Bistellar Cluster Algebras and Piecewise Linear Invariants
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911877094703104 |
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| author | Darby, Alastair Li, Fang Lu, Zhi |
| author_facet | Darby, Alastair Li, Fang Lu, Zhi |
| contents | Inspired by the ideas and techniques used in the study of cluster algebras we construct a new class of algebras, called bistellar cluster algebras, from closed oriented triangulated even-dimensional manifolds by performing middle-dimensional bistellar moves. This class of algebras exhibit the algebraic behaviour of middle-dimensional bistellar moves but do not satisfy the classical cluster algebra axiom: "every cluster variable in every cluster is exchangeable". Thus the construction of bistellar cluster algebras is quite different from that of a classical cluster algebra. Secondly, using bistellar cluster algebras and the techniques of combinatorial topology, we construct a direct system associated with a set of PL homeomorphic PL manifolds of dimension 2 or 4, and show that the limit of this direct system is a PL invariant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_09100 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bistellar Cluster Algebras and Piecewise Linear Invariants Darby, Alastair Li, Fang Lu, Zhi Algebraic Topology Commutative Algebra 57Q15, 13F60, 57Q25 Inspired by the ideas and techniques used in the study of cluster algebras we construct a new class of algebras, called bistellar cluster algebras, from closed oriented triangulated even-dimensional manifolds by performing middle-dimensional bistellar moves. This class of algebras exhibit the algebraic behaviour of middle-dimensional bistellar moves but do not satisfy the classical cluster algebra axiom: "every cluster variable in every cluster is exchangeable". Thus the construction of bistellar cluster algebras is quite different from that of a classical cluster algebra. Secondly, using bistellar cluster algebras and the techniques of combinatorial topology, we construct a direct system associated with a set of PL homeomorphic PL manifolds of dimension 2 or 4, and show that the limit of this direct system is a PL invariant. |
| title | Bistellar Cluster Algebras and Piecewise Linear Invariants |
| topic | Algebraic Topology Commutative Algebra 57Q15, 13F60, 57Q25 |
| url | https://arxiv.org/abs/2405.09100 |