Nonabelian omni-Lie algebras

Abstract

We introduce the notion of a nonabelian omni-Lie algebra associated to a Lie algebra. From a geometric point of view, it is the linearization of the Courant algebroid associated to a Poisson manifold. As an algebraic application, we show that its Dirac structures describe deformations of the Lie algebra.

VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

“…On the other hand, by Theorem 4.6, the generalized complex structure J will also induce a Lie algebroid structure on E by (13). By the equality…”

“…• Given a crossed module of Lie algebras (m, g), we get an m-Courant algebroid Hom(g, m) ⊕ g, which was given in [13] as a generalization of an omni-Lie algebra.…”

Poisson geometry and Lie $\bm{n}$-algebras

“…一个非交换的 omni-李代数可以看成由 Poisson 流形给出的 Courant 代数胚的线性化, 也可以看 成 omni-李代数的平凡形变, 有丰富的代数和几何性质. 更多内容参见文献 [28].…”

Omni n-Lie algebras and linearization of higher analogues of Courant algebroids

“…In [19], the authors introduced the notion of a nonabelian omni-Lie algebra (gl(g) ⊕ g, (·, ·) + , {·, ·} g ) associated to a Lie algebra (g, [·, ·] g ), which originally comes from the study of homotopy Poisson manifolds [18]. In particular, they showed that it is the linearization of the Courant algebroid T g * ⊕T * πg g * associated to the linear Poisson manifold (g * , π g ), where π g is the Lie-Poisson structure on g * .…”