The Kontsevich integral and quantized Lie superalgebras

Geer, Nathan

doi:10.2140/agt.2005.5.1111

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The Kontsevich integral and quantized Lie superalgebras

Nathan Geer

Algebraic & Geometric Topology 5 (2005) 1111–1139

DOI: 10.2140/agt.2005.5.1111

arXiv: math.GT/0411053

Abstract

Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R–matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that these two link invariants are the same. These constructions can be generalized to some classes of Lie superalgebras. In this paper we show that constructions 1) and 2) give the same invariants for the Lie superalgebras of type A–G. We use this result to investigate the Links–Gould invariant. We also give a positive answer to a conjecture of Patureau-Mirand’s concerning invariants arising from the Lie superalgebra $D (2, 1; α)$ .

Keywords

Vassiliev invariants, weight system, Kontsevich integral, Lie superalgebras, Links–Gould invariant, quantum invariants

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 17B65, 17B37

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Publication

Received: 6 May 2005

Accepted: 15 August 2005

Published: 11 September 2005

Authors

Nathan Geer
School of Mathematics Georgia Institute of Technology Atlanta GA 30332-0160 USA
http://www.math.gatech.edu/~geer/

Volume 5, issue 3 (2005)