Root Systems In Finite Symplectic Vector Spaces

TL;DR

This paper classifies symplectic root systems over the finite field F_2, showing that each Dynkin diagram has a unique minimal system and exploring their structure and applications in classifying Nichols algebras.

Contribution

It provides a complete classification of symplectic root systems over F_2, including their minimal forms and quotient relations, with explicit constructions for ADE types.

Findings

Every Dynkin diagram admits a unique minimal symplectic root system over F_2.

Non-minimal systems are quotients of minimal ones by a universal property.

Explicit constructions of symplectic root systems for ADE types.

Abstract

We study subsets in possibly degenerate symplectic vector spaces over finite fields, which are stable under a given Coxeter/Weyl reflection group. These symplectic root systems provide crucial combinatorical data to classify finite-dimensional Nichols algebras for nilpotent groups G over the complex numbers [Len13a], where the symplectic form is given by the group's commutator map. For example, the degree of degeneracy of the symplectic root system determines the size of the center of G. In this article we classify symplectic root systems over the finite field F_2, where symplectic just means isotropic. We prove that every Dynkin diagram admits, up to symplectic isomorphisms, a unique minimal symplectic root system over F_2 and thus requires a specific degree of degeneracy of the symplectic vector space. Any non-minimal symplectic root system turns out to be a quotient of a minimal one by a universal property. As examples and for further applications we explicitly construct all symplectic root systems for Cartan matrices resp. Dynkin diagrams of type ADE.