Vyjayanthi ChariDistinguished Professor of MathematicsMathematics Dept vyjayanc@ucr.edu(951) 827-6463
Quantum Affine Algebras: BGG reciprocity, Macdonald Polynomials, Schur Positivity
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AWARD NUMBER
006440-002
FUND NUMBER
21202
STATUS
Closed
AWARD TYPE
3-Grant
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AWARD EXECUTION DATE
9/5/2013
BEGIN DATE
9/15/2013
END DATE
8/31/2016
AWARD AMOUNT
$155,999
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Sponsor Information
SPONSOR AWARD NUMBER
SPONSOR
SPONSOR TYPE
FUNCTION
Organized Research
PROGRAM NAME
Proposal Information
PROPOSAL NUMBER
13040295
PROPOSAL TYPE
New
ACTIVITY TYPE
Basic Research
PI Information
PI
Chari, Vyjayanthi
PI TITLE
Other
PI DEPTARTMENT
Mathematics
PI COLLEGE/SCHOOL
College of Nat & Agr Sciences
CO PIs
Project Information
ABSTRACT
The proposal is on the interplay between the representation theory of affine algebras, its standard maximal parabolic subalgebra, namely the current algebra, and the quantum affine algebra associated to a simple Lie algebra. It focuses on the study of families of infinite-dimensional and level zero representations for each of these algebras and develops connections with Macdonald polynomials, Demazure characters and Schur positivity. One of the goals is to establish a Bernstein-Gelfand-Gelfand type duality principle for these categories and to investigate the combinatorial and homological consequences of having such a duality. Another goal of this proposal is to develop a connection between the homological properties of the category of infinite-dimensional representations of the quantum affine algebra and the tensor structure of this category. The existence of such a connection is unexpected and somewhat mysterious and is suggested by some recent work of the PI. It exists only at the quantum level and a deeper understanding of this should have a substantial impact on the study of this category. It should also also yield connections with recent work of others on cluster algebras and categorification.
The study of affine Lie algebras and their quantum analogs have long had remarkable connections to a number of different fields including string theory, conformal field theory, topological field theory, infinite dimensional geometry and mathematical physics. Many of the themes of the project are motivated by questions arising in solvable lattice models. Representations of affine Lie algebras and the standard maximal parabolic subalgebras will capture important physical information. The project will also provide a representation theoretic framework in which to understand various combinatorial problems.(Abstract from NSF)
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