On minimal symplectic alternating algebras

TL;DR

This paper investigates minimal symplectic alternating algebras, focusing on their structure and class, confirming a conjecture for algebras up to dimension 16 with specific minimal properties.

Contribution

It studies the dual subclass of nilpotent symplectic algebras of minimal class, confirming a conjecture for low-dimensional cases.

Findings

Minimal symplectic alternating algebras of dimension up to 16 have class confirming the conjecture.

The paper characterizes the structure of these minimal algebras.

Results support the conjecture for algebras of rank 2 with minimal nilpotency class.

Abstract

The structure of nilpotent symplectic algebras of maximal class has been studied in [8, 5]. In this paper, we study the dual subclass of algebras of minimal class. In particular, we show that symplectic alternating algebras of dimension up to $16$ that are minimal, in the sense that they are of rank $2$ with minimum nilpotency class, have a class that confirm a conjecture that has been raised in [3].