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Left-symmetric algebras, or pre-Lie algebras in geometry and physics
  • D. Burde
  • Physics, Mathematics
  • 10 September 2005
In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAsExpand
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Classification of Orbit Closures of 4-Dimensional Complex Lie Algebras
Abstract Let L n( C ) be the variety of complexn-dimensional Lie algebras. The groupGLn( C ) acts on it via change of basis. An orbitO(μ) under this action consists of all structures isomorphic to μ.Expand
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On a refinement of Ado's theorem
Abstract. In this paper we study the minimal dimension $ \mu (g) $ of a faithful g-module for n-dimensional Lie algebras g. This is an interesting invariant of g which is difficult to compute. It isExpand
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ABSTRACT We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotentExpand
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Simple left-symmetric algebras with¶solvable Lie algebra
Abstract:Left-symmetric algebras (LSAs) are Lie admissible algebras arising from geometry. The leftinvariant affine structures on a Lie group {G} correspond bijectively to LSA-structures on its LieExpand
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Degenerations of nilpotent Lie algebras
In this paper we study degenerations of nilpotent Lie algebras. If ; are two points in the variety of nilpotent Lie algebras, then is said to degenerate to , !deg , if lies in the Zariski closure ofExpand
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We investigate the existence of affine structures on nilmanifolds Γ\G in the case where the Lie algebra g of the Lie group G is filiform nilpotent of dimension less or equal to 11. Here we obtainExpand
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Left-Invariant Affine Structures on Reductive Lie Groups
We describe left-invariant affine structures (that is, left-invariant flat torsion-free affine connections ∇) on reductive linear Lie groupsG. They correspond bijectively to LSA-structures on the LieExpand
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Left-invariant affine structures on nilpotent Lie groups
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Jacobi–Jordan algebras
Abstract We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification inExpand
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