Yangian deformations of $\mathcal{S}$-commutative quantum vertex algebras and Bethe subalgebras

We construct a new class of quantum vertex algebras associated with the normalized Yang $R$-matrix. They are obtained as Yangian deformations of certain $\mathcal{S}$-commutative quantum vertex algebras and their $\mathcal{S}$-locality takes the form of a single $RTT$-relation. We establish some preliminary results on their representation theory and then further investigate their braiding map. In particular, we show that its fixed points are closely related with Bethe subalgebras in the Yangian quantization of the Poisson algebra $\mathcal{O}(\mathfrak{gl}_N((z^{-1})))$, which were recently introduced by Krylov and Rybnikov. Finally, we extend this construction of commutative families to the case of trigonometric $R$-matrix of type $A$.