Projection method and a universal weight function for the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_{N+1})$

We calculate the projection of the product of the Drinfeld currents on the intersection of the different Borel subalgebras in the current realization of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_{N+1})$. This projection yields a universal weight function and has the structure of nested Bethe vectors.