Unimodularity of Poincare polynomials of Lie algebras for semisimple singularities

TL;DR

This paper identifies classes of semisimple singularities whose associated Lie algebra Poincaré polynomials have roots on the unit circle, revealing specific spectral properties related to their algebraic structure.

Contribution

It characterizes classes of semisimple singularities with Poincaré polynomial roots on the unit circle, expanding understanding of their spectral properties.

Findings

Roots of Poincaré polynomials lie on the unit circle for a large class of singularities.

Some singularities have exactly four roots outside the unit circle.

The results relate algebraic properties to spectral distribution of roots.

Abstract

We single out a large class of semisimple singularities with the property that all roots of the Poincar\'e polynomial of the Lie algebra of derivations of the corresponding suitably (not necessarily quasihomogeneously) graded moduli algebra lie on the unit circle; for a still larger class there might occur exactly four roots outside the unit circle.