Gradings by Groups on Melikyan Algebras

Gradings on the Lie algebra $D_4$ revisited

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).

Gradings on simple algebras of finitary matrices

We describe gradings by finite abelian groups on the associative algebras of infinite matrices with finitely many nonzero entries, over an algebraically closed field of characteristic zero.

Group gradings on filiform Lie algebras

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.

Jordan gradings on exceptional simple Lie algebras

Models of all the gradings on the exceptional simple Lie algebras induced by Jordan subgroups of their groups of automorphisms are provided.

Non-semigroup gradings of associative algebras

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.

Gradings on semisimple algebras

The classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field is reduced, by means of the use of loop algebras, to the corresponding problem for simple algebras. This requires a good definition of (free) products of group-gradings.

Structural matrix algebras, generalized flags and gradings

We show that a structural matrix algebra $A$ is isomorphic to the endomorphism algebra of an algebraic-combinatorial object called a generalized flag. If the flag is equipped with a group grading, an algebra grading is induced on $A$. We classify the gradings obtained in this way as the orbits of the action of a double semidirect product on a certain set. Under some conditions on the associated graph, all good gradings on $A$ are of this type. As a bi-product, we obtain a new approach to compute the automorphism group of a structural matrix algebra.

Automorphisms, derivations and gradings of the split quartic Cayley algebra

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.

代数上的几乎精细分次与分次同构分类

Almost fine gradings on algebras and classification of gradings up to isomorphism

本文针对有限维代数上的群分次，引入几乎精细分次类，使得每个分次可本质上唯一地由其导出，并给出阿贝尔群情形下的构造方法，应用于半单李代数，关联根系并构造适应分次。

Gradings and Symmetries on Heisenberg type algebras

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 associative algebras with involution and real forms of classical simple Lie algebras

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.

Gradings on the algebra of triangular matrices as a Lie algebra: revisited

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…

Gradings on the Albert Algebra and on $f_4$

We study group gradings on the Albert algebra and on the simple exceptional Lie algebra $\frak{f}_4$ over algebraically closed fields of characteristic zero. There are eight nontoral nonequivalent gradings on the Albert algebra (three of them being fine) and nine on $\frak{f}_4$ (also three of them fine).

Group Gradings on Simple Lie Algebras of Type "A"

In this paper we describe all group gradings by a finite abelian group G of any Lie algebra L of the type "A" over algebraically closed field F of characteristic zero.

Classification of division gradings on finite-dimensional simple real algebras

We classify, up to isomorphism and up to equivalence, division gradings (by abelian groups) on finite-dimensional simple real algebras. Gradings on finite-dimensional simple algebras are determined by division gradings, so our results give the classification, up to isomorphism, of (not necessarily division) gradings on such algebras. Linear algebra over the field of two elements plays an interesting role in the proofs.

Gradings on the simple real Lie algebras of types $G_2$ and $D_4$

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.

Fine gradings on simple classical Lie algebras

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.

Gradings by Groups on Graded Cartan Type Lie Algebras

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.

Gradings on algebras over algebraically closed fields

The classification, both up to isomorphism or up to equivalence, of the gradings on a finite dimensional nonassociative algebra A over an algebraically closed field F, such that its group scheme of automorphisms is smooth, is shown to be equivalent to the corresponding problem for the scalar extension A_K for any algebraically closed field extension K.

Group gradings on simple Lie algebras of type A in positive characteristic

In this paper we consider gradings by a finite abelian group $G$ on the Lie algebra $\mathfrak{sl}_n(F)$ over an algebraically closed field $F$ of characteristic different from 2 and not dividing $n$.

Group gradings on the Jordan algebra of upper triangular matrices

Let G be an arbitrary group and let K be a field of characteristic different from 2. We classify the G-gradings on the Jordan algebra of upper triangular matrices of order n over K. It turns out that there are, up to a graded isomorphism, two families of gradings: the elementary gradings (analogous to the ones in the associative case), and the so called mirror type (MT) gradings. Moreover we prove that the G-gradings on this algebra are uniquely determined, up to a graded isomorphism, by the graded identities they satisfy.

Fine gradings on the simple Lie algebras of type $E$

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.

Arithmetically-free group-gradings of Lie algebras: II

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.

Gradings on the exceptional Lie algebras $F_4$ and $G_2$ revisited

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 $F_4$.

Arithmetically-free group-gradings of Lie algebras

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.

Gradings on classical central simple real Lie algebras

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).

可三角化代数上的群分次

Group gradings on triangularizable algebras

本研究旨在对Mesyan在2019年定义的既非单也非有限维的可三角化代数上的群分次进行同构类分类，揭示了无限维代数拓扑与幂等元在理解代数结构中的关键作用。

Abelian gradings in Lie algebras

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.

Group gradings on finitary simple Lie algebras

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.

On the Graded Identities for Elementary Gradings in Matrix Algebras over Infinite Fields

We find a basis for the $G$-graded identities of the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of characteristic $p>0$ with an elementary grading such that the neutral component corresponds to the diagonal of $M_n(K)$.

Group gradings on the Lie and Jordan algebras of block-triangular matrices

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 on tensor products of composition algebras and on the Smirnov algebra

We give classifications of group gradings, up to equivalence and up to isomorphism, on the tensor product of a Cayley algebra $\mathcal{C}$ and a Hurwitz algebra over a field of characteristic different from 2. We also prove that the automorphism group schemes of $\mathcal{C}^{\otimes n}$ and $\mathcal{C}^n$ are isomorphic. On the other hand, we prove that the automorphism group schemes of a Smirnov algebra (a $35$-dimensional simple exceptional structurable algebra constructed from a Cayley algebra $\mathcal{C}$) and $\mathcal{C}$ are isomorphic. This is used to obtain classifications, up to …

有限维关联代数上的群分次

Group gradings on finite dimensional incidence algebras

本文在特定域条件下分类了有限维关联代数的群分次，研究了阿贝尔群下的分次双模结构，得到更深刻的分次结构结果并分类了分次同构类。

结合代数上的精细分次与自同构群

Fine gradings and automorphism groups on associative algebras

本文研究结合代数的精细分次与自同构群，给出了分次精细的充要条件，并计算了相关自同构群，特别在实数域上有限维单代数情形下给出具体结果。

$\mathbb{Z}$-gradings of full support on the Grassmann algebra

Let $E$ be the infinite dimensional Grassmann algebra over a field $F$ of characteristic zero. In this paper we investigate the structures of $\mathbb{Z}$-gradings on $E$ of full support. Using methods of elementary number theory, we describe the $\mathbb{Z}$-graded polynomial identities for the so-called $2$-induced $\mathbb{Z}$-gradings on $E$ of full support. As a consequence of this fact we provide examples of $\mathbb{Z}$-gradings on $E$ which are PI-equivalent but not $\mathbb{Z}$-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent and no…

Weyl groups of fine gradings on simple Lie algebras of types A, B, C and D

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…

An overview of fine gradings on simple Lie algebras

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".

Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra

Given a grading $Γ: A=\oplus_{g\in G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a subspace) and the automorphisms that permute the components. By the Weyl group of $Γ$ 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 comp…

Group gradings on finite dimensional Lie algebras

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 $G$.

Non-group gradings on 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.

On the Number of Gradings on Matrix Algebras

We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$ of isomorphism classes of elementary $G$-gradings on the algebra $M_{m}(\mathbb{F})$ of $m\times m$ matrices with entries in a field $\mathbb{F}$ characterizes $G$. A formula for the number of isomorphism classes of gradings by a finite abelian group on an algebra of upper block-triangular matrices over an algebraically closed field, with mild restrictions on…

Classification of group gradings on simple Lie algebras of types A, B, C and D

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.

On group gradings on PI-algebras

We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] \leq exp(W)^K$. A G-grading $W = \bigoplus_{g \in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} \neq 0$ for any $r \geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$.

$2$-Graded Identities for the Tensor Square of the Grassmann Algebra

We consider the algebra $E\otimes E$ over an infinite field equipped with a $\mathbb{Z}_2$-grading where the canonical basis is homogeneous and prove that in various cases the graded identites are just the ordinary ones. If the grading is a non-canonical grading obtained as a quotient grading of the natural $\mathbb{Z}_2\times\mathbb{Z}_2$-grading we exhibit a basis for the graded identities.

Gradings of non-graded Hamiltonian Lie algebras

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras $H(2\colon\n;ω_2)$ (of dimension one less than a power of $p$) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of $H(2\colon\n;ω_2)$ with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).

Group gradings on restricted Cartan type Lie algebras

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.

Gradings on Incidence Algebras and their Graded Polynomial Identities

Let P a locally finite partially ordered set, F a field, G a group, and I(P,F) the incidence algebra of P over F. We describe all the inequivalent elementary G-gradings on this algebra. If P is bounded, F is a infinite field of characteristic zero, and A, B are both elementary G-graded incidence algebras satisfying the same G-graded polynomial identities, and the automorphisms group of P acts transitively on the maximal chains of P , we show that A and B are graded isomorphic.

Grading switching for modular non-associative algebras

We describe a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. This is inspired by a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. We trace the development of grading switching, from an early version based on taking the Artin-Hasse exponential of a nilpotent derivation, to a more general version which uses certain generalized Laguerre polynomials playing the role of general…

Fine gradings on Kantor systems of Hurwitz type

We give a classification up to equivalence of the fine group gradings by abelian groups on the Kantor pairs and triple systems associated to Hurwitz algebras (i.e., unital composition algebras), under the assumption that the base field is algebraically closed of characteristic different from 2. The universal groups and associated Weyl groups are computed. We also determine, in the case of Kantor pairs, the induced (fine) gradings on the associated Lie algebras given by the Kantor construction.

Graded Identities and Isomorphisms on Algebras of Upper Block-Triangular Matrices

Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\o…

Graded identities of simple real graded division algebras

Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed fie…

Fine gradings on $\mathfrak{e}_6$

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.

More non semigroup Lie gradings

This note is devoted to the construction of two very easy examples, of respective dimensions 4 and 6, of graded Lie algebras whose grading is not given by a semigroup, the latter one being a semisimple algebra. It is shown that 4 is the minimal possible dimension.

分次的自同构群概形与Weyl群

Automorphism group schemes and Weyl groups of gradings

本文提出了代数上分次的自同构群的概形理论版本，并将经典结果（特征0代数闭域上分次的自同构群是其对角群的正规化子）推广到任意域上的概形设定中，同时证明了自同构群与分次稳定子的商构成常值群概形（称为分次的Weyl群概形），但在任意域上该常值群概形不一定对应于普通Weyl群。