Cristina Draper, Alberto Elduque
This paper presents a survey of the results and ideas behind the classification of the fine gradings, up to equivalence, on the simple finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It provides an expanded version of the mini course delivered by the second author at the Conference "Advances in Group Theory and Applications AGTA-2015".
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We introduce grading on certain finite dimensional simple Lie superalgebras of type $P(t)$ by elementary abelian 2-group. This grading gives rise to Pauli matrices and is a far generalization of $(\mathbb Z_2\times \mathbb Z_2)$-grading on Lie algebra of $(2\times 2)$-traceless matrices.We use this grading for studying numerical invariants of polyomial identities of Lie superalgebras. In particular, we compute graded PI-exponent corresponding to Pauli grading.
Jordan gradings on exceptional simple Lie algebras
给出了例外单李代数的自同构群的若当子群诱导的所有分次的模型。
We classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank. In case of rank 0, we describe conditions to obtain non trivial $\Z_k$-gradings.
We classify group gradings on the simple Lie algebra $L$ of type $D_4$ over an algebraically closed field of characteristic different from 2: fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism. For each $G$-grading on $L$, we also study graded $L$-modules (assuming characteristic 0).
The fine abelian group gradings on the simple exceptional classical Lie superalgebras over algebraically closed fields of characteristic 0 are determined up to equivalence.
We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$, $n \geq 2$, over an algebraically closed field of characteristic different from $2$ (and not dividing $n+1$ in the Lie case): fine gradings up to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism.
In this paper we describe all gradings by abelian groups without elements of order p, where p > 2 is the characteristic of the base field, on the simple graded Cartan type Lie algebras.
We classify group gradings on the simple Lie algebras of types $G_2$ and $D_4$ over the field of real numbers (or any real closed field): fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism.
We describe two constructions of a certain $\mathbb{Z}_4^3$-grading on the so-called Brown algebra (a simple structurable algebra of dimension 56 and skew-dimension 1) over an algebraically closed field of characteristic different from 2 and 3. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types E6, E7 and E8.
Group Gradings on Simple Lie Algebras of Type "A"
本文描述了特征为零的代数闭域F上任意A型李代数L的有限阿贝尔群G的群分次。
We study group-graded Lie algebras L with finite support X. We show that L is nilpotent of |X|-bounded class if X is arithmetically-free. Conversely: we show that Y supports the grading of a non-nilpotent Lie algebra if Y is not arithmetically-free.
Some fine gradings on the exceptional Lie algebras $\mathfrak{e}_6$, $\mathfrak{e}_7$ and $\mathfrak{e}_8$ are described. This list tries to be exhaustive.
Group gradings on simple Lie algebras of type A in positive characteristic
本文考虑特征不等于2且不整除$n$的代数闭域$F$上的李代数$\mathfrak{sl}_n(F)$上的有限阿贝尔群群$G$的分次.
The fine abelian group gradings on the simple classical Lie algebras (including D4) over algebraically closed fields of characteristic 0 are determined up to equivalence. This is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them.
More non semigroup Lie gradings
本文讨论了两个非常简单的例子的构造，它们分别是4维和6维的分次李代数，其分次不是由半群给出的，后者是半单代数。它表明，4是最小可能的尺寸。
A Lie algebra L is known to be nilpotent if it admits a grading by (Zp, +) with support X not containing 0. It is also known that the class of L can be bounded by some explicit function of |X|. We generalise this and other classical results to gradings of Lie algebras by arbitrary groups with arithmetically-free support. We then apply these results to automorphisms of groups satisfying an identity.
For any abelian group $G$, we classify up to isomorphism all $G$-gradings on the classical central simple Lie algebras, except those of type $D_4$, over the field of real numbers (or any real closed field).
Given a Lie algebra $L$ graded by a group $G$, if $L$ is does not contain orthogonal graded ideals and $G$ is generated by the support of $L$, then $G$ is an abelian group.
In this paper we define the so-called twisted Heisenberg superalgebras over the complex number field by adding derivations to Heisenberg superalgebras. We classify the fine gradings up to equivalence on twisted Heisenberg superalgebras and determine the Weyl groups of those gradings.
All gradings by abelian groups are classified on the following algebras over an algebraically closed field of characteristic not 2: the simple Lie algebra of type $G_2$ (characteristic not 3), the exceptional simple Jordan algebra, and the simple Lie algebra of type $F_4$.
We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.
We investigate the group gradings on the algebra of upper triangular matrices over an arbitrary field, viewed as a Lie algebra. These results were obtained a few years early by the same authors. We provide streamlined proofs, and present a complete classification of isomorphism classes of the gradings. We also provide a classification of the practical isomorphism classes of the gradings, which is a better alternative way to consider these gradings up to being essentially the same object. Finally, we investigate in details the case where the characteristic of the base field is $2$, a topic that…
We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.
A set grading on the split simple Lie algebra of type $D_{13}$, that cannot be realized as a group-grading, is constructed by splitting the set of positive roots into a disjoint union of pairs of orthogonal roots, following a pattern provided by the lines of the projective plane over $GF(3)$. This answers in the negative Question 1.11 in Elduque-Kochetov monograph (2013). Similar non-group gradings are obtained for types $D_n$ with $n$ congruent to 1 modulo 12, by substituting the lines in the projective plane by blocks of suitable Steiner systems.
We study gradings by noncommutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if $L$ is gradeg by a non-abelian finite group $G$ then the solvable radical $R$ of $L$ is $G$-graded and there exists a Levi subalgebra $B=H_1\oplus\cdots\oplus H_m$ homogeneous in $G$-grading with graded simple summands $H_1, \ldots, H_m$. All supports $Supp~H_i, i=1\ldots, m$, are commutative subsets of $G$.
Group gradings on classical Lie superalgebras
本文在特征零代数闭域上对非例外经典单李超代数的群分次进行了同构分类，通过研究满足降链条件的分次单结合超代数，实现了对有限维单结合超代数阿贝尔群分次的分类。
Given a grading on a nonassociative algebra by an abelian group, we have two subgroups of automorphisms attached to it: the automorphisms that stabilize each homogeneous component (as a subspace) and the automorphisms that permute the components. By the Weyl group of the grading we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the automorphism group of the root system, i.e., the so-called extended Weyl group. A grading is called fine if it cannot be refined. We compute the Weyl groups of all fine gradi…
For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple Lie algebras of types A_n (n >= 1), B_n (n >= 2), C_n (n >= 3) and D_n (n > 4), in terms of numerical and group-theoretical invariants. The ground field is assumed to be algebraically closed of characteristic different from 2.
We classify up to isomorphism all gradings by an arbitrary group $G$ on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic $0$. It turns out that the support of such a grading always generates an abelian subgroup of $G$. Assuming that $G$ is abelian, our technique also works to obtain the classification of $G$-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic $0$ and $…
Gradings of non-graded Hamiltonian Lie algebras
薄李代数是在满足一定窄性条件的正整数上分次的李代数。我们描述了几个循环分次的模块化的哈密尔顿李代数$H(2\colon\n;ω_2)$（尺寸小于一个权力$p$），我们从构建无限维薄李代数。在这个过程中，我们提供了一个明确的识别$H(2\colon\n;ω_2)$与块代数。我们还计算了它的二阶上同调群和它的导子代数（在任意素特征下）。
For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple restricted Lie algebras of types W(m;1) and S(m;1) (m>=2), in terms of numerical and group-theoretical invariants. Our main tool is automorphism group schemes, which we determine for the simple restricted Lie algebras of types S(m;1) and H(m;1). The ground field is assumed to be algebraically closed of characteristic p>3.
A Lie grading which is not a semigroup grading
本文讨论了一类分次李代数的构造，它的分次不是由半群给出的。
Group gradings on exceptional simple Lie superalgebras
本文在特征0代数闭域上同构分类了例外单李超代数$G(3)$、$F(4)$、$D(2,1;α)$及$A(1,1)$上的任意群分次，核心方法是将Elduque等的新方法应用于已知的精细分次等价分类。
We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n \geq 2$, and on the simple associative superalgebras $M(m,n)$, $m, n \geq 1$, over an algebraically closed field: fine gradings up to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism. As a corollary, we also classify up to isomorphism the $G$-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from $G$-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be $0$.
Given a fine abelian group grading on a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero, with universal grading group $G$, it is shown that the induced grading by the free group $G/\tor(G)$ is a grading by a (not necessarily reduced) root system. Some consequences for the classification of fine gradings on the exceptional simple Lie algebras are drawn.
Graded contractions of the $\mathbb{Z}_2^3$-gradings on the exceptional Lie algebras coming from octonions
本研究对源自八元数的例外李代数（不含$\mathfrak{g}_2$）上的$\mathbb{Z}_2^3$分次做分次收缩，得到860个不同维数的非同构分次李代数，证明其均为通用型，以广义优美集为分类工具，通过多种结构特征区分所得代数。
Gradings on symmetric composition algebras
对任意域上对称复合代数的群分次进行了分类。本文还讨论了这一结果在例外单李代数分次上的应用。
Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field $\mathbb{F}$ (assuming $\mathrm{char} \mathbb{F}\ne 2$ in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type $B_r$ (the answer is just $r+1$), but involves counting orbits of certain finite groups in the case of Series $A$, $C$ and $D$. For $X\in\{A,C,D\}$, we determine the exact number of fine gradings, $N_X(r)$, on the simple Lie algebras of type $X_r…
Computing subalgebras and $\mathbb{Z}_2$-gradings of simple Lie algebras over finite fields
本文提出两种新算法，分别用于构造有限域上李代数的$\mathbb{Z}_2$-分次和计算所有子代数，并应用于$\mathbb{F}_2$上低维单李代数，确定了其分次结构、子代数及单子商。
Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$
本文研究了简单李代数 $E_6$、$E_7$、$E_8$ 上的特殊纯分级，通过二元二次型分类其等价类和同构类。
The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of…
For any grading by an abelian group $G$ on the exceptional simple Lie algebra $\mathcal{L}$ of type $E_6$ or $E_7$ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of $G$-graded simple $\mathcal{L}$-modules, as well as necessary and sufficient conditions for an $\mathcal{L}$-module to admit a $G$-grading compatible with the given $G$-grading on $\mathcal{L}$.
It is known that there are Lie algebras with non-semigroup gradings, i.e. such that the binary operation on the grading set is not associative. We provide a similar example in the class of associative algebras.
We present a constructive approach to torsion-free gradings of Lie algebras. Our main result is the computation of a maximal grading. Given a Lie algebra, using its maximal grading we enumerate all of its torsion-free gradings as well as its positive gradings. As applications, we classify gradings in low dimension, we consider the enumeration of Heintze groups, and we give methods to find bounds for non-vanishing $\ell^{q,p}$ cohomology.
Universal enveloping of a graded Lie algebra
本文构造了非阿贝尔群阶化下的李代数阶化泛包络代数，证明其存在唯一性，推导了Witt定理与Ado定理的阶化版本，并探讨了李阶化与阿贝尔阶化的等价性问题。
In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. We give a different proof that the existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier. Similarly, we prove that given a grading of one these types and a finite group of automorphisms, there always exist a positive grading which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on th…
There are fourteen fine gradings on the exceptional Lie algebra $\frak e_6$ over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that they are all.
The split quartic Cayley algebra is a structurable algebra which has been used to give constructions of Lie algebras of type D4. Here, we calculate its group of automorphisms, its algebra of derivations and its gradings.
Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D4, where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.
We describe the fine (group) gradings on the Heisenberg algebras, on the Heisenberg superalgebras and on the twisted Heisenberg algebras. We compute the Weyl groups of these gradings. Also the results obtained respect to Heisenberg superalgebras are applied to the study of Heisenberg Lie color algebras.
Gradings on the Albert Algebra and on $f_4$
研究了特征为零的代数闭域上的Albert代数和单例外李代数$\frak{f}_4$的群分次。阿尔伯特代数有八个非环非等价的等级（其中三个是好的），$\frak{f}_4$有九个（其中三个也是好的）。
We study the existence of various types of gradings on Lie algebras, such as Carnot gradings or gradings in positive integers, and prove that the existence of such gradings is invariant under extensions of scalars. As an application, we prove that if G is a finitely generated nilpotent group, its systolic growth is asymptotically equivalent to its word growth if and only if the Malcev completion of G is Carnot. We also characterize when G is non-cohopfian, in terms of the existence of a non-trivial grading in non-negative integers, and deduce that this property only depends on its real (or eve…
Short (SL2xSL2)-structures on Lie algebras
本文研究李代数上未被根系分次涵盖的短 (SL2xSL2)-结构，其约化群为最简单的半单非单约化群，并指出此类结构由带非平凡幂等元的Allison J-三元代数坐标化。
English: This work is a doctoral thesis in mathematics by compendium of four articles. Here we explain, using a language as simple as possible, the results achieved in those articles. The general objective is the classification of gradings on simple real Lie algebras. The text is written in both english and spanish. ----- Español: Esta obra es una tesis doctoral en matemáticas por compendio de cuatro artículos. Aquí explicamos, utilizando un lenguaje lo más sencillo posible, los resultados alcanzados en esos artículos. El objetivo general es la clasificación de las graduaciones en las álgebras…
Fine Gradings on the exceptional Lie algebra $\mathfrak d_4$
我们描述了李代数$\mathfrak d_4$上的所有细群分次，直到等价为止。这个问题等价于求$\mathfrak d_4$的自同构群的极大交换可对角化子群。我们用两种不同的观点证明了有十四个。第一种方法是计算的：我们得到一个完整的描述，通过使用一个特定的实现的Dynkin图的$\mathfrak d_4$和一些代数群的东西的自同构群的分级。第二种方法，更定性，强调一些代数方面，如triality，它主要致力于涉及三阶外自同构的分级。
We give a classification up to equivalence of the fine group gradings by abelian groups on the Jordan pairs and triple systems of types bi-Cayley and Albert, under the assumption that the base field is algebraically closed of characteristic different from $2$. The associated Weyl groups are computed. We also determine, for each fine grading on the bi-Cayley and Albert pairs, the induced grading on the exceptional simple Lie algebra given by the Tits-Kantor-Koecher construction.
$|3|-$gradings of complex simple Lie algebras
本文研究复单李代数的 $|3|-$ 阶化结构，完全确定可能的约化代数 $\mathfrak{n}_0$，并证明其负部中唯一出现的三阶自由幂零李代数来自例外李代数 $\mathfrak{g}_2$ 的通常 $|3|-$ 阶化。
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。
A Lax operator algebra is constructed for an arbitrary semi-simple Lie algebra over $\mathbb C$ equipped with a $\mathbb Z$-grading, and arbitrary compact Riemann surface with marked points. In this set-up, a treatment of almost graded structures, and classification of the central extensions of Lax operator algebras are given. A relation to the earlier approach based on the Tyurin parameters is established.
Fine Gradings of Low-Rank Complex Lie Algebras and of Their Real Forms
在这篇评论文章中，我们处理的主题，精细分级的李代数。这个概念不仅对研究代数的结构性质很重要，而且在此之上，精细分次通常被用作研究代数的分次压缩或变形的起点。一个基本的问题解决的工作是术语之间的关系“分级”和“组分级”。虽然这些条款最初声称符合简单李代数，后来发现，证明这一主张是不正确的。因此，关于精细分级和MAD组之间一一对应的关键陈述必须进行修改，并重新制定精细组分级。然而，仍然有一个假设，即“分次”和“群分次”这两个术语在简单复李代数中是一致的。我们使用MAD-群作为寻找复李代数$A_3 \cong D_3$、$B_2 \cong C_2$和$D_2$的精细群分次的主要工具。此外，我们还开发了其他的方法来寻找细（组）分级。它们对于复代数的实形式特别有用，在这方面它们提供了比MAD-群更丰富的结果。系统地使用了三个李代数的$4\times 4$矩阵的忠实表示：$A_3 = sl(4,\mathbb C)$，$C_2 = sp(4,\mathbb C)$，$D_2 = o(4,\mathbb C)$。内含物$sl(4,\mathbb C)\supset sp(4,\mathbb C)$和$sl(4,\mathbb C) \supset o(4,\mathbb C)$在我们的介绍中是很重要的，因为它们允许采用的方法之一，大大简化了计算时，发现精细群分级的子代数$sp(4,\mathb…
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