Semisimple algebras of vector fields on C^N of maximal rank

TL;DR

This work provides a local classification of finite dimensional complex semisimple Lie algebras of analytic vector fields on $\mathbb{C}^N$ that admit a Cartan subalgebra of dimension $N$. The authors leverage the local canonical form from ABM, together with the fact that complex Cartan subalgebras are split, to show that the simple factors must be of type $A_{\ell}$ and that their ranks sum to $N$, i.e., $\mathcal{G} \cong \bigoplus_i \mathfrak{sl}(\ell_i+1,\mathbb{C})$ with $\sum_i \ell_i = N$. The argument hinges on expressing root spaces as exponential-linear vector fields and ruling out non-$A_\ell$ types by embedding obstructions into low-dimensional spaces, while using ABM’s corollaries to characterize faithful representations of $A_k$ at maximal rank. An application to Levi decompositions shows that on $\mathbb{C}^2$ any nontrivial Levi decomposition forces the Levi part to be $\mathfrak{sl}(2,\mathbb{C})$, reducing the classification on $\mathbb{C}^3$ and for algebras with radicals to low-rank representations. Overall, the results connect analytic local structure with classical root-system classifications and align with Vinberg-type principles for algebraic groups.

Abstract

A classification of semisimple algebras of vector fields on C^N that have a Cartan subalgebra of dimension N is given. The proof uses basic representation theory and the local canonical form of semisimple Lie algebras of vector fields.

Theorems & Definitions (15)

• Theorem 1.1
• Lemma 2.1
• proof
• Definition 2.2
• Theorem 2.3
• Proposition 2.4
• proof
• Lemma 2.5
• proof
• Proposition 2.6