Modular group algebras with maximal Lie nilpotency indices
英文摘要:
In the present paper we give the full description of the Lie nilpotent group algebras which have maximal Lie nilpotency indices.
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Victor Bovdi, Ernesto Spinelli
In the present paper we give the full description of the Lie nilpotent group algebras which have maximal Lie nilpotency indices.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G'|+1, where |G'| is the order of the commutator subgroup. The authors have previously determined the groups G for which this index is maximal and here they determine the G for which it is `almost maximal', that is the next highest possible value, namely |G'|-p+2.
Lower and Upper Lie Nilpotency Indices of Modular Group Algebras
该研究解决了模群代数下与上李幂零指数问题的剩余情形,在特征2和3下证明两指数相等,完成有限域上素数幂阶有限群的相关问题,引入终端限制PBW构造完成论证。
Let $KG$ be the modular group algebra of an arbitrary group $G$ over a field $K$ of characteristic $p>0$. It is seen that if $KG$ is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least $p+1$. The classification of group algebras $KG$ with upper Lie nilpotency index $t^{L}(KG)$ upto $9p-7$ have already been determined. In this paper, we classify the modular group algebra $KG$ for which the upper Lie nilpotency index is $10p-8$.
In this paper, we classify the modular group algebra $KG$ of a group $G$ over a field $K$ of characteristic $p>0$ having upper Lie nilpotency index $t^{L}(KG)= \vert G^{\prime}\vert - k(p-1) + 1$ for $k=14$ and $15$. Group algebras of upper Lie nilpotency index $\vert G^{\prime}\vert - k(p-1) + 1$ for $k\leq 13$, have already been characterized completely.
Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie nilpotency index is at most |G'|+1, where |G'| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal have already been determined. Here we determine G for which upper (or lower) Lie nilpotency index is the next highest possible.
Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G'|+1, where |G'| is the order of the commutator subgroup. Previously we determined the groups G for which the upper/lower nilpotency index is maximal or the upper nilpotency index is `almost maximal' (that is, of the next highest possible value, namely |G'|-p +2). Here we determine the groups for which the lower nilpotency index is `almost maximal'.
Let $G$ be a group and let $F$ be a field of characteristic different from 2. Denote by $(FG)^+$ the set of symmetric elements and by $\mathcal{U}^+(FG)$ the set of symmetric units, under an oriented classical involution of the group algebra $FG$. We give some lower and upper bounds on the Lie nilpotency index of $(FG)^+$ and the nilpotency class of $\mathcal{U}^+(FG)$.
In this article, we show that if $KG$ is Lie nilpotent group algebra of a group $G$ over a field $K$ of characteristic $p>0$, then $t_{L}(KG)=k$ if and only if $t^{L}(KG)=k$, for $k\in\{5p-3, 6p-4\}$, where $t_{L}(KG)$ and $t^{L}(KG)$ are the lower and the upper Lie nilpotency indices of $KG$, respectively.
Nilpotency indices for quantum Schubert cell algebras
研究量子 Schubert 胞腔代数中 Lusztig 根向量的 $q$-伴随幂零指数,通过引入基于 Bruhat 序的等价关系对根向量对进行分类,并计算各类别的幂零指数。
On the Modular Isomorphism Problem for groups of class 3 and obelisks
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Let $p$ be a prime number, $G$ be a finite $p$-group and $K$ be a field of characteristic $p$. The Modular Isomorphism Problem (MIP) asks whether the group algebra $KG$ determines the group $G$. Dealing with MIP, we investigated a question whether the nilpotency class of a finite $p$-group is determined by its modular group algebra over the field of $p$ elements. We give a positive answer to this question provided one of the following conditions holds: (i) $\exp G=p$; (ii) $\cl(G)=2$; (iii) $G'$ is cyclic; (iv) $G$ is a group of maximal class and contains an abelian subgroup of index $p$.
Local derivations and automorphisms of nilpotent Lie algebra
本文研究幂零李代数的局部导子与自同构,证明幂零指数为3和4时存在非导子的局部导子及非自同构的局部自同构,并给出存在性充分条件及一个局部导子空间与导子空间重合的非结合代数例子。
Fibonacci Lie algebra revisited
本文以自包含方式研究Fibonacci李代数,证明其增长函数多项式行为非一致、非PI,并给出泛包络代数增长及幂零指数的界,同时探讨相关代数同调与Euler特征的无限性及混沌行为。
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