Related papers: Symmetries on the real form $\mathfrak e_{6,-26}$

The Z_2^n gradings of the classical Lie algebras are described. To elucidate the grading, the classical Lie algebras are expressed in terms of matrix algebras over one of eight fields or Clifford algebras which carry gradings ranging from…

The group gradings on the symmetric composition algebras over arbitrary fields are classified. Applications of this result to gradings on the exceptional simple Lie algebras are considered too.

For commutative rings, we introduce the notion of a {\em universal grading}, which can be viewed as the "largest possible grading". While not every commutative ring (or order) has a universal grading, we prove that every {\em reduced order}…

Automorphisms of order $2$ are studied in order to understand generalized symmetric spaces. The groups of type $E_6$ we consider here can be realized as both the group of linear maps that leave a certain determinant invariant, and also as…

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…

We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.

We classify all integrable complex structures on 6-dimensional Lie algebras of the form $\mathfrak{g}\times\mathfrak{g}$.

A new combinatorial object, called generalised nice set, is classified up to collineations of the Fano plane. This classification is necessary to find the graded contractions of all the exceptional complex Lie algebras of dimension at least…

In this paper we investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on a simple Lie algebras of type $E_6$, $F_4$ and $G_2$. The methods for classifying the orbits for…

We give explicit structure of the graded ring of modular forms with respect to Gamma(N) (N=1,2,3,4,5,6,7,8,9,10,12,16,18) and for some other congruence groups. We also study the modular forms of half-integer weight for certain groups.

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…

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…

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 consider the grading of $sl(n,\mathbb{C})$ by the group $\Pi_n$ of generalized Pauli matrices. The grading decomposes the Lie algebra into $n^2-1$ one--dimensional subspaces. In the article we demonstrate that the normalizer of grading…

We classify gradings on matrix algebras by a finite abelian group. A grading is called good if all elementary matrices are homogeneous. For cyclic groups, all gradings on a matrix algebra over an algebraically closed field are good. We can…

Consider the smooth quadric Q_6 in P^7. The middle homology group H_6(Q_6,Z) is two-dimensional with a basis given by two classes of linear subspaces. We classify all threefolds of bidegree (1,p) inside Q_6.

A new structure, based on joining copies of a group by means of a \emph{twist}, has recently been considered to describe the brackets of the two exceptional real Lie algebras of type $G_2$ in a highly symmetric way. In this work we show…

We classify, up to isomorphism, the group gradings on the non-exceptional classical simple Lie superalgebras, except for type A(1,1), over an algebraically closed field of characteristic zero. To this end, we study graded-simple and…

Consider a Leibniz superalgebra $\mathfrak L$ additionally graded by an arbitrary set $I$ (set grading). We show that $\mathfrak L$ decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case…

We classify up to equivalence the gradings on Hurwitz superalgebras and on symmetric composition superalgebras, over any field. Also, classifications up to isomorphism are given in case the field is algebraically closed. By grading, here we…