A new family of positively based algebras ${\mathcal{H}}_n$

In this paper, we introduce a new family of algebras ${\mathcal{H}}_n$, which are generated by three generators $x,y,z$, with the following relations: (1) $x^{2n}=1,y^2=xy+y,xy=yx;$ and (2) $z^2=z,xz=zx=z,zy=2z.$ First, it shows that ${\mathcal{H}}_n$ is a positively based algebra. Then, all the indecomposable modules of ${\mathcal{H}}_n$ are constructed. Additionally, it shows that the dimension of each indecomposable ${\mathcal{H}}_n$-module is at most $2$. Finally, all the left (right) cells and left (right) cell modules of ${\mathcal{H}}_n$ are described, and the decompositions of the decomposable left cell modules are also obtained.