Abstract
<abstract><p>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.</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
Reference19 articles.
1. Z. Arad, E. Fisman, M. Muzychuk, Generalized table algebras, Isr. J. Math., 114 (1999), 29–60. https://doi.org/10.1007/BF02785571
2. I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of associative algebras volume 1: Techniques of representation theory, Cambridge University Press, 2006. https://doi.org/10.1017/CBO9780511614309
3. H. I. Blau, Table algebras, Eur. J. Combin., 30 (2009), 1426–1455. https://doi.org/10.1016/j.ejc.2008.11.008
4. L. F. Cao, H. X. Chen, L. B. Li, The cell modules of the Green algebra of Drinfel'd quantum double $D(H_4)$, Acta Math. Sin.-English Ser., 38 (2022), 1116–1132. https://doi.org/10.1007/s10114-022-9046-8
5. J. L. Chen, S. L. Yang, D. G. Wang, Y. J. Xu, On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras, arXiv preprint, 2018. https://doi.org/10.48550/arXiv.1809.00514