Singular quadratic Lie superalgebras

Minh Thanh Duong (1) , Rosane Ushirobira (2, 3)

1 Ho Chi Minh City University of Pedagogy
2 NON-A - Non-Asymptotic estimation for online systems
3 IMB - Institut de Mathématiques de Bourgogne [Dijon]

In this paper, we generalize some results on quadratic Lie algebras to quadratic Lie superalgebras, by applying graded Lie algebras tools. We establish a one-to-one correspondence between non-Abelian quadratic Lie superalgebra structures and nonzero even super-antisymmetric 3-forms satisfying a structure equation. An invariant number of quadratic Lie superalgebras is then defined, called the dup-number. Singular quadratic Lie superalgebras (i.e. those with nonzero dup-number) are studied. We show that their classification follows the classifications of O(m)-adjoint orbits of o(m) and Sp(2n)-adjoint orbits of sp(2n). An explicit formula for the quadratic dimension of singular quadratic Lie superalgebras is also provided. Finally, we discuss a class of 2-nilpotent quadratic Lie superalgebras associated to a particular super-antisymmetric 3-form.

Mots clés: 17B30; 17B05; Adjoint orbits 2000 MSC: 15A63; Invariant; 17B70; Double extensions; Super Poisson bracket; Quadratic Lie superalgebras; Generalized double extensions

Domaines: Théorie des représentations [math.RT]

Minh Thanh Duong, Rosane Ushirobira. Singular quadratic Lie superalgebras. Journal of Algebra, 2014, 407, pp.372 - 412. ⟨10.1016/j.jalgebra.2014.02.034⟩. ⟨hal-01114188⟩