Paul J. Sanders
For a prime p and natural number n with p greater than or equal to n, we establish the existence of a non-functorial one-to-one correspondence between isomorphism classes of groups of order p^n whose derived subgroup has exponent dividing p, and isomorphism classes of nilpotent p^n-element Lie algebras L over the truncated polynomial ring F_p[T]/(T^n) in which T[L,L]=0.
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A Lazard correspondence for post-Lie rings and skew braces
本文通过研究Lazard对应在李环半直和上的行为，建立了后李环与满足自然完备性条件的斜括号之间的对应关系，并推广了Smoktunowicz关于阶为$p^k$的斜括号与左幂零后李环对应的结果。
In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite Fp-braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent Fp-braces generated by one element of cardinality p^4 is obtained, answering a question posed by Leandro Vendramin. It is also shown that the sum of a finite number of left nilpotent ideals in a left brace is a left…
Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras
该研究将群论中的三个经典结果推广到乘性李代数的非交换二阶上同调场景，证明了对应定理、Wells定理并建立一维Lyndon-Hochschild-Serre正合序列，特殊情形下可复现已有结论。
In this paper we extend the notion of CP covers for groups to the field of Lie algebras, and show that despite the case of groups, all CP covers of a Lie algebra are isomorphic. Finally we show that CP covers of groups and Lie rings which are in Lazard correspondence, are in Lazard correspondence too, and the Bogomolov multipliers are isomorphic.
A formal Lie correspondence
本文在零特征下建立了形式 nilpotent 李代数与指数群的等价关系，扩展了 Mal'cev 等人的等价关系，并应用于广义形式级数的组合和代数自同构。
We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in [1], in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced, and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.
Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-$p$ groups of nilpotency class smaller than $p$ and finitely generated nilpotent $\mathbb{Z}_p$-Lie algebras of nilpotency class smaller than $p$. Denote by $H_{Gr}^i$ and $H_{Lie}^i$ the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for $i=0$, $1$ and $1$, and for a given category of modules the cohomology functors $H_{Gr}^i\circ \textbf{exp}$ and $H^i_{Lie}$ are naturally equivalent. A similar result is proven for $i=3$ and the r…
Let $N$ be a normal subgroup of a finite group $G$ and $V$ be a fixed finite-dimensional $G$-module. The Poincaré series for the multiplicities of induced modules and restriction modules in the tensor algebra $T(V)=\oplus_{k \geq 0}V^{\otimes k}$ are studied in connection with the McKay-Slodowy correspondence. In particular, it is shown that the closed formulas for the Poincaré series associated with the distinguished pairs of subgroups of $\mathrm{SU}_2$ give rise to the exponents of all untwisted and twisted affine Lie algebras except ${\rm A}_{2n}^{(1)}$.
We classify the nilpotent Lie rings of order $p^8$ with maximal class for $p \ge 5$. This also provides a classification of the groups of order $p^8$ with maximal class for $p \ge 11$ via the Lazard correspondence.
We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.
The family of quaternionic quasi-unitary (or quaternionic unitary Cayley--Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_{N+1}, as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature. This common framework allows to perform the study of many properties for all these Lie algebras simult…
On some connections between braces and pre-Lie rings outside of the context of Lazard's correspondence
本文研究Lazard对应之外的brace与pre-Lie环的联系，在特定条件下证明因子brace可由左幂零pre-Lie环的流群公式构造。
As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type $A_{n-1}$, split or twisted. Key to the proofs is the study of certain characters of SL$_n(q)$ and SU$_n(q)$ related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.
In this paper, we establish the inductive Alperin weight condition for the finite simple groups of Lie type $\mathsf A$, contributing to the program to prove the Alperin weight conjecture by checking the inductive condition for all finite simple groups.
We notice that one of the Diophantine equations, $knm=2kn+2km+2nm$, arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In both cases it describes "regular polyhedrons" with $k$ edges in each vertex, $n$ edges of each face, with total number of edges $|m|$, and Euler characteristics $χ=\pm 2$. In the case of negative $m$ this equation corresponds to $χ=2$ and describes true regular polyhedrons, Platonic solids. The case with positive $m$ corresponds to Euler characteristic $χ=-2$…
A short note on coproducts of abelian pro-Lie groups
本文引入了阿贝尔 pro-Lie 群范畴中的条件余积概念，并证明任意阿贝尔 pro-Lie 群族的笛卡尔积可由条件余积的泛性质刻画。
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