Geometry & Topology, Vol. 6 (2002) Paper no. 20, pages 563--607.

Caracteres sur l'algebre de diagrammes trivalents Lambda

Bertrand Patureau-Mirand

Abstract. The theory of Vassiliev invariants deals with many modules of diagrams on which the algebra Lambda defined by Pierre Vogel acts. By specifying a quadratic simple Lie superalgebra, one obtains a character on Lambda. We show the coherence of these characters by building a map of graded algebras beetwen Lambda and a quotient of a ring of polynomials in three variables; all the characters induced by simple Lie superalgebras factor through this map. In particular, we show that the characters for the Lie superalgebra f(4) with dimension 40 and for sl(3) are the same.

Resume. De nombreux modules de diagrammes sont utilises dans la theorie des invariants de Vassiliev. Pierre Vogel a definit une algebre Lambda qui agit sur ces espaces. Les superalgebres de Lie simples quadratiques fournissent des caracteres sur Lambda. On montre leur coherence en construisant un morphisme d'algebre graduee, entre Lambda et un quotient d'un anneau de polyneme en trois variables, qui factorise tous ces caracteres. En particulier, on montre que le caractere associe a la superalgebre de Lie f(4) de dimension 40 coincide avec celui associe a sl(3).

Keywords. Finite type invariants, weight system, representation theory

AMS subject classification. Primary: 57M27. Secondary: 57M25 17B10.

DOI: 10.2140/gt.2002.6.563

E-print: arXiv:math.GT/0107137

Submitted to GT on 4 July 2001. Paper accepted 28 October 2002. Paper published 1 December 2002.

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Bertrand Patureau-Mirand
L.M.A.M. Universite de Bretagne-Sud, Centre de Recherche
Campus de Tohannic, BP 573, F-56017 Vannes, France
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