International Workshop on Leavitt Path Algebras (IWLPA)

Organised by North-Eastern Hill University and Centre for Research in Mathematics and Data Science, Western Sydney University
April 7, 2025 - April 11, 2025

Overview

The International Workshop on Leavitt Path Algebras (IWLPA) was held at North-Eastern Hill University (NEHU), Shillong, from April 7th to 11th, 2025. The event was jointly organized by NEHU and the Centre for Research in Mathematics and Data Science, Western Sydney University. This workshop explored K-theoretic tools in LPAs and Representation theoretic tools in LPAs. Tailored for graduate students, early-career researchers, and established mathematicians, it offered a blend of lectures, problem-solving sessions, and interdisciplinary discussions.

Organizers

Participants

Lectures

*Few lectures were online

Roozbeh Hazrat (Western Sydney University)

Title: Classification of Leavitt path algebras in relation to symbolic dynamics

Abstract: The classification of Leavitt path algebras remains a central problem in algebra and operator theory. These algebras, constructed from directed graphs, are deeply connected to symbolic dynamics (shifts of finite type) and C-algebras, where similar classification programs have had transformative effects. This talk will survey ongoing efforts to classify Leavitt path algebras using invariants inspired by symbolic dynamics and K-theory. Current advances highlight surprising parallels with the classification of Cuntz–Krieger algebras, while also presenting distinct challenges in the purely algebraic setting. A number of open problems and promising new directions will be discussed, illustrating how techniques from dynamics, category theory, and noncommutative geometry come together in this classification program.

Praneet Srivastava (University of Bonn) and Joshua Graham (University of New South Wales)

Title: Higher K-theoretic tools in Leavitt path algebras

Abstract: Algebraic K-theory provides a powerful set of invariants to study rings, algebras, and categories. In this two-part workshop, we begin with a modern introduction to higher algebraic K-theory, following Quillen’s approach via the Q-construction, and connect it with ideas from homotopy theory and ∞-categories. Particular emphasis will be on understanding group completion and computing higher K-groups for finite fields. Building on this foundation, we turn to Leavitt path algebras, which serve as rich testing grounds for these tools. Explicit long exact sequences relating the K-theory of finite quivers to field invariants will be presented. Applications include concrete computations of K-theory for Leavitt path algebras over finite fields, giving new insight into their structure and laying a path toward broader classification.

Mohan R (Azim Premji University)

Title: Leavitt path algebras of bi-separated graphs and its module theory

Abstract: Bi-separated graphs were introduced to unify various generalizations of Leavitt algebras, producing Cohn–Leavitt path algebras with rich combinatorial and algebraic behavior. This talk has two parts: We examine structural properties of these algebras, focusing on their V-monoids and how order-ideals correspond directly to graph-theoretic data. A refined class of bi-separated graphs will also be described, showcasing elegant algebra–combinatorics interplay. Building on Raimund Presser’s recent work, we turn to the module theory of these algebras. In particular, we explore the emergence of simple modules, including how the well-known Cuntz–Krieger biseparation naturally produces Chen simple modules. This framework suggests new ways of understanding representation theory in the setting of generalized Leavitt path algebras.

Guillermo Cortiñas (University of Buenos Aires)

Title: Bivariant algebraic K-theory and Leavitt path algebras

Abstract: Classification often reduces to finding invariants that capture essential features of a structure while being computable. Bivariant algebraic K-theory, denoted kk, provides a universal invariant satisfying natural functoriality, excision, and stability conditions. This talk introduces kk as the “finest” such invariant and demonstrates its relevance for Leavitt path algebras. In particular, we discuss how kk unifies various classification results and allows comparison with invariants from both algebra and topology. Applications range from distinguishing non-isomorphic algebras to establishing Morita equivalences, showing the depth of interaction between K-theory and graph algebras.

Murad Ozaydin (University of Oklahoma)

Title: Leavitt Path Algebras, Categorically speaking I, II, III

Abstract: Category theory offers a natural language to study path algebras and their representations. Beginning with path categories of directed graphs, this tutorial series introduces participants to quiver representations and their classification problems, highlighting both tractable and wild cases. We then shift to Leavitt path algebras, whose representation theory forms a distinguished subcategory of quiver representations with powerful structural constraints. Remarkably, unlike the general case, finite-dimensional representations of Leavitt path algebras are fully classifiable. These lectures also connect to applications in ring theory, Lie theory, physics, and topological data analysis, demonstrating the wide reach of quiver representations. Joint work on classification with Ayten Koç and collaborators will be presented, along with discussions of the IBN property, irreducible representations, and quotient algebras.

Promit Mukerjee (Jadavpur University)

Title: A talented monoid perspective on higher-rank graphs and their Kumjian–Pask algebras

Abstract: Higher-rank (k-)graphs generalize directed graphs to higher dimensions, giving rise to Kumjian–Pask algebras—algebraic analogues of graph C*-algebras. To classify these algebras, one seeks invariants sensitive to both algebraic and combinatorial data. This talk introduces the Talented monoid TΛ, a Zk-commutative monoid associated to a k-graph Λ, and explores its surprising ability to encode features like cofinality, aperiodicity, and simplicity. We will see how TΛ arises from both graded K-theory and groupoid perspectives, and how it provides powerful classification criteria. Joint work with Roozbeh Hazrat, David Alan Pask, and Sujit Kumar Sardar will be highlighted, with examples demonstrating how the talented monoid bridges algebra and geometry in higher-rank graph theory.

Sujit Kumar Sardar (Jadavpur University)

Title: An Introductory Survey on Boolean Inverse Semigroups

Abstract: Boolean inverse semigroups extend Boolean algebras to a non-commutative framework, replacing Stone spaces with ample groupoids. This extension connects algebraic semigroup theory with operator algebras, especially Steinberg algebras and Leavitt path algebras. We begin with examples such as symmetric inverse monoids (partial bijections on sets) to illustrate the fundamental ideas. The concept of the type monoid is then introduced, with a focus on Kudryavtseva’s result relating the type monoid of the tight Booleanization of a graph inverse semigroup to the graph monoid of its underlying directed graph. The survey aims to equip participants with both basic definitions and deeper insights into how Boolean inverse semigroups serve as algebraic models for groupoid-based operator algebras.

Alfilgen Sebandal (LNU, Sweden & CVIF, Philippines)

Title: A confirmation to the Leavitt path algebra (finite) graded classification conjecture

Abstract: W. Leavitt’s 1960s work on algebras lacking the Invariant Basis Number (IBN) property initiated the field of Leavitt algebras, later extended to graph-based Leavitt path algebras (LPAs). In 2013, Roozbeh Hazrat proposed the Graded Classification Conjecture, asserting that the talented monoid serves as a graded Morita invariant for LPAs. This talk confirms the conjecture in the finite-dimensional case, viewing LPA modules as a special form of quiver representations with relations. We will see how representation-theoretic and monoid-theoretic techniques combine to validate the conjecture. The work represents a collaboration with Wolfgang Bock, Roozbeh Hazrat, and Jocelyn Vilela, and points toward a broader resolution of the classification problem.

Contributory Lectures

Sumanta Das (Calcutta University)

Title: Leavitt path algebra for power graph and its supergraph

Abstract: This work investigates Leavitt path algebras associated with power graphs of semigroups and their order supergraphs. Specifically, the algebraic properties and stable ranks of LPAs arising from directed power graphs of finite groups are analyzed. Further, the structure of LPAs attached to torsion group supergraphs and punctured order supergraphs is explored, revealing connections between group-theoretic properties and algebraic invariants of the corresponding LPAs. This contributes to a growing body of literature linking combinatorial graph constructions with algebraic invariants.

Soumitra Das (IISER Berhampur)

Title: Pure-direct-injectivity in categories of sheaves

Abstract: Injectivity is a central concept in homological algebra, and its adaptations to sheaf theory enrich categorical approaches in geometry. We introduce pure-direct-injective sheaves, a new class defined with respect to geometrical purity. The behavior of these objects under functors such as restriction, extension by zero, and direct image is examined. Several illustrative examples show how pure-direct-injective sheaves differ from pure-injective ones. Importantly, coherent sheaves over Noetherian schemes are always pure-injective in their subcategory, while quasi-coherent sheaves over Artinian principal ideal rings are pure-direct-injective. These results provide new perspectives for understanding sheaf-theoretic homological structures, with implications for both algebraic geometry and categorical representation theory.