Consequences of the Grunewald–O’Halloran conjecture for algebras of pseudoquonic operators

Investigating a recent positive solution of a conjecture of Grunewald and O’Halloran for complex finite-dimensional nilpotent Lie algebras, we find results of existence and uniqueness for the construction of complex nilpotent Lie algebras of arbitrary dimension via pseudobosonic operators. We involve the theory of the deformation of Lie algebras of Gerstenhaber, in order to prove our main results. There is not a generalized version of the Grunewald–O’Halloran Conjecture when we consider pseudoquonic operators, which specialize to pseudobosonic operators, so we prove results of uniqueness for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> {C*} -algebras of pseudoquonic operators leveraging on different methods of functional analysis and operator theory.