Projective class ring of a restricted quantum group ${\overline{U}}_q({\mathfrak{s}\mathfrak{l}}_2^{\ast })$

In this paper, we compute the projective class ring of the new type restricted quantum group ${\overline{U}}_q({\mathfrak{s}\mathfrak{l}}_2^{\ast })$. First, we describe the principal indecomposable projective ${\overline{U}}_q({\mathfrak{s}\mathfrak{l}}_2^{\ast })$-modules and study their radicals, composition series, Cartan matrix of ${\overline{U}}_q({\mathfrak{s}\mathfrak{l}}_2^{\ast })$ and so on. Then, we deconstruct the tensor products between two simple modules, two indecomposable projective modules and a simple module and an indecomposable projective module, into direct sum of some indecomposable representations. At last, we characterize the projective class ring by generators and relations explicitly.