τHKR, and\n \τL of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko,\n respectively. In particular, we discuss explicitly how \τL specializes\nto \τRT for semisimple categories and to \τHKR for Tannakian\ncategories. We give arguments for that \τL is the most general invariant\nthat stems from an extended TQFT. We introduce a canonical, central element,\n sf Q, for a quasi-triangular Hopf algebra, A, that allows us to apply the\nHennings algorithm directly, in order to compute \τRT, which is\noriginally obtained from the semisimple trace-subquotient of A-mod.\nMoreover, we generalize Hennings' rules to the context of cobordisms, in order\nto obtain a TQFT for connected surfaces compatible with \τHKR ,. As an\napplication we show that, for lens spaces and A=Uq(sl2) ,, the ratio of\n\τHKR and \τRT is the order of the first homology group. In the\ncourse of this paper we also outline the topology and the algebra that enter\ninvariance proofs, which contain no reference to 2-handle slides, but to other\nmoves that are local. Finally, we give a list of open questions regarding\ncellular invariants, as defined by Turaev-Viro, Kuperberg, and others, their\nrelations among each other, and their relations to the surgical invariants from\nabove.\n