Khoroshkin–Tolstoy approach to quantum superalgebras

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 the completed tensor product of two copies of a 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 use of Lusztig automorphisms, and the other is based on the concepts of normal ordering and $q$-commutator. In the case of a quantum superalgebra, we cannot use the first approach because we do not know an explicit expression for the Lusztig automorphisms. The second approach can be used, although it requires some modifications. In this article, we present the necessary modification of the method and use it to find the $R$-operator for a quantum integrable system related to the quantum superalgebra $\mathrm{U}_q(\mathcal{L}(\mathfrak{sl}_{M | N}))$.