Khoroshkin-Tolstoy approach for quantum superalgebras

Abstract

The central object of the quantum algebraic approach to the study of quantum integrable models is the universal $R$-matrix, which is an element of a completed tensor product of two copies of quantum algebra. Various integrability objects are constructed by choosing representations for the factors of this tensor product. There are two approaches to constructing explicit expressions for the universal $R$-matrix. One is based on the quantum double construction, and the other is based on the concept of the $q$-commutator. In the case of a quantum superalgebra, we cannot use the first approach, since we do not know an explicit expression for the Lusztig automorphisms. One can use the second approach, but it requires some modifications related to the presence of isotropic roots. In this article, we provide the necessary modification of the method and use it to find an $R$-operator for quantum integrable systems related to the quantum superalgebra $\mathrm U_q(\mathcal{L}(\mathfrak{sl}_{M | N}))$.