In this talk we will define a Brauer configuration and a Brauer configuration algebra, and also we will compute the dimension of one of the invariants of a Brauer configuration algebra: the dimension of its center. Brauer configuration algebras were introduced by Green and Schroll in 2015 and it is expected that the Brauer configuration also will encode the representation theory of Brauer configuration algebras. As the Brauer graph algebras, the Brauer configuration algebras have additional structure arising from combinatorial data, called a Brauer configuration. Brauer configuration algebras are a generalization of Brauer graph algebras, in the sense that every Brauer graph is a Brauer configuration and every Brauer graph algebra is a Brauer configuration algebra. One of the principal features of a Brauer graph algebra is that the associated Brauer graph encodes its representation theory. In 2015 Schroll showed that the class of symmetric special biserial algebras coincides with the class of Brauer graph algebras. This class is that of the symmetric special biserial algebras. Whitin the vast class of biserial algebras there is a particular one that has the feature that its representation theory is largely controlled by the uniserial modules. It was until 1961 when Tachikawa gave the defining property of biserial algebras, but was Fuller who coined the name of “biserial algebra” in 1979. Serial algebras were introduced by Nakayama in 1941 and they were the first non-semisimple algebras with only finitely many indecomposable modules.
NAKAYAMA AUTOMORPHISM SERIAL
Biserial algebras are a slightly generalization of what is called a serial algebra.