Creating quantum projective spaces by deforming q-symmetric algebras
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915022359232512 |
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| author | Matviichuk, Mykola Pym, Brent Schedler, Travis |
| author_facet | Matviichuk, Mykola Pym, Brent Schedler, Travis |
| contents | We construct a large collection of "quantum projective spaces", in the form of Koszul, Calabi-Yau algebras with the Hilbert series of a polynomial ring. We do so by starting with the toric ones (the q-symmetric algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. We then prove that these algebras are identified with the canonical quantizations of corresponding families of quadratic Poisson structures, in the sense of Kontsevich. In this way, we obtain the first broad class of quadratic Poisson structures for which his quantization can be computed explicitly, and shown to converge, as he conjectured in 2001. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10425 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Creating quantum projective spaces by deforming q-symmetric algebras Matviichuk, Mykola Pym, Brent Schedler, Travis Quantum Algebra Algebraic Geometry Rings and Algebras Symplectic Geometry We construct a large collection of "quantum projective spaces", in the form of Koszul, Calabi-Yau algebras with the Hilbert series of a polynomial ring. We do so by starting with the toric ones (the q-symmetric algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. We then prove that these algebras are identified with the canonical quantizations of corresponding families of quadratic Poisson structures, in the sense of Kontsevich. In this way, we obtain the first broad class of quadratic Poisson structures for which his quantization can be computed explicitly, and shown to converge, as he conjectured in 2001. |
| title | Creating quantum projective spaces by deforming q-symmetric algebras |
| topic | Quantum Algebra Algebraic Geometry Rings and Algebras Symplectic Geometry |
| url | https://arxiv.org/abs/2411.10425 |