The Structure of Biserial Algebras

By an algebra Λ we mean an associative k ‐algebra with identity, where k is an algebraically closed field. All algebras are assumed to be finite dimensional over k (except the path algebra kQ ). An algebra is said to be biserial if every indecomposable projective left or right Λ‐module P contains uniserial submodules U and V such that U + V =Rad( P ) and U ∩ V is either zero or simple. (Recall that a module is uniserial if it has a unique composition series, and the radical Rad( M ) of a module M is the intersection of its maximal submodules.) Biserial algebras arose as a natural generalization of Nakayama's generalized uniserial algebras [ 2 ]. The condition first appeared in the work of Tachikawa [ 6 , Proposition 2.7], and it was formalized by Fuller [ 1 ]. Examples include blocks of group algebras with cyclic defect group; finite dimensional quotients of the algebras (1)–(4) and (7)–(9) in Ringel's list of tame local algebras [ 4 ]; the special biserial algebras of [ 5 , 8 ] and the regularly biserial algebras of [ 3 ]. An algebra Λ is basic if Λ/Rad(Λ) is a product of copies of k . This paper contains a natural alternative characterization of basic biserial algebras, the concept of a bisected presentation. Using this characterization we can prove a number of results about biserial algebras which were inaccessible before. In particular we can describe basic biserial algebras by means of quivers with relations.