Balanced Deep Matrix Algebras

This paper continues the study of associative and Lie deep matrix algebras, DM(X,mathbbK){mathcal{DM}}(X,{mathbb{K}}) and mathfrakgld(X,mathbbK){mathfrak{gld}}(X,{mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X,mathbbK){mathcal{BDM}}(X,{mathbb{K}}) and mathfrakbld(X,mathbbK){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, mathfrakbld(X,mathbbK){mathfrak{bld}}(X,{mathbb{K}}) is shown to be semisimple. The Lie algebra mathfrakbld(X,mathbbK){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 mathfraksl_n{mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on mathfraksl₂mathfrakd{mathfrak{sl}_2mathfrak{d}} (a natural depth analogue of mathfraksl₂{mathfrak{{sl}_2}}) and mathfrakbld{mathfrak{bld}}.