# zbMATH — the first resource for mathematics

$$R$$-matrice universelle pour $$U_h(D(2,1,x))$$ et invariant d’entrelacs associé. (Universal $$R$$-matrix for $$U_h(D(2,1,x))$$ and link invariant arising from it). (French) Zbl 1013.17007
Lie superalgebras were introduced by Kac and were subsequently quantised by a number of authors. A universal $$R$$-matrix has been found for all of the quantised versions, with the exception to date of the family of algebras $$D(2,1,x)$$, depending on one continuous parameter $$x$$.
In this paper the author uses the method of quantum doubles to construct a universal $$R$$-matrix for these algebras also, and much of the paper is devoted to the details. He goes on to construct a knot invariant from a 6-dimensional module $$M$$ of one of these algebras, using the $$R$$-matrix and a ribbon element in the sense of Turaev. This invariant is shown to be the evaluation of Kauffman’s Dubrovnik polynomial $$D(a,z)$$ for the knot, with $$a=-q^{-1}$$ and $$z=q-q^{-1}$$. The result stems from the fact that the module $$M$$ is self-dual, and its $$R$$-matrix has minimal polynomial of degree 3.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 16W35 Ring-theoretic aspects of quantum groups (MSC2000)
Full Text: