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, ..., a_t where a₁, ..., a_s are the primitive idempotent generators and a_{s + 1}, ..., a_t are the nonidempotent generators. Each nonidempotent generator, a_k must have the property that a_i * a_k * a_j = a_k for specific idempotent generators a_i and a_j. The projective indecomposable modules have the form P_i = a_i .A for i = 1, ..., s and the simple modules have the form S_i = P_i/Rad(P_i), where Rad(P_i) is the radical of P_i. [Next][Prev] [Right] [____] [Up] [Index] [Root]