Intrinsic functions on real semisimple algebras

Realizations of real semisimple low-dimensional Lie algebras Realizations of real semisimple low-dimensional Lie algebras

A complete set of inequivalent realizations of threeand four-dimensional real unsolvable Lie algebras in vector fields on a space of an arbitrary (finite) number of variables is obtained. Representations of Lie algebras by vector fields are widely applicable e.g. in integrating of ordinary differential equations, group classification of partial differential equations, the theory of differential...

Notes on semisimple algebras

(1.5) Proposition Let R be a semisimple ring. Then R is isomorphic to a finite direct product ∏s i=1 Ri, where each Ri is a simple ring. (1.6) Proposition Let R be a simple ring. Then there exists a division ring D and a positive integer n such that R ∼= Mn(D). (1.7) Definition Let R be a ring with 1. Define the radical of R to be the intersection of all maximal left ideals of R. The above defi...

Small Semisimple Subalgebras of Semisimple Lie Algebras

The goal of Section 2 is to provide a proof of Theorem 2.0.1. Section 3 introduces the necessary facts about Lie algebras and representation theory, with the goal being the proof of Proposition 3.5.7 (ultimately as an application of Theorem 2.0.1), and Proposition 3.3.1. In Section 4 we prove the main theorem, using Propositions 3.3.1 and 3.5.7. In Section 5, we apply the theorem to the special...

Real Representations of Semisimple Lie Algebras Have Q-forms

We prove tha t each real semisimple Lie algebra g has a Q-form go, such that every real representation of go can be realized over Q. This was previously proved by M. S. Raghunathan (and rediscovered by P. Eberlein) in the special case where g is compact.

Almost semisimple algebras