Non-integrable distributions with simple infinite-dimensional Lie (super)algebras of symmetries

TL;DR

This paper classifies non-integrable distributions with simple infinite-dimensional Lie superalgebras of symmetries over complex numbers and positive characteristic fields, identifying new algebraic structures and distributions.

Contribution

It provides a comprehensive classification of such distributions and their symmetry algebras, including new examples in positive characteristic fields.

Findings

15 series of symmetry algebras identified over c

7 exceptional Lie superalgebras classified

New distributions and algebraic structures found in characteristic 2 and 3

Abstract

Under usual locality assumptions, we classify all non-integrable distributions with simple infinite-dimensional Lie superalgebra of symmetries over $\mathbb{C}$: we single out 15 series (containing 2 analogs of contact series and one family of deformations of their divergence-free subalgebras), and 7 exceptional Lie superalgebras. Over algebraically closed fields~$\mathbb{K}$ of characteristic $p>0$, we classify the W-gradings (corresponding to a maximal subalgebra of finite codimension) of the known simple vectorial Lie (super)algebras with unconstrained shearing vector of heights of the indeterminates, distinguish W-gradings of (super)algebras preserving non-integrable distributions. For $p>3$, we get analogs of the result over $\mathbb{C}$. For $p=3$, of all possible W-gradings (12 of Skryabin algebras, 3 of superized Melikyan algebras, and 4 of Bouarroudj superalgebras) most are new, together with the corresponding distributions. For $p=2$, we also get several new examples of distributions and their Lie (super)algebras of symmetries.