The classification of simple complex Lie superalgebras of polynomial vector fields and their deformations

TL;DR

This paper reviews the classification of simple infinite-dimensional complex Lie superalgebras of polynomial growth, corrects previous results, and introduces new classifications and conjectures about their deformations and rigidity.

Contribution

It classifies simple W-graded and W-filtered vectorial Lie superalgebras, corrects prior classifications, and explores deformation parameters and rigidity conjectures.

Findings

Classified simple W-graded and W-filtered vectorial Lie superalgebras.

Corrected several previously published classification results.

Described the supervariety of deformation parameters and conjectured rigidity of exceptional superalgebras.

Abstract

We overview classifications of simple infinite-dimensional complex $\mathbb{Z}$-graded Lie (super)algebras of polynomial growth, and their deformations. A subset of such Lie (super)algebras consist of vectorial Lie (super)algebras whose elements are vector fields with polynomial, or formal power series, or divided power coefficients. A given vectorial Lie (super)algebra with a (Weisfeiler) filtration corresponding to a maximal subalgebra of finite codimension is called W-filtered; the associated graded algebra is called W-graded. Here, we correct several published results: (1) prove our old claim "the superization of \'E. Cartan's problem (classify primitive Lie algebras) is wild", (2) solve a tame problem: classify simple W-graded and W-filtered vectorial Lie superalgebras, (3) describe the supervariety of deformation parameters for the serial W-graded simple vectorial superalgebras, (4) conjecture that the exceptional simple vectorial superalgebras are rigid. We conjecture usefulness of our method in classification of simple infinite-dimensional vectorial Lie (super)algebras over fields of positive characteristic.