The Drinfeld associator of gl(1 | 1)

Abstract

We determine explicitly a rational even Drinfeld associator Φ in a completion of the universal enveloping algebra of the Lie superalgebra gl(1 | 1)⊕3. More generally, we define a new algebra of trivalent diagrams that has a unique even horizontal group-like Drinfeld associator Φ. The associator Φ is mapped to Φ by a weight system. As a related result of independent interest, we show how O. Viro’s generalization Δ1 of the multi-variable Alexander polynomial can be obtained from the universal Vassiliev invariant of trivalent graphs. We determine Φ by using the invariant Δ1(tetrahedron) of a planar tetrahedron.