The Kontsevich integral and quantized Lie superalgebras

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 $\mathcal{D}\left(2,1;\alpha \right)$.