Maximal subalgebras of finite-dimensional algebras

We study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

If [Formula: see text] is a semisimple Lie algebra of vector fields on [Formula: see text] with a split Cartan subalgebra [Formula: see text], then it is proved here that the dimension of the generic orbit of [Formula: see text] coincides with the dimension of [Formula: see text]. As a consequence one obtains a local canonical form of [Formula: see text] in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates — for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on [Formula: see text] and to determine all representations of [Formula: see text] as local vector fields on [Formula: see text]. These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits [Formula: see text] as its symmetry algebra and for a system of two second-order ordinary differential equations that admits [Formula: see text] as its symmetry algebra.