Lucas Buzaglo

Building on our previous work, we classify group gradings of three-variable multi-parameter skew-polynomial rings using superpotential techniques. We also compute the invariant rings of the gradings by non-abelian groups, and determine when they are Artin-Schelter regular.

We generalize previous results on subalgebras of the one-sided Witt algebra to the setting of vector field Lie algebras on affine curves. In particular, we show that an infinite-dimensional Lie subalgebra of vector fields on an affine curve $C$ is isomorphic to a finite codimension subalgebra of the Lie algbebra of vector fields on some other affine curve $D$. The curve $D$ is canonically constructed from the subalgebra one starts with. As a consequence of this result, we prove that the enveloping algebra of any infinite-dimensional subalgebra of vector fields on a curve is never noetherian. Other applications of our main result are discussed.

We classify group gradings on three-variable skew-polynomial rings by way of the Manin universal coacting Hopf algebra. In particular, we classify the cocommutative Hopf quotients of the Manin universal coacting Hopf algebra of skew polynomial rings, yielding the classification of gradings as a corollary.

We commence the systematic study of the boundary conformal Carrollian algebra (BCCA), a Lie algebra first discovered in a physical context in this paper. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is an infinite-dimensional Lie algebra which is filtered but not graded. In this paper, we establish some structural Lie algebraic properties and construct and study Whittaker modules for this Lie algebra.

We prove that enveloping algebras of derivations of finitely generated associative algebras are not noetherian over a field of characteristic zero. This extends a result of Sierra and Walton on the Witt algebra, as well as a previous result of mine on Krichever-Novikov algebras. We highlight that the result applies to derivations of both commutative and noncommutative algebras without restriction on their growth.

Lie algebras formed via semi-direct sums of the Witt algebra and its modules have become increasingly prominent in both physics and mathematics in recent years. We compute central extensions, derivations, and automorphisms of semi-direct sums of the Witt algebra with its intermediate series modules, which are graded modules with one-dimensional graded components.

We study subalgebras of general Cartan type Lie algebras $\operatorname{Der}(\Bbbk[x_1,\dots,x_n])$ of GK-dimension $n$. For $n = 1$, we completely classify such subalgebras. As a consequence of the classification, we (re-)prove one case of a conjecture of Kaiming Zhao with links to the Jacobian conjecture. For arbitrary $n$, we prove that any such subalgebra has a non-noetherian enveloping algebra.

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.

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.

This thesis is devoted to studying infinite-dimensional Lie algebras of derivations, with a focus on the noetherianity of their universal enveloping algebras. We survey the current state of knowledge on the noetherianity enveloping algebras of infinite-dimensional Lie algebras and prove that various classes of Lie algebras of derivations have non-noetherian enveloping algebras. We finish by studying some Lie algebraic properties of subalgebras of the Witt algebra.

This is a short (roughly 2 pages) and self-contained proof of the non-noetherianity of the universal enveloping algebra of the Witt algebra. The proof consists of an explicit construction of an ideal which is not finitely generated as a left or right ideal, based on work of Sierra and Špenko.