Pseudo-Euclidean Novikov Superalgebras: Structure and Properties

TL;DR

This paper studies the structure of pseudo-Euclidean Novikov superalgebras, introduces Milnor superalgebras, and provides classification results for low-dimensional cases.

Contribution

It introduces Milnor superalgebras, develops a double extension method, and classifies low-dimensional pseudo-Euclidean Novikov superalgebras.

Findings

Any such superalgebra with a non-degenerate ideal is a Milnor superalgebra.

Every superalgebra can be constructed via double extensions from a Milnor superalgebra.

Complete classification of superalgebras of dimension at most four.

Abstract

A pseudo-Euclidean Novikov superalgebra $A$ is a Novikov superalgebra endowed with a non-degenerate symmetric bilinear form $\langle,\rangle$ such that all left multiplication operators are $\langle,\rangle$-antisymmetric. In this case, the associated Lie superalgebra $(A^{-},$\langle,\rangle$)$ is a flat pseudo-Euclidean Lie superalgebra. In this paper, we investigate the structure of pseudo-Euclidean Novikov superalgebras. In particular, we introduce a distinguished subclass, called Milnor superalgebras, and prove that any pseudo-Euclidean Novikov superalgebra whose two-sided ideal is non-degenerate belongs to this class. We provide a method for constructing pseudo-Euclidean Novikov superalgebras. We also introduce a double extension procedure for pseudo-Euclidean Novikov superalgebras and show that every such superalgebra with a degenerate two-sided ideal can be obtained via this method. Furthermore, we establish that any pseudo-Euclidean Novikov superalgebra is either a Milnor superalgebra or can be obtained by a sequence of double extensions starting from a Milnor superalgebra. As an application, we provide a complete classification of pseudo-Euclidean Novikov superalgebras of total dimension at most four.