Introduction

A basic algebra is a finite dimensional algebra A over a field, all of whose simple modules have dimension one. In the literature such an algebra is known as a "split" basic algebra. Every algebra is Morita equivalent to a basic algebra, though a field extension may be necessary to obtain the split basic algebra. Magma has several functions that create the basic algebras corresponding to algebras of different types.

The type AlgBas in Magma is optimized for the purposes of doing homological calculations. A basic algebra A is generated by elements a1, a2, ..., at where a1, ..., as are the primitive idempotent generators and as + 1, ..., at are the nonidempotent generators. Each nonidempotent generator, ak must have the property that ai * ak * aj = ak for specific idempotent generators ai and aj. The projective indecomposable modules have the form Pi = ai .A for i = 1, ..., s and the simple modules have the form Si = Pi/Rad(Pi), where Rad(Pi) is the radical of Pi.

V2.28, 13 July 2023