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The Heisenberg double
of involutory Hopf algebras and invariants of closed
$3$–manifolds
Serban Matei Mihalache, Sakie Suzuki and Yuji Terashima |
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Algebraic & Geometric Topology 24 (2024) 3669–3691
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Abstract |
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We construct an invariant of closed oriented –manifolds using a finite-dimensional involutory unimodular and counimodular Hopf algebra . We use the framework of normal o–graphs introduced by R Benedetti and C Petronio, in which one can represent a branched ideal triangulation via an oriented virtual knot diagram. We assign a copy of the canonical element of the Heisenberg double of to each real crossing, which represents a branched ideal tetrahedron. The invariant takes values in the cyclic quotient , which is isomorphic to the base field. In the construction we use only the canonical element and structure constants of , and not any representations of . This, together with the finiteness and locality conditions of the moves for normal o–graphs, makes the calculation of our invariant rather simple and easy to understand. When is the group algebra of a finite group, the invariant counts the number of group homomorphisms from the fundamental group of the –manifold to the group. |
Keywords
invariants of $3$–manifolds, Hopf algebras, Heisenberg
doubles
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Mathematical Subject Classification
Primary: 57K31
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References |
Publication
Received: 4 May 2022
Revised: 4 December 2022
Accepted: 19 December 2022
Published: 9 December 2024
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Authors |
| Serban Matei Mihalache | |
| Department of Mathematics Tohoku University Sendai Japan |
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| Sakie Suzuki | |
| Department of Mathematical and
Computing Science Tokyo Institute of Technology Tokyo Japan |
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| Yuji Terashima | |
| Department of Mathematics Tohoku University Sendai Japan |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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