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Cohomology and Support Theory for Quantum Groups Nakano, Daniel
Description
Quantum groups are a fertile area for explicit computations of cohomology and support varieties because of the availability of geometric methods involving complex algebraic geometry. Ginzburg and Kumar have shown that for l>h (l is order of the root of unity and h is the Coxeter number), the cohomology ring identifies with the coordinate algebra of the nilpotent cone of the underlying Lie algebra g=Lie(G). Bendel, Pillen, Parshall and the speaker have determined the cohomology ring when l is less than or equal to h and have shown that in most cases this identifies with the coordinate algebra of a G-invariant irreducible subvariety of the nilpotent cone. The latter computation employs vanishing results on partial flag variety G/P via the Grauert-Riemenschneider theorem. Support varieties have been determined for tilting modules (by Bezrukavinov), induced/Weyl modules (by Ostrik and Bendel-Nakano-Pillen-Parshall), and simple modules (by Drupieski-Nakano-Parshall). The calculations for tilting modules and simple modules employed the deep fact that the Lusztig Character Formula holds for quantum groups when l>h. In this talk, I will survey several of the main results of the topic and indicate the combinatorial and geometric techniques necessary to make such calculations. Open problems will also be presented.
Item Metadata
Title |
Cohomology and Support Theory for Quantum Groups
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2016-02-08T10:30
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Description |
Quantum groups are a fertile area for explicit computations of cohomology and support varieties
because of the availability of geometric methods involving complex algebraic geometry.
Ginzburg and Kumar have shown that for l>h (l is order of the root of unity and h is the Coxeter number), the cohomology ring
identifies with the coordinate algebra of the nilpotent cone of the underlying Lie algebra g=Lie(G). Bendel, Pillen, Parshall
and the speaker have determined the cohomology ring when l is less than or equal to h and have shown that in most cases this identifies with the coordinate algebra of a G-invariant irreducible subvariety of the nilpotent cone. The latter computation employs vanishing results on partial
flag variety G/P via the Grauert-Riemenschneider theorem.
Support varieties have been determined for tilting modules (by Bezrukavinov), induced/Weyl modules (by Ostrik and Bendel-Nakano-Pillen-Parshall),
and simple modules (by Drupieski-Nakano-Parshall). The calculations for tilting modules and simple modules employed the deep fact that the Lusztig Character Formula holds for quantum groups when l>h.
In this talk, I will survey several of the main results of the topic and indicate the combinatorial and geometric techniques necessary to make such
calculations. Open problems will also be presented.
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Extent |
49 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Georgia
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Series | |
Date Available |
2016-08-09
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0307419
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International