basic algebra

Let A be a finite dimensional, unital algebra over a field k. By Krull-Schmidt Theorem A can be decomposed as a (right) A-module as follows:


where each Pi is an indecomposable moduleMathworldPlanetmath and this decomposition is unique.

Definition. The algebraMathworldPlanetmathPlanetmath A is called (right) basic if Pi is not isomorphicPlanetmathPlanetmathPlanetmath to Pj when ij.

Of course we may easily define what does it mean for algebra to be left basic. Fortunetly these properties coincide. Let as state some known facts (originally can be found in [1]):


  1. 1.

    A finite algebra A over a field k is basic if and only if the algebra A/radA is isomorphic to a productMathworldPlanetmathPlanetmathPlanetmath of fields k××k. Thus A is right basic iff it is left basic;

  2. 2.

    Every simple module over a basic algebra is one-dimensional;

  3. 3.

    For any finite-dimensional, unital algebra A over k there exists finite-dimensional, unital, basic algebra B over k such that the categoryMathworldPlanetmath of finite-dimensional modules over A is k-linear equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the category of finite-dimensional modules over B;

  4. 4.

    Let A be a finite-dimensional, basic and connected (i.e. cannot be written as a product of nontrivial algebras) algebra over a field k. Then there exists a bound quiver (Q,I) such that AkQ/I;

  5. 5.

    If (Q,I) is a bound quiver over a field k, then both kQ and kQ/I are basic algebras.


Title basic algebra
Canonical name BasicAlgebra
Date of creation 2013-03-22 19:17:10
Last modified on 2013-03-22 19:17:10
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Definition
Classification msc 13B99
Classification msc 20C99
Classification msc 16S99