Highest weight theory for finite-dimensional graded algebras with triangular decomposition

Finite dimensional graded simple algebras

Let R be a finite-dimensional algebra over an algebraically closed field F graded by an arbitrary group G. We prove that R is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite subgroup of G. If the characteristic of F is zero or char F does not divide the order of any finite subgroup of G then we prove that R is graded simple if and only if it i...

Graded triangular algebras

Ela Graded Triangular Algebras

The structure of graded triangular algebras T of arbitrary dimension are studied in this paper. This is motivated in part for the important role that triangular algebras play in the study of oriented graphs, upper triangular matrix algebras or nest algebras. It is shown that T decomposes as T = U + ( ∑ i∈I Ti), where U is an R-submodule contained in the 0-homogeneous component and any Ti a well...

Highest-weight Theory for Truncated Current Lie Algebras

Let g be a Lie algebra over a field k of characteristic zero, and a fix positive integer N. The Lie algebra ĝ = g ⊗k k[t]/t N+1 k[t] is called a truncated current Lie algebra. In this paper a highest-weight theory for ĝ is developed when the underlying Lie algebra g possesses a triangular decomposition. The principal result is the reducibility criterion for the Verma modules of ĝ for a wide cla...

Algebras with Locally Finite Decomposition

We introduce the notion of locally finite decomposition rank, a structural property shared by many stably finite nuclear C∗-algebras. The concept is particularly relevant for Elliott’s program to classify nuclear C∗algebras by K-theory data. We study some of its properties and show that a simple unital C∗-algebra, which has locally finite decomposition rank, real rank zero and which absorbs the...