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In continuation of our article in Colloq. Math. 116.1, we give a complete description of the symmetric algebras of strictly canonical type by quivers and relations, using Brauer quivers. Introduction and the main result. Throughout the article, K will denote a fixed algebraically closed field. By an algebra is meant an associative, finite-dimensional K-algebra with an identity, which we shall assume (without loss of generality) to be basic and indecomposable. For an algebra A, we denote by modA the category of finite-dimensional (over K) right A-modules and by D : modA → modAop the standard duality HomK(−,K). In [21] we describe the structure and homological properties of the Auslander–Reiten quivers of selfinjective algebras of strictly canonical type, that is, the orbit algebras B̂/G, where B̂ is the repetitive algebra of a branch extension (equivalently, branch coextension) B of a canonical algebra (in the sense of Ringel [31]) and G is an infinite cyclic group generated by a strictly positive automorphism of B̂. Here, we are concerned with the classification of the symmetric algebras of strictly canonical type. Recall that an algebra A is called symmetric if A and D(A) are isomorphic as A-A-bimodules. By a classical result due to Nakayama (see [23], [24]), an algebra A is symmetric if and only if there exists a nondegenerate, symmetric, associative, K-bilinear form (−,−) : A×A→ K. The classical examples of symmetric algebras are provided by the group algebras KG of finite groups G, or more generally their blocks. In the representation theory of blocks of group algebras, a prominent role is played by the Brauer tree algebras. Namely, by the deep theorem due to Dade, Janusz and Kupisch (see [1], [13], [18], [19], [20]), every block of finite representation type of a group algebra is Morita equivalent to a Brauer tree algebra. More2000 Mathematics Subject Classification: 16D50, 16G10.

A classification of symmetric algebras of strictly canonical type