# 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.

Definition. The algebra $A$ is called (right) basic if $P_{i}$ is not isomorphic to $P_{j}$ when $i\neq j$.

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]):

Proposition.

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

• 1 I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras, vol 1., Cambridge University Press 2006, 2007
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Type of Math Object:
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## Mathematics Subject Classification

13B99 None of the above, but in MSC2010 section 13Bxx
20C99 None of the above, but in MSC2010 section 20Cxx
16S99 None of the above, but in MSC2010 section 16Sxx

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Owner: joking
Added: 2011-02-22 - 04:52
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