The main objects of interest in my research are universal enveloping algebras of infinite-dimensional Lie algebras. These are rather mysterious rings which we have only begun to understand in the last ten years. One of the most important open questions about these is the question of noetherianity: it is widely believed that these rings are never (left or right) noetherian, but there are very few examples whose (non-)noetherianity is known. This is a long-standing question which first appeared in print nearly fifty years ago in Amayo and Stewart's book on infinite-dimensional Lie algebras. Currently, there are only two general classes of Lie algebras for which the non-noetherianity of their enveloping algebras has been established: graded simple Lie algebras of polynomial growth, and Lie algebras of derivations on affine varieties. The former was proved by Susan Sierra and Chelsea Walton, while the latter is my own work.
I am also interested in Lie theoretic properties of infinite-dimensional Lie algebras. This includes, among other things, representation theory, derivations, automorphisms, and cohomology.
Most of my research is focused on Lie algebras related to the one-sided Witt algebra. This is the Lie algebra of derivations of the polynomial ring in one variable (equivalently, the Lie algebra of vector fields on the affine line).
My MathSciNet profile can be found here.
Work in progress
|Low-dimensional cohomology of semi-direct sums of the Witt algebra with its intermediate series modules, with Girish Vishwa
|We compute some cohomology groups of semi-direct sums of the Witt algebra with its intermediate series modules. These are graded representations such that each graded piece is one-dimensional.
|Maximal dimensional subalgebras of general Cartan type Lie algebras, with Jason Bell
|Submitted to the Bulletin of the London Mathematical Society.
|We study subalgebras of general Cartan type Lie algebras Der(k[x1,...,xn]) of GK-dimension n. For n = 1, we completely classify such subalgebras. For arbitrary n, we prove that any such subalgebra has a non-noetherian enveloping algebra.
|Derivations, extensions, and rigidity of subalgebras of the Witt algebra, Journal of Algebra (2024)
|We study Lie algebraic properties of subalgebras of the Witt algebra and the one-sided Witt algebra: we compute derivations, one-dimensional extensions, and automorphisms of these subalgebras. In particular, all these properties are inherited from the full Witt algebra (e.g. derivations of subalgebras are simply restrictions of derivations of the Witt algebra). We also prove that any isomorphism between subalgebras of finite codimension extends to an automorphism of the Witt algebra. We explain this "rigid" behavior by proving a universal property satisfied by the Witt algebra as a completely non-split extension of any of its subalgebras of finite codimension. This is a purely Lie algebraic property which we introduce in the paper.
|Enveloping algebras of Krichever-Novikov algebras are not noetherian, Algebras and Representation Theory (2022)
|This work is part of the overarching question of whether it is possible for the universal enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. The main result of this paper is that the universal enveloping algebra of any Krichever-Novikov algebra is not noetherian. A Krichever-Novikov algebra is the Lie algebra of derivations of an affine curve. The second part of the paper focuses on Lie subalgebras of the one-sided Witt algebra: we construct new families of subalgebras which were previously unknown and make significant progress in the classification of subalgebras.