Special symplectic Lie groups and hypersymplectic Lie groups

Abstract

A special symplectic Lie group is a triple $(G,\omega ,\nabla )$ such that $G$ is a finite-dimensional real Lie group and $\omega$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $\nabla$. In this paper starting from a special symplectic Lie group we show how to ``deform" the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure $\nabla$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.

32 pages

References in corpus (6)

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• Special complex manifolds
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Cited by in corpus (3)

• Symmetric and skew-symmetric complex structures
• Flat affine symplectic Lie groups
• On $k$-para-Kähler Lie algebras a subclass of $k$-symplectic Lie algebras