Fine Gradings on Exceptional Simple Lie Superalgebras

Abstract

The fine abelian group gradings on the simple exceptional classical Lie superalgebras over algebraically closed fields of characteristic 0 are determined up to equivalence.

On the Structure of Graded Lie Triple Systems

“…For instance, in the case of Lie algebras, we can cite many works related to theory of strings, to color supergravity, to Walsh functions, to electroweak interactions or to gauge models [1,4,9,10,17,18,20,22,25,29,34]. In the case of Lie superalgebras, we can also cite several works modelling continuous suppersymmetry transformations between bosons and fermions or conformal field theory [3,5,19,26,31]. Finally, as it is pointed out in [24], we note that Lie triple systems are well related to the theory of Quantum Mechanics with P T -symmetric Hamiltonians and Krein space-related models in general, by identifying this underlying structure in the recognizing of P T -like involutory structures in physical models.…”

Gradings and symmetries on Heisenberg type algebras

“…The fine group gradings on the real forms of g 2 and f 4 , also simple algebras, have been classified by the three first authors [11]. Gradings have also been considered in certain Z 2 -graded structures (for instance superalgebras), so the three first authors have studied the case of the Jordan superalgebra K 10 (see [12]), and the second and third authors together with Elduque have classified the fine gradings on exceptional Lie superalgebras in [18]. In relation with other Lie structures, the Lie triple systems of exceptional type have also been considered from the viewpoint of gradings (see [13]).…”

“…, a t l } a block B II l 2 (u, α t ) for each t ≤ r, chosen as in Eq. (18). Now the union of these blocks as in Eq.…”

Some properties of the family Γ of modular Lie superalgebras

“…Many important results for modular Lie superalgebras have been obtained (see [2], [4], [13], [14], [22]- [33]). The study of graded Lie superalgebras also have got several deep results in recent years (see [3], [5]). This paper is devoted to investigating the filtration structures of the family Γ of modular Lie superalgebras by the method of minimal dimension of image spaces and then some properties are discussed.…”