AI 中文总结
本文建立特征0代数闭域上有限维半单代数正则量子交换分解的充要准则，应用于群代数得到峰群相关结论，还构造了不可解峰群。
AI 中文摘要
本文中，我们建立了特征为0的代数闭域上的有限维半单代数容许正则量子交换分解的充要准则。将该特征化结果应用于群代数，证明有限群代数容许此类分解当且仅当基础群是峰群，即其不可约特征标度数关于整除序存在最大元的有限群。我们研究峰群的结构，并依据其最大不可约特征标度数的素因子建立若干可解性准则，特别表明若干重要群类为峰群，而仅超可解性不蕴含峰性质，最终在Frobenius群类中构造出一个不可解峰群。
英文摘要
In this paper, we establish a necessary and sufficient criterion for a finite-dimensional semisimple algebra over an algebraically closed field of characteristic $0$ to admit a regular quantum commutative decomposition. We apply this characterization to group algebras and show that a finite group algebra admits such a decomposition if and only if the underlying group is a peak group, that is, a finite group whose irreducible character degrees have a greatest element with respect to the divisibility ordering. We investigate the structure of peak groups and establish several criteria for their solvability in terms of the prime divisors of their largest irreducible character degree. In particular, we show that several important classes of groups are peak groups, while supersolvability alone does not imply the peak property. Finally, we construct a non-solvable peak group within the class of Frobenius groups.
Comments14 pages
