Abstract
In this paper, we study the Green ring and the stable Green ring of a finite dimensional Hopf algebra by means of bilinear forms. We show that the Green ring of a Hopf algebra of finite representation type is a Frobenius algebra over \(\mathbb {Z}\) with a dual basis associated to almost split sequences. On the stable Green ring we define a new bilinear form which is more accurate to determine the bi-Frobenius algebra structure on the stable Green ring. We show that the complexified stable Green algebra is a group-like algebra, and hence a bi-Frobenius algebra, if the bilinear form on the stable Green ring is non-degenerate.
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The authors would like to thank the referees for their careful reading of this article and for valuable comments. The first author was funded by Postdoctoral Science Foundation of China (Grant No. 2017M610316), the second author was funded by the NSF of China (Grant No. 11871063).
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Presented by: Sarah Witherspoon
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Wang, Z., Li, L. & Zhang, Y. Bilinear Forms on the Green Rings of Finite Dimensional Hopf Algebras. Algebr Represent Theor 22, 1569–1598 (2019). https://doi.org/10.1007/s10468-018-9832-2
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DOI: https://doi.org/10.1007/s10468-018-9832-2