Universal R-matrices for quantum groups associated to simple Lie superalgebras
Let \(G\) be a simple Lie superalgebra of type \(A\)-\(G\). Let \(\Pi\) and \(p:\Pi\to\{0,1\}\) be a set of simple roots and a parity function of \(G\). We assume that \((\Pi,p)\) is of the distinguished type if \(G\) is of type \(F_ 4\) or \(G_ 3\). We define an \(h\)-deformation \(U_ h(\Pi,p)\) of \(U(G)\) by Chevalley generators and defining relations. We prove that \(U_ h(\Pi,p)\) is a topologically free \(Ch\)-Hopf superalgebra which goes to \(U(G)\) when \(h\to 0\). (Moreover, letting \(h\to 0\) in the defining relations, we obtain defining relations of \(U(G)\). This result also seems to be new.) A remarkable fact is that the defining relations are not exhausted by binary relations such as the \(q\)-Serre relations; we also need trinomial and quadrinomial relations. Let \(\sigma\) be the involution of \(U_ h(\Pi,p)\) defined by \(\sigma(x)=x\) (resp. \(-x)\) for an even (resp. odd) element \(x\). We introduce the ``Hopf algebraization'' \(U_ h(\Pi,p)^ \sigma\) of \(U_ h(\Pi,p)\); as an algebra, \(U_ h(\Pi,p)^ \sigma\) is the twisted product of \(U_ h(\Pi,p)\) and \(\langle\sigma\rangle\). Our main result explicitly describes the universal \(R\)-matrices \(\tilde R\) of \(U_ h(\Pi,p)^ \sigma\) by using Lusztig type \(q\)-root vectors and the \(q\)- exponential. If \((\Pi,p)\) is of type \(A\) and \(\pi\) is the fundamental representation of \(U_ h(\Pi,p)^ \sigma\), then \(\pi\otimes\pi(\tilde R)\) is the (constant) \(R\)-matrix of Perk and Schultz.
scientific article; zbMATH DE number 29810