Lie algebras of derivations and their universal enveloping algebras
Creators: name: Buzaglo, Lucas
Publisher: name: The University of Edinburgh
Year: 2024
Subjects: Lie algebra
infinite-dimensional
universal enveloping algebra
noetherian
derivation
Krichever-Novikov

description: Let k be a field of characteristic zero. In this thesis, we study Lie algebras of derivations and answer various Lie theoretic and ring theoretic questions about them. One of the main fundamental questions which motivates this thesis is whether it is possible for the universal enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. We show that the universal enveloping algebra of the Lie algebra of derivations of any infinite-dimensional affine commutative algebra is not noetherian. This extends a result of Sierra and Walton on the Witt algebra (the Lie algebra of derivations of the ring of Laurent polynomials). The most difficult case is when we have a Lie algebra of derivations on an integral domain of Krull dimension 1. Such a Lie algebra is known as a Krichever–Novikov algebra, and can be studied using techniques from algebraic geometry. We then focus on subalgebras of the general Cartan type Lie algebras (also known as Cartan type W Lie algebras), defined as Wₙ = Der(k[x₁, . . . , xₙ]). In particular, we study Lie subalgebras g of Wₙ of maximal Gelfand–Kirillov (GK) dimension, that is, with GKdim(g) = n. For n = 1, we completely classify such g. We first define a new family of subalgebras of W₁ (denoted by L(f), where f ∈ k[t]), and then prove that, up to a finite-dimensional vector space, all infinite-dimensional subalgebras of W₁ are of the form L(f) for some appropriate choice of f. As a corollary of the classification, we obtain a new proof that W₁ satisfies the Dixmier conjecture, in other words that End(W₁) \ {0} = Aut(W₁), a result first shown by Du. Our proof is more conceptual, following from the classification of subalgebras by showing that W₁ does not have any proper simple subalgebras. For arbitrary n, we show that if g is a GK-dimension n subalgebra of Wn, then U(g) is not (left or right) noetherian. For n = 1, this is a consequence of the classification of subalgebras. For n ≥ 2, our method is different: in this case the key is to introduce a chain of subalgebras gij of g and study the dimensions of successive quotients gᵢⱼ/gᵢ, ⱼ₊₁. We finish by analysing Lie theoretic properties of subalgebras of the Witt algebra W = Der(k[t, t⁻¹]) and the one-sided Witt algebra W₁ = Der(k[t]). We first study finite codimension subalgebras of W₁ by computing their derivations and one-dimensional extensions. These correspond to Ext¹ᵤ₍₉₎(M, g), where g is a subalgebra of W₁ and M is a one-dimensional representation of g. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W₁ extends to an automorphism of W₁. Finally, we explain the observed rigidity by defining a notion of “completely non-split extension” and proving that W1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W₁ as abstract Lie algebras, they remember that they are contained in W₁. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations.