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.