On the tensor system of a semisimple Lie algebra

Murphy, Timothy C.

doi:10.1017/s0305004100050465

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1976

2015

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“…An important class of weight systems is obtained as follows from the structure constants of metric Lie algebras, which roots in papers of Penrose [14] and Murphy [13], and the relevance for knot theory was pioneered by Bar-Natan [2,3] and Kontsevich [12]. (In this paper, all Lie algebras are finite-dimensional and complex.)…”

Section: Introductionmentioning

confidence: 99%

On Lie algebra weight systems for 3-graphs

Schrijver

2015

Journal of Pure and Applied Algebra

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A 3-graph is a connected cubic graph such that each vertex is equipped with a cyclic order of the edges incident with it. A weight system is a function f on the collection of 3-graphs which is antisymmetric: f (H) = −f (G) if H arises from G by reversing the orientation at one of its vertices, and satisfies the IHX-equation:Key instances of weight systems are the functions ϕ g obtained from a metric Lie algebra g by taking the structure tensor c of g with respect to some orthonormal basis, decorating each vertex of the 3-graph by c, and contracting along the edges. We give equations on values of any complex-valued weight system that characterize it as complex Lie algebra weight system. It also follows that if f = ϕ g for some complex metric Lie algebra g, then f = ϕ g for some unique complex reductive metric Lie algebra g . Basic tool throughout is geometric invariant theory.

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Section: Introductionmentioning

confidence: 99%