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Research Statement Garrett Johnson
 

Summary: Research Statement
Garrett Johnson
My research focuses mainly on quantum groups and noncommutative ring theory. More specifically, I
study the structure and representation theory of the noncommutative algebras arising in these areas, such
as quantized universal enveloping algebras and quantized coordinate rings. In this research statement I will
summarize my past and current research in these areas as well as state some open problems I will pursue in
future research.
1 Double Affine Hecke Algebras and R-Matrices
For a vector space V over a field k and a linear operator R End(V V ), we define R12 := R1 End(V 3
)
and R23 := 1 R End(V 3
). The solutions to the braided quantum Yang-Baxter (QYB) equation
R12R23R12 = R23R12R23 are called R-matrices and first gained importance in physical problems, more
specifically in the quantum inverse scattering method, and have since played a large role in quantum groups,
topology, representation theory, and statistical mechanics.
In [9], I give a new interpretation of the so-called Cremmer-Gervais R-matrices in terms of the representa-
tion theory of the double affine Hecke algebras (DAHAs). The DAHAs are algebras related to combinatorics
and were introduced by Cherednik [3] in the early 1990's. For a reductive algebraic group G defined over a
field k, the DAHA HHG
q,t associated to G is, informally, a deformed version of the group algebra k[W (P P)],

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics