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.Comment: 32 page

On para-Kähler and hyper-para-Kähler Lie algebras

“…Proposition 3.3. With the hypothesis above, the product on Φ(U) given by (5) is Lie-admissible if and only if ρ(X, α)Y = ρ(Y, α)X and ρ * (α, X)β = ρ * (β, X)α (7) for any X, Y ∈ U and any α, β ∈ U * . Moreover, this product is left symmetric if and only if ρ(X, α) = 0, for any X ∈ U and any α ∈ U * .…”

“…Proposition 3.4. The product on Φ(U) given by (5) is Lie-admissible if and only if ξ is a 1-cocycle of (U, [ , ]) with respect to the representation Ψ U and µ is a 1-cocycle of (U * , [ , ]) with respect to the representation Ψ U * , i.e., for any…”

“…This implies that (U, [ , ], ω) is a symplectic Lie algebra. Symplectic left symmetric algebras, called in [5] special symplectic Lie algebras, play a central role in the study of hyper-para-Kähler Lie algebras (see Section 5).…”

“…We recover also the notion of hyper-para-Kähler Lie algebra when we start from a left invariant hyper-para-Kähler structure on a Lie group. Para-Kähler and hyper-para-Kähler Lie algebras has been studied by many authors [1,2,4,5,6]. This paper is devoted to the study of para-Kähler and hyper-para-Kähler Lie algebras.…”

Symmetric and skew-symmetric complex structures

“…On the other hand, over the last years, a great deal of research has been devoted to pseudo-Kähler and complex symplectic structures on nilpotent and solvable Lie algebras, see for instance [5,8,9,10,24]. Hypersymplectic structures on nilpotent and solvable Lie algebras have been studied in [1,3,15,29]. Such structures at the Lie algebra level provide left-invariant structures on the corresponding connected, simply connected Lie groups, and on compact quotients thereof.…”