Weak Hopf algebras corresponding to Cartan matrices

We replace the group of group-like elements of the quantized enveloping algebra $U_q(\mathfrak{g})$ of a finite dimensional semisimple Lie algebra $\mathfrak{g}$ by some regular monoid and get the weak Hopf algebra ${\mathfrak{w}}_q^{\mathsf{d}}(\mathfrak{g})$. It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of ${\mathfrak{w}}_q^{\mathsf{d}}(\mathfrak{g})$ and determine the group of weak Hopf algebra automorphisms of ${\mathfrak{w}}_q^{\mathsf{d}}(\mathfrak{g})$ when $q$ is not a root of unity.