A Two Parameter Deformation of the Su(1/1) Superalgebra and the Xy Quantum Chain in a Magnetic Field

We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the SU(1/1) superalgebra. One is led to an extension of the braid group and the Hecke algebra which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. CERN-TH.6299/91 October 1991 ∗Permanent address: Universität Bonn, Physikalisches Institut, Nussallee 12, W-5300 Bonn 1, FRG There were several attempts to extend the one-parameter quantum algebras to multiparameter ones [1]. As shown however by Reshetikhin [2] the link polynomials depend only on one parameter. One can state this result in a different way: if one has a one-dimensional quantum chain which is invariant under a multiparameter quantum algebra, one can do a similarity transformation which eliminates all the parameters but one. As will be shown in this paper, the situation is different in the case of quantum superalgebras. We will start with a physical example. Consider the quantum chain