The Jacobi Identity beyond Lie Algebras

zbMATH Open1182.51005arXiv0904.1155MaRDI QIDQ3651348

Hirokazu Nishimura

Publication date: 9 December 2009

Abstract: Frolicher and Nijenhuis recognized well in the middle of the previous century that the Lie bracket and its Jacobi identity could and should exist beyond Lie algebras. Nevertheless the conceptual meaning of their discovery has been obscured by the messy techniques they exploited. The principal objective in this paper is to show that the double dualization functor in a cartesian closed category as well as synthetic differential geometry provides an adequate framework, in which their discovery's conceptual meaning appears lucid.

Full work available at URL: https://arxiv.org/abs/0904.1155

zbMATH Keywords

Jacobi identityLie algebrasynthetic differential geometrySchwarz distributiondouble dualization functorgeneral Jacobi identity

Mathematics Subject Classification ID

Lie algebras of vector fields and related (super) algebras (17B66) Lie algebras and Lie superalgebras (17B99) Synthetic differential geometry (51K10) Topos-theoretic approach to differentiable manifolds (58A03)

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