Left-symmetric algebroids

Abstract

In this paper, we introduce a notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie algebroid. We construct left-symmetric algebroids from $\mathcal{O}$-operators on Lie algebroids. We study phase spaces of Lie algebroids in terms of left-symmetric algebroids. Representations of left-symmetric algebroids are studied in detail. At last, we study deformations of left-symmetric algebroids, which could be controlled by the second cohomology class in the deformation cohomology.

21 pages

References in corpus (3)

• Field Algebras
• A Unified Algebraic Approach to Classical Yang-Baxter Equation
• Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation

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• Cohomology and Deformations of left-symmetric Rinehart Algebras
• Koszul-Vinberg structures and compatible structures on left-symmetric algebroids
• Extensions and crossed modules of $n$-Lie Rinehart algebras
• Homotopy Poisson algebras, Maurer-Cartan elements and Dirac structures of CLWX 2-algebroids
• On Nash resolution of (singular) Lie algebroids
• Left-symmetric bialgebroids and their corresponding Manin triples