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\(R\)-matrice universelle pour \(U_h(D(2,1,x))\) et invariant d’entrelacs associ√©. (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)
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