Special symplectic Lie groups and hypersymplectic Lie groups

Details Special symplectic Lie groups and hypersymplectic Lie groups Special symplectic Lie groups and hypersymplectic Lie groups

2010-10-15

arxiv.org/abs/1010.3160v1

Manuscripta Math. 133 (2010) 373-408

Physics

Mathematical Physics

32 pages

Scientific paper

10.1007/s00229-010-0375-z

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.

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