Leavitt Path Algebra over Kronecker Square of Quivers and Cross product algebra
Jehan Alarfaj, Dolores Martín Barquero, Ashish K. Srivastava
TL;DR
The paper initiates and develops the study of Leavitt path algebras over the Kronecker square $\widehat{Q}$ of a quiver, linking them to cross products $L_{ ext{K}}(Q) \otimes_G L_{ ext{K}}(Q)$ and Hayashi's face algebra $\mathcal{H}_{\mathbb{K}}(Q)$. It establishes several positive results showing graded isomorphisms $L_{ ext{K}}(\widehat{Q}) \cong L_{ ext{K}}(Q) \otimes_G L_{ ext{K}}(Q)$ in certain regimes (e.g., line or no-sink quivers), and provides counterexamples demonstrating limits of this correspondence in general. A structural bridge is built by identifying $\mathcal{A}_{\mathbb{K}}(Q) \cong L_{\mathbb{K}}(\widehat{Q})$, enabling categorical and K-theoretic analysis via Hayashi's face algebra and graded Grothendieck groups; Hazrat's shift-equivalence framework is leveraged to compare Kronecker-square LPAs under graded Morita equivalence. The work also analyzes direct limits, GK-dimension behavior, and conditions under which $L_{ ext{K}}(Q)$ sits as a graded ideal in $L_{ ext{K}}(\widehat{Q})$, offering a nuanced view of invariants across Kronecker products and setting a conjectural exact sequence linking socles and crossed products. Overall, the paper broadens the algebraic landscape around Leavitt path algebras, their Kronecker-based constructions, and their connections to quantum-like face algebras and graded K-theory.
Abstract
In this paper, we initiate the study of Leavitt path algebra over Kronecker square of a quiver and show the similarities and contrast in the properties of Leavitt path algebra over a quiver and its Kronecker square. Furthermore, we discuss the connection of Leavitt path algebra over Kronecker square of a quiver with Hayashi's face algebra and the cross product algebra construction of the Leavitt path algebra over the original quiver.