ON THE RADICAL OF THE CENTER OF SMALL SYMMETRIC LOCAL ALGEBRAS

Year 2020, , 175 - 186, 14.07.2020

Pierre Landrock

https://doi.org/10.24330/ieja.768246

Cited By: 1

Abstract

This article is motivated by some results from Chlebowitz and Külshammer on how the structure of a symmetric local algebra is influenced by its center. They have shown that a symmetric local algebra is almost always commutative if its center is at most 5-dimensional. In this article we are interested in how the ideal property of the radical of the center of a symmetric local algebra is influenced by the dimension of the algebra itself. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

Keywords

Symmetric algebra, local algebra, radical of the center

References

• M. Chlebowitz and B. Külshammer, Symmetric local algebras with 5-dimensional center, Trans. Amer. Math. Soc., 329(2) (1992), 715-731.
• T. C. Craven and T. L. Smith, Symmetric algebras over rings and fields, Bull. Aust. Math. Soc., 89(3) (2014), 466-472.
• C. W. Eaton, Morita equivalence classes of blocks with elementary abelian defect groups of order 16, preprint, arXiv:1612.03485v4, (2019).
• R. Kessar, On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the Abelian Defect Group Conjecture, J. Lond. Math. Soc., 85(2) (2012), 491-510.
• B. Külshammer, Symmetric local algebras and small blocks of finite groups, J. Algebra, 88 (1984), 190-195.
• P. Landrock, Eine Klasse von Blocken mit Einer Elementarabelschen Defektgruppe der Ordnung 16, Ph.D. Thesis, Jena 2018.
• P. Landrock and B. Sambale, On centers of blocks with one simple module, J. Algebra, 472 (2017), 339-368.