2026-09-19T01:28:22Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/677902022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Pavel Etingof.
Jordan, David Andrew
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2011-12-19T18:51:45Z
2011-12-19T18:51:45Z
2011
2011
http://hdl.handle.net/1721.1/67790
767740932
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 109-112).
In this thesis, a new class of algebras called quantized multiplicative quiver varieties A (Q), is constructed, depending upon a quiver Q, its dimension vector d, and a certain "moment map" parameter . The algebras Ad(Q) are obtained via quantum Hamiltonian reduction of another algebra D,(Matd(Q)) relative to a quantum moment map pq, both of which are also constructed herein. The algebras Dq(Matd(Q)) and A (Q) bear relations to many constructions in representation theory, some of which are spelled out herein, and many more whose precise formulation remains conjectural. When Q consists of a single vertex of dimension N with a single loop, the algebra Dq(MatA(Q)) is isomorphic to the algebra of quantum differential operators on G = GLN. In this case, for any n E Z>o, we construct a functor from the category of Dq-modules to representations of the type A double affine Hecke algebra of rank n. This functor is an instance of a more general construction which may be applied to any quasi-triangular Hopf algebra H, and yields representations of the elliptic braid group of rank n.
by David Andrew Jordan.
Ph.D.
112 p.
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Quantized multiplicative quiver varieties and actions of higher genus braid groups
Quantized multiplicative quiver varieties
Thesis
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false
Publication
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<Abstract>In this thesis, a new class of algebras called quantized multiplicative quiver varieties A (Q), is constructed, depending upon a quiver Q, its dimension vector d, and a certain "moment map" parameter . The algebras Ad(Q) are obtained via quantum Hamiltonian reduction of another algebra D,(Matd(Q)) relative to a quantum moment map pq, both of which are also constructed herein. The algebras Dq(Matd(Q)) and A (Q) bear relations to many constructions in representation theory, some of which are spelled out herein, and many more whose precise formulation remains conjectural. When Q consists of a single vertex of dimension N with a single loop, the algebra Dq(MatA(Q)) is isomorphic to the algebra of quantum differential operators on G = GLN. In this case, for any n E Z>o, we construct a functor from the category of Dq-modules to representations of the type A double affine Hecke algebra of rank n. This functor is an instance of a more general construction which may be applied to any quasi-triangular Hopf algebra H, and yields representations of the elliptic braid group of rank n.</Abstract>
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