Trivial extensions of monomial algebras are symmetric fractional Brauer configuration algebras

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.

💡 Research Summary

The paper establishes a precise and far‑reaching correspondence between trivial extensions of arbitrary finite‑dimensional monomial algebras and a newly defined class of symmetric fractional Brauer configuration algebras (f‑s‑BCAs). After recalling the recent notion of a fractional Brauer configuration (f‑BC) introduced by Li and Xing—an object consisting of a set of “angles” equipped with a cyclic group action, two partitions (P for vertices, L for arrows), and an integer degree function d satisfying six combinatorial conditions—the authors explain how to associate a quiver Q_E and an ideal I_E generated by three families of relations (R1, R2, R3). They also provide an equivalent description of the relations in terms of “special paths” (R1′, R2′), and prove Lemma 2.6 that the two presentations generate the same ideal.

The central construction begins with a monomial algebra A = kQ/I, where I is generated by monomial relations. From the combinatorial data of Q and I the authors build a fractional Brauer configuration E_A of type S (i.e., symmetric, finite, and with integral fractional degrees). The construction assigns each vertex of Q to a ⟨g⟩‑orbit, each arrow to an L‑class, and defines the degree function d(e) as the length of the corresponding special path, guaranteeing that the resulting configuration is f‑degree‑free.

The main theorem (Theorem 4.4) shows that the fractional Brauer configuration algebra A_{E_A} = kQ_{E_A}/I_{E_A} is isomorphic to the trivial extension T(A) = A ⊕ D(A), where D(A) denotes the k‑dual of A. The proof constructs an explicit algebra isomorphism φ that matches vertices and arrows of Q_{E_A} with those of Q, and checks that φ respects all relations. Symmetry of the configuration (the Nakayama automorphism σ being the identity) ensures that A_{E_A} is a symmetric algebra, exactly the property required of a trivial extension.

In the second part of the paper the authors introduce admissible cuts for symmetric f‑s‑BCAs. An admissible cut selects, for each polygon (i.e., each P‑class), a single arrow to be “cut”, thereby imposing additional relations that delete that arrow from the quiver. The resulting cut algebra is again a monomial algebra, and the process can be reversed: starting from a monomial algebra, one recovers a symmetric f‑s‑BCA together with a free fractional‑degree function and an admissible cut. Corollary 5.6 formalises this bijection between isomorphism classes of monomial algebras and equivalence classes of triples (symmetric f‑s‑BCA, free fractional‑degree, admissible cut). This generalises earlier results: Schroll’s identification of trivial extensions of gentle algebras with Brauer graph algebras, and Green–Schroll’s extension to almost‑gentle algebras.

The paper includes several detailed examples (Sections 2.8–2.10 and Example 5.7) that illustrate how a concrete monomial algebra yields a specific fractional Brauer configuration, how the associated quiver and relations look, and how admissible cuts operate. These examples demonstrate that the construction works even when the resulting algebra is not multiserial, highlighting the flexibility of the f‑BC framework.

Finally, the authors discuss future directions: extending the correspondence to non‑monomial algebras, investigating skew‑Brauer configurations, and exploring infinite‑dimensional or non‑symmetric fractional Brauer configuration algebras. By unifying trivial extensions, symmetric algebras, and combinatorial configurations under the umbrella of fractional Brauer data, the work opens a new pathway for classifying and studying a broad family of finite‑dimensional algebras.