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:

 $A\simeq P_{1}\oplus\cdots\oplus P_{k}$

where each $P_{i}$ is an indecomposable module  and this decomposition is unique.

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.

A finite algebra $A$ over a field $k$ is basic if and only if the algebra $A/\mathrm{rad}A$ is isomorphic to a product    of fields $k\times\cdots\times k$. 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 category  of finite-dimensional modules over $A$ is $k$-linear equivalent     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 $A\simeq kQ/I$;

5. 5.

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

References

Title basic algebra BasicAlgebra 2013-03-22 19:17:10 2013-03-22 19:17:10 joking (16130) joking (16130) 5 joking (16130) Definition msc 13B99 msc 20C99 msc 16S99