Skew-adjoint maps and quadratic Lie algebras

Abstract

The procedure of double extension of vector spaces endowed with non-degenerate bilinear forms allows us to introduce the class of generalized $\mbK$-oscillator algebras over any arbitrary field $\mbK$. Starting from basic structural properties of such algebras and the canonical forms of skew-adjoint endomorphisms, we will proceed to classify the subclass of quadratic nilpotent algebras and characterize those algebras in the class with quadratic dimension 2. This will enable us to recover the well-known classification of real oscillator algebras, also known as Lorentzian algebras, given by Alberto Medina in 1985.

arXiv comment24 pages

References in corpus (5)

• Classification problems for system of forms and linear mappings
• On the structure of symmetric self-dual Lie algebras
• Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras
• Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form
• Metrics related to Oscillator algebras