Applications of the quaternionic Jordan form to hypercomplex geometry

TL;DR

This paper uses the quaternionic Jordan form to classify hypercomplex nilpotent almost abelian Lie algebras and constructs new hypercomplex solvmanifolds with specific properties.

Contribution

It provides a complete classification of 12-dimensional hypercomplex almost abelian Lie algebras and constructs infinitely many hypercomplex solvmanifolds from integer polynomials.

Findings

Classified all 12-dimensional hypercomplex almost abelian Lie algebras.

Determined which 12-dimensional hypercomplex Lie groups admit lattices.

Constructed infinitely many hypercomplex solvmanifolds from polynomial families.

Abstract

We apply the quaternionic Jordan form to classify the hypercomplex nilpotent almost abelian Lie algebras in all dimensions and to carry out the complete classification of 12-dimensional hypercomplex almost abelian Lie algebras. Moreover, we determine which 12-dimensional simply connected hypercomplex almost abelian Lie groups admit lattices. Finally, for each integer $n>1$ we construct infinitely many, up to diffeomorphism, $(4n+4)$-dimensional hypercomplex almost abelian solvmanifolds which are completely solvable. These solvmanifolds arise from a distinguished family of monic integer polynomials of degree $n$.