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

Shiyu Lin; Shilin Yang; Shiyu Lin; Shilin Yang

doi:10.3934/math.2024128

AIMS Mathematics

2024, Volume 9, Issue 2: 2602-2618. doi: 10.3934/math.2024128

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Research article

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

Shiyu Lin ,
Shilin Yang ^,

Department of Mathematics, Beijing University of Technology, Beijing 100124, China

Received: 25 October 2023 Revised: 04 December 2023 Accepted: 10 December 2023 Published: 26 December 2023
MSC : 16D80, 16G60

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.
- positively based algebra,
- indecomposable module,
- cell module
Citation: Shiyu Lin, Shilin Yang. A new family of positively based algebras $ {\mathcal{H}}_n $[J]. AIMS Mathematics, 2024, 9(2): 2602-2618. doi: 10.3934/math.2024128

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Abstract

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.

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