Cocommutative elements form a maximal commutative subalgebra in quantum matrices

In this paper we prove that the subalgebras of cocommutative elements in the quantized coordinate rings of $M_n$, $G{L}_n$ and $S{L}_n$ are the centralizers of the trace $x_{1,1}+\cdots +x_{n,n}$ in each algebra, for $q\in {\mathbb{C}}^{\times }$ being not a root of unity. In particular, it is not only a commutative subalgebra as it was known before, but it is a maximal one.