分次李代数与群的针织积

Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free $d$-generator Lie rings $\mathfrak{f}_{c,d}$ of nilpotency class $c$ for all $c\leq2$ and for $(c,d)\in\{(3,3),(3,2),(4,2)\}$. We apply our computations to obtain information about $\mathfrak{p}$-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.

5-Graded simple Lie algebras, structurable algebras, and Kantor pairs

Relying on the classification of simple Lie algebras over algebraically closed fields of characteristic $>3$, we show that any finite-dimensional central simple 5-graded Lie algebra over a field $k$ of characteristic $\neq 2,3$ is a simple Lie algebra of Chevalley type, i.e. a central quotient of the Lie algebra of a simple algebraic $k$-group. As a consequence, we prove that all central simple structurable algebras and Kantor pairs over $k$ arise from 5-gradings on simple Lie algebras of Chevalley type.

分次李代数的双自同构

Double automorphisms of graded Lie algebras

引入了A-分次李代数L的双自同构的概念。粗略地说，这是L的一个自同构，它也导出群A的一个自同构。显然，L的所有双重自同构的集合形成Aut（L）中的子群。本文证明了容许双自同构的有限群的分次李代数的几个幂零性准则。我们也给出了我们的结果的一个应用程序的群体承认一个弗罗贝尼乌斯群的自同构。

Constituents of graded Lie algebras of maximal class and chain lengths of thin Lie algebras

Thin Lie algebras are infinite-dimensional graded Lie algebras $L=\bigoplus_{i=1}^{\infty}$, with $\dim(L_1)=2$ and satisfying a covering property: for each $i$, each nonzero $z\in L_i$ satisfies $[zL_1]=L_{i+1}$. It follows that each homogeneous components $L_i$ is either one- or two-dimensional, and in the latter case is called a diamond. Hence $L_1$ is a diamond, and if there are no other diamonds then $L$ is a graded Lie algebra of maximal class. We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method,…

Coabelian ideals in $\mathbb{N}$-graded Lie algebras and applications to right angled Artin Lie algebras

We consider homological finiteness properties $FP_n$ of certain $\mathbb{N}$-graded Lie algebras. After proving some general results, see Theorem A, Corollary B and Corollary C, we concentrate on a family that can be considered as the Lie algebra version of the generalized Bestvina-Brady groups associated to a graph $Γ$. We prove that the homological finiteness properties of these Lie algebras can be determined in terms of the graph in the same way as in the group case. In the last version we have corrected some missprints, in particular the statement of Theorem D (from $n-1$-acyclicity to $n-…

从立方范数对到 $G_2$- 和 $F_4$- 分级的群和李代数

From cubic norm pairs to $G_2$- and $F_4$-graded groups and Lie algebras

本文提出了一种从立方范数对构造 $G_2$ 和 $F_4$ 分级李代数及群的方法，并揭示了其结构特性。