Three Ideals of Lie Superalgebras

We define perfect ideals, near perfect ideals and upper bounded ideals of a finite-dimensional Lie superalgebra, and study the properties of these three kinds of ideals through their relevant sequences. We prove that a Lie superalgebra is solvable if and only if its maximal perfect ideal is zero, or its quotient superalgebra by the maximal perfect ideal is solvable. We also show that a Lie superalgebra is nilpotent if and only if its maximal near perfect ideal is zero. Moreover, we prove that a nilpotent Lie superalgebra has only one upper bounded ideal, which is the nilpotent Lie superalgebra itself.

Holomorphic integer graded vertex superalgebras

In this paper, we study holomorphic [Formula: see text]-graded vertex superalgebras. We prove that all such vertex superalgebras of central charge [Formula: see text] and [Formula: see text] are purely even. For the case of central charge [Formula: see text] we prove that the weight-one Lie superalgebra is either zero, of superdimension [Formula: see text], or else is one of an explicit list of 1332 semisimple Lie superalgebras.

The ℤ2 × ℤ2-graded Lie superalgebras pso(2n + 1|2n) and pso(∞|∞), and parastatistics Fock spaces

The parastatistics Fock spaces of order p corresponding to an infinite number of parafermions and parabosons with relative paraboson relations are constructed. The Fock spaces are lowest weight representations of the ℤ2 × ℤ2-graded Lie superalgebra pso(∞|∞), with a basis consisting of row-stable Gelfand-Zetlin patterns.

Tensor Hierarchy Algebra Extensions of Over-Extended Kac–Moody Algebras

Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental rôle they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac–Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac–Moody algebra by a Virasoro derivation $$L_1$$ L 1 . A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.