Hypersymplectic structures on Courant algebroids

Abstract

We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a one-to-one correspondence between hypersymplectic and hyperkähler structures. This correspondence provides a simpler way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic structures on Courant algebroids encompass hyperkähler and hyperkähler structures with torsion on Lie algebroids. In the latter, the torsion existing at the Lie algebroid level is incorporated in the Courant structure. Cases of hypersymplectic structures on Courant algebroids which are doubles of Lie, quasi-Lie and proto-Lie bialgebroids are investigated. (2010). Primary 53D17; Secondary 53D18, 53C26. Mathematics Subject Classifications

Hypersymplectic structures with torsion on Lie algebroids

“…The next proposition was proved in [4] for the hypersymplectic case. The parahypersymplectic case has an analogous proof.…”

“…Applying Proposition 6.1 we conclude that (ω 1 , ω 2 , ω 3 ) is a (para-)hypersymplectic structure with torsion on (A, µ Ni ). The converse holds because the statements of Theorem 5.5 in [4] and Proposition 6.1 are equivalences.…”

“…From Proposition 6.1, the triplet (S 1 , S 2 , S 3 ) is a (para-)hypersymplectic structure on the pre-Courant algebroid (A ⊕ A * , µ + ψ), with ψ = ε k 2 [π k , π k ], for any k ∈ {1, 2, 3}. Using Proposition 5.4 in [4], we have C µ+ψ (T i , T j ) = 0, with T i given by (21), i.e.,…”

“…Let (ω 1 , ω 2 , ω 3 ) be a (para-)hypersymplectic structure with torsion on (A, µ) and fix i ∈ {1, 2, 3}. From Theorem 5.5 in [4] and Proposition 6.1 we have that (S 1 , S 2 , S 3 ) is a (para-)hypersymplectic structure on the pre-Courant algebroid (A ⊕ A * , (µ + ψ) Si ), with ψ = ε k 2 [π k , π k ], for any k ∈ {1, 2, 3}. In the computation that follows we consider k = i.…”

“…Hypersymplectic structures with torsion on Lie algebroids were introduced in [4], in relation with hypersymplectic structures on Courant algebroids, when these Courant algebroids are doubles of quasi-Lie and proto-Lie bialgebroids. In fact, while looking for examples of hypersymplectic structures on Courant algebroids we found in [4], in a natural way, hypersymplectic structures with torsion on Lie algebroids. The aim of this article is to study these structures on Lie algebroids.…”