 
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 RMatrices
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 YangBaxter (QYB) equation
R12R23R12 = R23R12R23 are called Rmatrices 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 socalled CremmerGervais Rmatrices 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)],
