The quantum double of the Yangian of the Lie superalgebra $A(m,n)$ and computation of the universal $R$-matrix

The Yangian double $DY(A(m,n))$ of the Lie superalgebra $A(m,n)$ is described in terms of generators and defining relations. We prove the triangular decomposition for Yangian $Y(A(m,n))$ and its quantum double $DY(A(m,n))$ as a corollary of the PBW theorem. We introduce normally ordered bases in the Yangian and its dual Hopf superalgebra in the quantum double. We calculate the pairing formulas between the elements of these bases. We obtain the formula for the universal $R$-matrix of the Yangian double. The formula for the universal $R$-matrix of the Yangian, which was introduced by V. Drinfel'd, is also obtained.