Related papers: Group gradings on classical Lie superalgebras

We describe the isomorphism classes of infinite-dimensional graded Lie algebras of maximal class, generated by elements of weight one, over fields of odd characteristic.

Superderivations for the eight families of finite or infinite dimensional graded Lie superalgebras of Cartan-type over a field of characteristic $p>3$ are completely determined by a uniform approach: The infinite dimensional case is reduced…

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial…

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.

Let $\mathbb F$ be an algebraically closed field, $G$ be an abelian group, and let $A$ and $B$ be arbitrary finite-dimensional $G$-graded simple algebras over $\mathbb F$. We prove that $A$ and $B$ are isomorphic if, and only if, they…

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all…

For each $n\ge2$ we classify all $n$-dimensional algebras over an arbitrary infinite field which have the property that the $n$-dimensional abelian Lie algebra is their only proper degeneration.

We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard…

For sufficiently high dimensions, the naturally graded nonsplit nilpotent Lie algebras with linear characteristic sequence are classified.

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.

We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are…

We describe the isomorphism classes of certain infinite-dimensional graded Lie algebras of maximal class, generated by an element of weight one and an element of weight two, over fields of odd characteristic.

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…

Simple Lie algebras of finite dimension over an algebraically closed field of characteristic 0 or $p> 3$ were recently classified. However, the problem over an algebraically closed field of characteristics 2 or 3 there exist only partial…

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…

Decomposition classes provide a way of partitioning the Lie algebras of an algebraic group into equivalence classes based on the Jordan decomposition. In this paper, we investigate the decomposition classes of the Lie algebras of connected…

In this expository article, we describe the classification of the subalgebras of the rank 2 semisimple Lie algebras. Their semisimple subalgebras are well-known, and in a recent series of papers, we completed the classification of the…

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.

We illustrate some simple ideas that can be used for obtaining a classification of small-dimensional solvable Lie algebras.Using these we obtain the classification of 3 and 4 dimensional solvable Lie algebras (over fields of any…

Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a finite abelian group. If $A=\oplus_{g\in G} A_g$ is a $G$-graded algebra, we study degree-inverting involutions on $A$, i.e., involutions $*$ on $A$ satisfying…