| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 10, 2017 |
| Latest Amendment Date: | May 22, 2019 |
| Award Number: | 1702254 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
James Matthew Douglass
mdouglas@nsf.gov (703)292-2467 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | June 1, 2017 |
| End Date: | May 31, 2020 (Estimated) |
| Total Intended Award Amount: | $318,090.00 |
| Total Awarded Amount to Date: | $318,090.00 |
| Funds Obligated to Date: |
FY 2018 = $125,823.00 FY 2019 = $73,254.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
1001 EMMET ST N CHARLOTTESVILLE VA US 22903-4833 (434)924-4270 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
PO Box 400137 Charlottesville VA US 22904-4137 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ALGEBRA,NUMBER THEORY,AND COM |
| Primary Program Source: |
01001819DB NSF RESEARCH & RELATED ACTIVIT 01001920DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The symmetries of a snowflake or a baseball are described by the notion of a group. Groups also describe more abstract symmetries, including supersymmetry in theoretical particle physics. Many important symmetries are described by continuous groups known as Lie groups, after mathematician Sophus Lie; these groups are generated by associated Lie algebras. Quantum groups, which are deformations of such symmetries, also serve as a shadow of higher structures. This research project aims to further develop a new approach, via so-called i-quantum groups, to representations of Lie algebras and Lie superalgebras. This new approach aims to uncover the underlying geometric and higher categorical structures of i-quantum groups. Results are expected also to have applications to knot theory.
Because of recently discovered connections to geometry of flag varieties, canonical bases, and categorification, i-quantum groups, which are co-ideal subalgebras of Drinfeld-Jimbo quantum groups, have been shown to play an increasingly important role in the theory of quantum groups and representations of Lie algebras and Lie superalgebras. In this project the investigator plans to develop the theory of canonical bases and categorical actions of i-quantum groups and their modules in the Kac-Moody setting. In particular, a categorification of affine -quantum groups will be formulated and applied to study the modular representation theory of quantum (super)groups of classical type at roots of unity and classical algebraic groups in prime characteristic. Character formulae in Bernstein-Gelfand-Gelfand category for exceptional Lie superalgebras will also be formulated.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Symmetries and groups are mathematical ways of describing the regularity and beauty of objects in the physical world, such as baseballs and snowflakes. More subtle yet no less intrinsic symmetries are formulated using the notion of quantum groups and variants.
The PI develops a program of generalizing various fundamental constructions from quantum groups (due to Drinfeld, Jimbo, Lusztig, Ringel, Bridgeland, and many others) to i-quantum groups arising from quantum symmetric pairs. He constructed canonical bases on i-quantum groups of Kac-Moody type and their tensor product modules. The new canoical basis arising from i-quantum groups of type AIII has a major application to the character formulas for Lie superalgebras of classical type. (A different approach to the character formulas for Lie superalgebras of exceptional type has also been developed.) The simplest canonical basis elements arising this way are known as i-divided powers, and they form a building blocks for i-quantum groups. The i-divided powers have been used in the formulation of a Serre presentation of i-quantum groups. Multi-parameter Schur dualities of finite and affine type between Hecke algebra of classical type and i-quantum groups are obtained, generalizing the Schur-Jimbo duality. A Hall algebra realization of i-quantum groups and braid group symmetries has been initiated, and this generalizes Bridgeland's Hall algebra construction of a whole quantum group. A geometric version of this Hall algebra construction yields a dual canonical basis of i-quantum groups with positivity.
Last Modified: 06/04/2020
Modified by: Weiqiang Wang
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