Some Consequences of the Grunewald-O'Halloran Conjecture for Pseudoquonic Operators

TL;DR

This paper explores the implications of the Grunewald-O'Halloran Conjecture for constructing complex nilpotent Lie algebras using pseudobosonic operators, revealing existence results but leaving open questions on uniqueness.

Contribution

It demonstrates the existence and provides a direct construction of pseudobosonic $O^*$-algebras, extending the conjecture's implications to these operators.

Findings

Existence of pseudobosonic $O^*$-algebras established

Construction method for these algebras provided

Uniqueness of the construction remains unresolved

Abstract

Investigating a recent positive solution of a conjecture of Grunewald and O'Halloran for complex finite dimensional nilpotent Lie algebras, we are in the position to find results of existence and uniqueness for the construction of complex nilpotent Lie algebras of arbitrary dimension via pseudobosonic operators. We involve the so-called theory of the deformation of Lie algebras of Gerstenhaber, in order to prove our main results. There isn't a generalized version of the Grunewald-O'Halloran Conjecture when we consider pseudoquonic operators, which specialize to pseudobosonic operators in many cirumstances. Therefore we prove a result of existence (and a direct construction) of pseudobosonic $O^*$-algebras of operators, but leave open the problem of the uniqueness of the construction.