Courant Algebroids and Poisson Geometry

Ricci flow, Courant algebroids, and renormalization of Poisson–Lie T-duality

“…If E → M is a CA and φ : M ′ → M a smooth map, the CA structures on φ * E satisfying the condition (13) were characterized by Li-Bland and Meinrenken [10] as follows: if ρ : φ * E → T M ′ is a vector bundle map then such a CA structure on φ * E with the anchor ρ exists (and moreover is unique) iff…”

Pseudo-Dirac structures

“…Let E = d × M with anchor map a = ρ and with the bundle metric coming from the metric on d. As shown in [34], the Lie bracket on constant sections d ⊆ C ∞ (M, d) = Γ (E) extends to a Courant bracket if and only if the action has coisotropic stabilizers, i.e. ker(ρ(·, m)) ⊆ d satisfies ker(ρ(·, m)) ⊥ ⊆ ker(ρ(·, m)) for any m ∈ M. Explicitly, for σ 1 , σ 2 ∈ Γ (E) = C ∞ (M, d) the Courant bracket reads (see [34,Section 4])…”

“…Suppose d is a quadratic Lie algebra which acts on a manifold M with coisotropic stabilizers. In this case, as explained in [34], the bundle d × M is naturally a Courant algebroid (see Example 2.8 for details). If h ⊆ d is any subalgebra, then…”

“…Therefore W ⊂ TM is just a Dirac structure. In fact it is the graph of the Poisson structure constructed by Lu and Yakimov for the action of the Manin triple (d, e, f) on M [38,34].…”

A Class of Lie 2-Algebras in Higher-Order Courant Algebroids

“…Equivalent definition was given by Roytenberg [2]. In resent years, with the development and exploration of the theory of categorified Lie algebras, or "Lie 2-algebras", Courant algebroids have been far and wide studied from several aspects and have been found many applications in the theory of Manin pairs and moment maps [3] [4]; generalized complex structures [5]; L ∞ -algebras and symplectic supermanifolds [2]; gerbes [6] as well as BV algebras and topological field theories.…”