[Paper Review] New strong HKT manifolds arising from quaternionic representations

M. L. Barberis, Anna Fino|arXiv (Cornell University)|May 15, 2008

Geometric Analysis and Curvature Flows18 references9 citations

TL;DR

This paper presents a construction method to generate 8n-dimensional HKT Lie algebras from 4n-dimensional ones using quaternionic representations. The procedure preserves the strong HKT condition, enabling the discovery of new compact examples of strong HKT manifolds, thus expanding the known class of such geometric structures.

ABSTRACT

We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively weak) condition is preserved by our construction. This allows to find new compact examples of strong HKT manifolds.

Motivation & Objective

• To develop a systematic method for constructing higher-dimensional HKT Lie algebras from lower-dimensional ones.
• To preserve the strong HKT condition during the construction process.
• To generate new compact examples of strong HKT manifolds, which are rare and geometrically significant.
• To explore the role of quaternionic representations in extending HKT structures beyond the original dimension.

Proposed method

• The construction begins with a 4n-dimensional HKT Lie algebra equipped with a quaternionic representation.
• A new 8n-dimensional Lie algebra is defined using the semidirect product construction involving the original algebra and its quaternionic representation.
• The HKT structure is extended to the new algebra by lifting the complex structures and Hermitian metrics via the representation space.
• The integrability and compatibility conditions of the HKT structure are verified using the representation's quaternionic properties.
• The strong HKT condition is shown to be preserved under this construction by analyzing the associated fundamental forms.
• The resulting Lie algebra is shown to admit a compact quotient, yielding a compact strong HKT manifold.

Experimental results

Research questions

• RQ1 Can a 4n-dimensional HKT Lie algebra be used to construct a higher-dimensional HKT Lie algebra while preserving the strong HKT condition?
• RQ2 What role do quaternionic representations play in extending HKT structures to higher dimensions?
• RQ3 Are there new compact examples of strong HKT manifolds that can be constructed via this method?
• RQ4 How does the dimension doubling via quaternionic representation affect the geometric and algebraic properties of the HKT structure?

Key findings

• The construction yields an 8n-dimensional HKT Lie algebra from a 4n-dimensional one using a quaternionic representation.
• The strong HKT condition is preserved under the proposed construction, ensuring the resulting structure remains strongly HKT.
• The method enables the explicit construction of new compact examples of strong HKT manifolds.
• The resulting HKT structures are compatible with the quaternionic action on the representation space.
• The procedure generalizes known constructions and provides a systematic way to generate higher-dimensional strong HKT manifolds.
• The compactness of the resulting manifolds is established via quotient construction on the Lie algebra level.