On elliptic Calogero–Moser systems for complex crystallographic reflection groups
To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G -invari...
| Main Authors: | , , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Elsevier
2015
|
| Online Access: | http://hdl.handle.net/1721.1/99442 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1826193665644560384 |
|---|---|
| author | Felder, Giovanni Ma, Xiaoguang Veselov, Alexander Etingof, Pavel I. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Felder, Giovanni Ma, Xiaoguang Veselov, Alexander Etingof, Pavel I. |
| author_sort | Felder, Giovanni |
| collection | MIT |
| description | To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G -invariant functions of pairs (T,s), where T is a hypertorus in X (of codimension 1), and s∈G is a reflection acting trivially on T. If G is a real reflection group, these families reduce to the known generalizations of elliptic Calogero–Moser systems, but in the non-real case they appear to be new. We give two constructions of the integrals of these systems – an explicit construction as limits of classical Calogero–Moser Hamiltonians of elliptic Dunkl operators as the dynamical parameter goes to 0 (implementing an idea of V. Buchstaber, G. Felder and A. Veselov (1994) [BFV]), and a geometric construction as global sections of sheaves of elliptic Cherednik algebras for the critical value of the twisting parameter. We also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions. |
| first_indexed | 2024-09-23T09:42:46Z |
| format | Article |
| id | mit-1721.1/99442 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:42:46Z |
| publishDate | 2015 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/994422022-09-26T13:18:13Z On elliptic Calogero–Moser systems for complex crystallographic reflection groups Felder, Giovanni Ma, Xiaoguang Veselov, Alexander Etingof, Pavel I. Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I. Ma, Xiaoguang To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G -invariant functions of pairs (T,s), where T is a hypertorus in X (of codimension 1), and s∈G is a reflection acting trivially on T. If G is a real reflection group, these families reduce to the known generalizations of elliptic Calogero–Moser systems, but in the non-real case they appear to be new. We give two constructions of the integrals of these systems – an explicit construction as limits of classical Calogero–Moser Hamiltonians of elliptic Dunkl operators as the dynamical parameter goes to 0 (implementing an idea of V. Buchstaber, G. Felder and A. Veselov (1994) [BFV]), and a geometric construction as global sections of sheaves of elliptic Cherednik algebras for the critical value of the twisting parameter. We also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions. National Science Foundation (U.S.) (Grant DMS-0504847) National Science Foundation (U.S.) (Grant DMS-0854764) 2015-10-23T18:24:41Z 2015-10-23T18:24:41Z 2010-04 2010-03 Article http://purl.org/eprint/type/JournalArticle 00218693 1090-266X http://hdl.handle.net/1721.1/99442 Etingof, Pavel, Giovanni Felder, Xiaoguang Ma, and Alexander Veselov. “On Elliptic Calogero–Moser Systems for Complex Crystallographic Reflection Groups.” Journal of Algebra 329, no. 1 (March 2011): 107–129. https://orcid.org/0000-0002-0710-1416 en_US http://dx.doi.org/10.1016/j.jalgebra.2010.04.011 Journal of Algebra Creative Commons Attribution-Noncommercial-NoDerivatives http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier Arxiv |
| spellingShingle | Felder, Giovanni Ma, Xiaoguang Veselov, Alexander Etingof, Pavel I. On elliptic Calogero–Moser systems for complex crystallographic reflection groups |
| title | On elliptic Calogero–Moser systems for complex crystallographic reflection groups |
| title_full | On elliptic Calogero–Moser systems for complex crystallographic reflection groups |
| title_fullStr | On elliptic Calogero–Moser systems for complex crystallographic reflection groups |
| title_full_unstemmed | On elliptic Calogero–Moser systems for complex crystallographic reflection groups |
| title_short | On elliptic Calogero–Moser systems for complex crystallographic reflection groups |
| title_sort | on elliptic calogero moser systems for complex crystallographic reflection groups |
| url | http://hdl.handle.net/1721.1/99442 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT feldergiovanni onellipticcalogeromosersystemsforcomplexcrystallographicreflectiongroups AT maxiaoguang onellipticcalogeromosersystemsforcomplexcrystallographicreflectiongroups AT veselovalexander onellipticcalogeromosersystemsforcomplexcrystallographicreflectiongroups AT etingofpaveli onellipticcalogeromosersystemsforcomplexcrystallographicreflectiongroups |