Volume 5, issue 3 (2005)

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The Kontsevich integral and quantized Lie superalgebras

Nathan Geer

Algebraic & Geometric Topology 5 (2005) 1111–1139
 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 $\mathsc{D}\left(2,1;\alpha \right)$.

Keywords
Vassiliev invariants, weight system, Kontsevich integral, Lie superalgebras, Links–Gould invariant, quantum invariants
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 17B65, 17B37
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/