My research interests lie primarily in the field of non-commutative algebra.
I'm particularly interested in Hopf algebras of finite GK-dimension, quantum groups,
quantum homogeneous spaces and Poisson algebraic groups.
My published research papers in this field are listed below.
We study algebraic and homological properties of two classes of infinite
dimensional Hopf algebras over an algebraically closed field k of characteristic zero.
The first class consists of those Hopf k-algebras that are connected graded as
algebras, and the second class are those Hopf k-algebras that are connected as
coalgebras. For many but not all of the results presented here, the Hopf algebras
are assumed to have finite Gel'fand-Kirillov dimension. It is shown that if the
Hopf algebra H is a connected graded algebra of finite Gel'fand-Kirillov
dimension n, then H is a noetherian domain which is Cohen-Macaulay,
Artin-Schelter regular and Auslander regular of global dimension n.
It has S^2 = Id_H, and is Calabi-Yau. Detailed information is also provided
about the Hilbert series of H. Our results leave open the possibility that
the first class of algebras is (properly) contained in the second. For this
second class, the Hopf k-algebras of finite Gel'fand-Kirillov dimension n
with connected coalgebra, the underlying coalgebra is shown to be
Artin-Schelter regular of global dimension n. Both these classes of Hopf
algebra share many features in common with enveloping algebras of finite
dimensional Lie algebras. For example, an algebra in either of these classes
satisfies a polynomial identity only if it is a commutative polynomial
algebra. Nevertheless, we construct, as one of our main results, an
example of a Hopf k-algebra H of Gel'fand-Kirillov dimension 5, which is
connected graded as an algebra and connected as a coalgebra, but is not
isomorphic as an algebra to U(g) for any Lie algebra g.
Let k be a field and let H denote a pointed Hopf k-algebra with
antipode S. We are interested in determining the order of S. Building on the
work done by Taft and Wilson , we define an invariant for H, denoted mH,
and prove that the value of this invariant is connected to the order of S.
In the case where chark=0, it is shown that if S has finite order then it is
either the identity or has order 2mH. If in addition H is assumed to be
coradically graded, it is shown that the order of S is finite if and only
if mH is finite. We also consider the case where chark=p>0, generalising
the results of  to the infinite-dimensional setting.
Let H be a connected Hopf k-algebra of finite Gel'fand–Kirillov dimension
over an algebraically closed field k of characteristic 0. The objects of
study in this paper are the left or right coideal subalgebras T of H.
They are shown to be deformations of commutative polynomial k-algebras.
A number of well-known homological and other properties follow immediately
from this fact. Further properties are described, examples are considered,
invariants are constructed and a number of open questions are
This is a brief survey of some recent developments in the study of infinite dimensional
Hopf algebras which are either noetherian or have finite Gelfand-Kirillov dimension. A
number of open questions are listed.