Dorfman connections and Courant algebroids

Autor: M. Jotz Lean
Rok vydání: 2018
Předmět:
Zdroj: Journal de Mathématiques Pures et Appliquées. 116:1-39
ISSN: 0021-7824
DOI: 10.1016/j.matpur.2018.06.016
Popis: We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. We illustrate this analogy with examples. In particular, we study horizontal spaces in the standard Courant algebroids over vector bundles: A linear connection ∇ : X ( M ) × Γ ( E ) → Γ ( E ) on a vector bundle E over a smooth manifold M is tantamount to a linear splitting T E ≃ T q E E ⊕ H ∇ , where T q E E is the set of vectors tangent to the fibres of E. Furthermore, the curvature of the connection measures the failure of the horizontal space H ∇ to be integrable. We extend this classical result by showing that linear horizontal complements to T q E E ⊕ ( T q E E ) ∘ in T E ⊕ T ⁎ E can be described in the same manner via a certain class of Dorfman connections Δ : Γ ( T M ⊕ E ⁎ ) × Γ ( E ⊕ T ⁎ M ) → Γ ( E ⊕ T ⁎ M ) . Similarly to the tangent bundle case, we find that, after the choice of such a linear splitting, the standard Courant algebroid structure of T E ⊕ T ⁎ E → E can be completely described by properties of the Dorfman connection. As a corollary, we find that the horizontal space is a Dirac structure if and only if Δ is the dual derivation to a Lie algebroid structure on T M ⊕ E ⁎ . We use this to study splittings of T A ⊕ T ⁎ A over a Lie algebroid A and, following Gracia-Saz and Mehta, we compute the representations up to homotopy defined by any linear splitting of T A ⊕ T ⁎ A and the linear Lie algebroid T A ⊕ T ⁎ A → T M ⊕ A ⁎ . We characterize VB- and LA-Dirac structures in T A ⊕ T ⁎ A via Dorfman connections.
Databáze: OpenAIRE
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