The Heisenberg double of involutory Hopf algebras and invariants of closed $3$-manifolds
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arXiv
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| Format: | Preprint |
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2021
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| author | Mihalache, Serban Matei Suzuki, Sakie Terashima, Yuji |
| author_facet | Mihalache, Serban Matei Suzuki, Sakie Terashima, Yuji |
| contents | We construct an invariant of closed oriented $3$-manifolds using a finite dimensional, involutory, unimodular and counimodular Hopf algebra $H$. 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 a canonical element of the Heisenberg double $\mathcal{H}(H)$ of $H$ to each real crossing, which represents a branched ideal tetrahedron. The invariant takes values in the cyclic quotient $\mathcal{H}(H)/{[\mathcal{H}(H),\mathcal{H}(H)]}$, which is isomorphic to the base field. In the construction we use only the canonical element and structure constants of $H$ and we do not use any representations of $H$. 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 $H$ is the group algebra of a finite group, the invariant counts the number of group homomorphisms from the fundamental group of the $3$-manifold to the group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_03037 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Heisenberg double of involutory Hopf algebras and invariants of closed $3$-manifolds Mihalache, Serban Matei Suzuki, Sakie Terashima, Yuji Geometric Topology 57K31, 16T05 We construct an invariant of closed oriented $3$-manifolds using a finite dimensional, involutory, unimodular and counimodular Hopf algebra $H$. 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 a canonical element of the Heisenberg double $\mathcal{H}(H)$ of $H$ to each real crossing, which represents a branched ideal tetrahedron. The invariant takes values in the cyclic quotient $\mathcal{H}(H)/{[\mathcal{H}(H),\mathcal{H}(H)]}$, which is isomorphic to the base field. In the construction we use only the canonical element and structure constants of $H$ and we do not use any representations of $H$. 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 $H$ is the group algebra of a finite group, the invariant counts the number of group homomorphisms from the fundamental group of the $3$-manifold to the group. |
| title | The Heisenberg double of involutory Hopf algebras and invariants of closed $3$-manifolds |
| topic | Geometric Topology 57K31, 16T05 |
| url | https://arxiv.org/abs/2104.03037 |