Trivial extensions of monomial algebras are symmetric fractional Brauer configuration algebras
Yuming Liu, Bohan Xing
TL;DR
This work extends the connection between monomial algebras and Brauer-type configurations to the framework of fractional Brauer configuration algebras of type S. For any finite-dimensional monomial algebra $A$, the authors construct an $f_s$-BC $E_A$ whose associated algebra $A_E$ is isomorphic to the trivial extension $T(A)$, effectively realizing trivial extensions as symmetric $f_s$-BCAs. They establish a bijection between isomorphism classes of monomial algebras and equivalence classes of pairs $( ext{a symmetric } f_s ext{-BCA with a free } f ext{-degree}, ext{an admissible cut})$, enabling reconstruction of $A$ as a cut algebra from its trivial extension. The framework generalizes previous results for gentle and almost gentle algebras to broader monomial settings and highlights a deep link between trivial extensions, fractional Brauer configurations, and admissible cuts.”
Abstract
By providing equivalent definitions of fractional Brauer configuration algebras in certain special cases, we associate to each monomial algebra some combinatorial data called a fractional Brauer configuration, from which we construct a corresponding fractional Brauer configuration algebra. We show that this algebra is isomorphic to the trivial extension of the given monomial algebra. Furthermore, we establish a one-to-one correspondence between the isomorphism classes of monomial algebras and the equivalence classes of pairs consisting of a symmetric fractional Brauer configuration algebra of type S with a free fractional-degree function and an admissible cut on it.