Balanced Deep Matrix Algebras

Abstract

This paper continues the study of associative and Lie deep matrix algebras, \({\mathcal{DM}}(X,{\mathbb{K}})\) and \({\mathfrak{gld}}(X,{\mathbb{K}})\), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, \({\mathcal{BDM}}(X,{\mathbb{K}})\) and \({\mathfrak{bld}}(X,{\mathbb{K}})\), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, \({\mathfrak{bld}}(X,{\mathbb{K}})\) is shown to be semisimple. The Lie algebra \({\mathfrak{bld}}(X,{\mathbb{K}})\) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of \({\mathfrak{{sl}_n}}\)’s. We classify all associative bilinear forms on \({\mathfrak{sl}_2\mathfrak{d}}\) (a natural depth analogue of \({\mathfrak{{sl}_2}}\)) and \({\mathfrak{bld}}\).