4-Manifold Invariants From Hopf Algebras
| Autor: | Julian Chaidez, Jordan Cotler, Shawn X Cui |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Geometric Topology
Mathematics::Quantum Algebra Mathematics::Rings and Algebras Mathematics - Quantum Algebra FOS: Mathematics Quantum Algebra (math.QA) FOS: Physical sciences Geometric Topology (math.GT) Geometry and Topology Mathematical Physics (math-ph) Mathematics::Geometric Topology Mathematical Physics |
| Popis: | The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants. 59 pages, many figures and diagrams; v3 to appear in Algebraic and Geometric Topology |
| Databáze: | OpenAIRE |
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