Applications of the quaternionic Jordan form to hypercomplex geometry

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 $n$.

Final version, to appear in J. Algebra

References in corpus (12)

• Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures
• Hyperkähler torsion structures invariant by nilpotent Lie groups
• Generalized Kähler almost abelian Lie groups
• Holonomy of the Obata connection on SU(3)
• Existence of HKT metrics on hypercomplex manifolds of real dimension 8
• Explicit harmonic and spectral analysis in Bianchi I-VII type cosmologies
• Hypercomplex almost abelian solvmanifolds
• Subvarieties of hypercomplex manifolds with holonomy in SL(n,H)
• The structure of almost Abelian Lie algebras
• Almost Abelian Lie groups, subgroups and quotients
• Locally conformally product structures on solvmanifolds
• Harmonic $G_2$-structures on almost Abelian Lie groups