Semisimple Triangular AF Algebras

The Topology of Ideals in Some Triangular Af Algebras

A set of meet-irreducible ideals is described for a class of maximal triangular almost finite algebras. This set forms a topological space under the hull-kernel closure, and there is a one-to-one correspondence between closed sets in this space and ideals in the AF algebra. Some connections with nest representations and nestprimitive ideals are also described.

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...

Almost semisimple algebras

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...

Representations of Semisimple Lie Algebras

Let L be a finite-dimensional, semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let H be a fixed Cartan subalgebra of L, and Φ be the root system. Fix a base ∆ = {α1, · · · , αl} of Φ. Let Λ denote the set of dominant, integral linear functions on H. Theorem 0.1. There is a one-to-one correspondence Λ ∼ −→ {isomorphism classes of finite-dimensional irreducible L-...