Show simple item record

dc.contributor.advisorPavel Etingof.en_US
dc.contributor.authorTsymbaliuk, Oleksandren_US
dc.contributor.otherMassachusetts Institute of Technology. Department of Mathematics.en_US
dc.date.accessioned2014-09-19T21:45:11Z
dc.date.available2014-09-19T21:45:11Z
dc.date.copyright2014en_US
dc.date.issued2014en_US
dc.identifier.urihttp://hdl.handle.net/1721.1/90192
dc.descriptionThesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.en_US
dc.descriptionCataloged from PDF version of thesis.en_US
dc.descriptionIncludes bibliographical references (pages 183-186).en_US
dc.description.abstractIn the first part of this thesis, we obtain some new results about infinitesimal Cherednik algebras. They have been introduced by Etingof-Gan-Ginzburg in [EGG] as appropriate analogues of the classical Cherednik algebras, corresponding to the reductive groups, rather than the finite ones. Our main result is the realization of those algebras as particular finite W-algebras of associated semisimple Lie algebras with nilpotent 1-block elements. To achieve this, we prove its Poisson counterpart first, which identifies the Poisson infinitesimal Cherednik algebras introduced in [DT] with the Poisson algebras of regular functions on the corresponding Slodowy slices. As a consequence, we obtain some new results about those algebras. We also generalize the classification results of [EGG] from the cases GL, and SP2n to SOl. In the second part of the thesis, we discuss the loop realization of the affine Yangian of gl₁. Similar objects were recently considered in the work of Maulik-Okounkov on the quantum cohomology theory, see [MO]. We present a purely algebraic realization of these algebras by generators and relations. We discuss some families of their representations. A similarity with the representation theory of the quantum toroidal algebra of gl₁ is explained by adapting a recent result of Gautam-Toledano Laredo, see [GTL], to the local setting. We also discuss some aspects of those two algebras such as the degeneration isomorphism, a shuffle presentation, and a geometric construction of the Whittaker vectors.en_US
dc.description.statementofresponsibilityby Oleksandr Tsymbaliuk.en_US
dc.format.extent186 pagesen_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsM.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.en_US
dc.rights.urihttp://dspace.mit.edu/handle/1721.1/7582en_US
dc.subjectMathematics.en_US
dc.titleThe affine Yangian of gl₁, and the infinitesimal Cherednik algebrasen_US
dc.typeThesisen_US
dc.description.degreePh. D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematics
dc.identifier.oclc890211772en_US


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record