Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities
Original manuscript July 10 2009
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Walter de Gruyter
2013
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| Online Access: | http://hdl.handle.net/1721.1/79893 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1811097706267934720 |
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| author | Schedler, Travis Etingof, Pavel I. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Schedler, Travis Etingof, Pavel I. |
| author_sort | Schedler, Travis |
| collection | MIT |
| description | Original manuscript July 10 2009 |
| first_indexed | 2024-09-23T17:03:37Z |
| format | Article |
| id | mit-1721.1/79893 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T17:03:37Z |
| publishDate | 2013 |
| publisher | Walter de Gruyter |
| record_format | dspace |
| spelling | mit-1721.1/798932022-10-03T10:06:05Z Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities Schedler, Travis Etingof, Pavel I. Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I. Schedler, Travis Original manuscript July 10 2009 Let X ⊂ ℂ[superscript 3] be a surface with an isolated singularity at the origin, given by the equation Q(x, y, z) = 0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = ℂ[superscipt 2]/G for G < SL[subscript 2](ℂ) finite. Let Y ≔ S[superscript n]X be the n-th symmetric power of X. We compute the zeroth Poisson homology HP[subscript 0](𝒪[subscript Y]), as a graded vector space with respect to the weight grading, where 𝒪[subscript Y] is the ring of polynomial functions on Y. In the Kleinian case, this confirms a conjecture of Alev, that HP[subscript 0] (𝒪 [G [superscipt n]⋊ S[subscript n]over ℂ[2n]) ≃ HH [subscript 0] (Weyl (𝒪 [G [superscipt n]⋊ S[subscript n]over ℂ[2n]), where Weyl[subscript 2n] is the Weyl algebra on 2n generators. That is, the Brylinski spectral sequence degenerates in degree zero in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, A[subscript γ], for all but countably many parameters γ in the elliptic curve. As a consequence, we deduce a bound on the number of irreducible finite-dimensional representations of all quantizations of Y. This includes the noncommutative spherical symplectic reflection algebras associated to G[superscript n] ⋊ S[subscript n]. National Science Foundation (U.S.) (Grant DMS-0504847) American Institute of Mathematics (Fellowship) Massachusetts Institute of Technology. Undergraduate Research Opportunities Program 2013-08-21T15:55:23Z 2013-08-21T15:55:23Z 2012-06 2010-02 Article http://purl.org/eprint/type/JournalArticle 1435-5345 0075-4102 http://hdl.handle.net/1721.1/79893 Etingof, Pavel, and Travis Schedler. “Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities.” Journal für die reine und angewandte Mathematik (Crelles Journal) 2012, no. 667 (January 2012). https://orcid.org/0000-0002-0710-1416 en_US http://dx.doi.org/10.1515/crelle.2011.124 Journal fur die reine und angewandte Mathematik (Crelles Journal) Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Walter de Gruyter arXiv |
| spellingShingle | Schedler, Travis Etingof, Pavel I. Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities |
| title | Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities |
| title_full | Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities |
| title_fullStr | Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities |
| title_full_unstemmed | Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities |
| title_short | Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities |
| title_sort | zeroth poisson homology of symmetric powers of isolated quasihomogeneous surface singularities |
| url | http://hdl.handle.net/1721.1/79893 https://orcid.org/0000-0002-0710-1416 |
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