$q$-deformation of Aomoto complex
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917669509267456 |
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| author | Yoshinaga, Masahiko |
| author_facet | Yoshinaga, Masahiko |
| contents | A degree one element of the Orlik-Solomon algebra of a hyperplane arrangement defines a cochain complex known as the Aomoto complex. The Aomoto complex can be considerd as the ``linear approximation'' of the twisted cochain complex with coefficients in a complex rank one local system.
In this paper, we discuss $q$-deformations of the Aomoto complex. The $q$-deformation is defined by replacing the entries of representation matrices of the coboundary maps with their $q$-analogues. While the resulting maps do not generally define cochain complexes, for certain special basis derived from real structures, the $q$-deformation becomes again a cochain complex. Moreover, it exhibits universality in the sense that any specialization of $q$ to a complex number yields the cochain complex computing the corresponding local system cohomology group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00810 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $q$-deformation of Aomoto complex Yoshinaga, Masahiko Geometric Topology Combinatorics Quantum Algebra 52C35, 32S22, 55N25 A degree one element of the Orlik-Solomon algebra of a hyperplane arrangement defines a cochain complex known as the Aomoto complex. The Aomoto complex can be considerd as the ``linear approximation'' of the twisted cochain complex with coefficients in a complex rank one local system. In this paper, we discuss $q$-deformations of the Aomoto complex. The $q$-deformation is defined by replacing the entries of representation matrices of the coboundary maps with their $q$-analogues. While the resulting maps do not generally define cochain complexes, for certain special basis derived from real structures, the $q$-deformation becomes again a cochain complex. Moreover, it exhibits universality in the sense that any specialization of $q$ to a complex number yields the cochain complex computing the corresponding local system cohomology group. |
| title | $q$-deformation of Aomoto complex |
| topic | Geometric Topology Combinatorics Quantum Algebra 52C35, 32S22, 55N25 |
| url | https://arxiv.org/abs/2401.00810 |