MAGMA Computational Algebra System

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. The type in Magma is optimized for the purposes of doing homological calculations. The 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 are of 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.