Talks

Roozbeh HAZRAT

Western Sydney University

Monday 13:00 - 14:00, June 16, 2025

The talented monoid associated to higher rank graphs.

Abstract: To a higher rank graph we associate a monoid, the so-called talented monoid, and study how this monoid can capture the properties of the graph and the algebraic properties of the Kumjian-Pask algebra associated to the higher rank graph.

Søren EILERS

University of Copenhagen

Monday 14:00 - 15:00, June 16, 2025

Structure-preserving isomorphisms of Leavitt path algebras

Abstract: Following recent work in the study of graph C*-algebras, I will present and compare eight notions of sameness of Leavitt path algebras that, to varying degree, require isomorphisms to preserve additional structure.

Exactly half of these notions are known to be identical to their C*-algebraic counterparts, allowing direst transfer of results obtained there. For the other half, the relations are not fully understood, but partial results can still be obtained. In some cases, one can infer new classification results for unqualified isomorphism by showing structure-preserving isomorphism this way.

I intend to give an overview of what such methods yield at this time, but also emphasize examples currently outside reach. Joint work with Efren Ruiz.

Piotr M. HAJAC

IMPAN, Warsaw, Poland

Monday 15:00 - 16:00, June 16, 2025

Relation morphisms of directed graphs

Abstract: Associating graph algebras to directed graphs leads to both covariant and contravariant functors from suitable categories of graphs to the category k-Alg of algebras and algebra homomorphims. As both functors are often used at the same time, one needs a new category of graphs that allows a “common denominator” functor unifying the covariant and contravariant constructions. In this talk, I will show how to solve this problem by first introducing the relation category of graphs RG, and then determining the concept of admissible graph relations that yields a subcategory of RG admitting a contravariant functor to k-Alg simultaneously generalizing the aforementioned covariant and contravariant functors. I will illustrate relation morphisms of graphs by many naturally occurring examples, including Cuntz algebras, quantum spheres and quantum balls. Although I will focus on Leavitt path algebras and graph C*-algebras, time permitting, I will unravel functors given by path algebras, Cohn path algebras and Toeplitz graph C*-algebras from suitable subcategories of RG to k-Alg. Based on joint work with Gilles G. de Castro and Francesco D'Andrea.

Ayten KOÇ

Gebze Technical University

Monday 16:30 - 17:30, June 16, 2025

Ideal spaces and quotients of Leavitt path algebras

Abstract: In joint work with Murad Özaydın we provide a complete answer to the question "When is a quotient of a Leavitt path algebra isomorphic to a Leavitt path algebra?" in terms of the interaction of the kernel of the quotient homomorphism with the cycles of the digraph. A key ingredient is the characterization of finitely generated projective (Leavitt path algebra) modules whose endomorphism algebras are finite dimensional. As a consequence of our characterization we get that any quotient of a Leavitt path algebra divided by its Jacobson radical is a Leavitt path algebra if the coefficient field is large enough. We define a stratification and a parametrization of the ideal space of a Leavitt path algebra (initially in terms of the digraph, later algebraically) and show that a generic quotient of a Leavitt path algebra is a Leavitt path algebra. Contrary to most algebraic properties of Leavitt path algebras, our criterion for a quotient to be isomorphic to a Leavitt path algebra is not independent of the field of coefficients.

*This study was supported by Scientific and Technological Research Council of Türkiye (TÜBİTAK) under the grants 124F214 and 122F414.

Ashish K. SRIVASTAVA

Saint Louis University

Monday 17:30 - 18:30, June 16, 2025

Zhang twist of Leavitt path algebras

Abstract: We construct graded automorphisms of Leavitt path algebras and as an application, we study Zhang twist of Leavitt path algebras and describe new classes of irreducible representations of Leavitt path algebras of the rose graph. (This is a joint work with T. G. Nam and N. T. Vien)

Guillermo CORTIÑAS

Universidad de Buenos Aires

Monday 18:30 - 19:30, June 16, 2025

K-theory and homology of Exel-Pardo algebras.

Abstract: A seminar paper by Ruy Exel and Enrique Pardo associates an algebra L(G, E) to a self-similar action of a group G on a graph E. In the talk we'll discuss the K-theory and the Hochschild homology of such algebras and of their twists.

Elizabeth PACHECO

Western Sydney University

Tuesday 13:00 - 14:00, June 17, 2025

New representations of Leavitt path algebras

Abstract: A particular class of irreducible representations of Leavitt path algebras were defined by Chen in 2013. These representations have since been generalized, but until recently these (as well as projective modules based at sinks) were the only known irreducible Leavitt path algebra modules. In this talk we will discuss non-Chen irreducible representations of Leavitt path algebras, as well some background material in symbolic dynamics. This is joint work with Murad Özaydın.

Alfilgen SEBANDAL

Linnaeus University - Research Center for Theoretical Physics, Jagna Bohol

Tuesday 14:00 - 15:00, June 17, 2025

Applications of representation theory to the classification of Leavitt path algebras

Abstract: In 2013, R. Hazrat formulated the Graded Classification Conjecture for Leavitt path algebras which claims that its monoid of graded finitely generated projective modules is a graded Morita invariant. Motivated by the works of A. Koç and M. Özaydın, we shall utilize representations of the Leavitt path algebra as a tool to solve this conjecture and see how classifications in the monoid reflect to (graded) categorical classifications in the algebra. More concretely, we will provide evidences and a confirmation in the finite-dimensional case of the conjecture. This is a compilation of joint works with Wolfgang Bock, Roozbeh Hazrat, and Jocelyn Vilela.

Arnaud BROTHIER

University of Trieste - University of New South Wales

Tuesday 15:00 - 16:00, June 17, 2025

From Jones' technology to representations of Leavitt path algebras

Abstract: In his quest in constructing conformal field theories from subfactors Vaughan Jones found an unexpected connection with Richard Thompson's group. This led to Jones' technology: a powerful method for constructing actions of certain groups.

I will introduce the great lines of this fascinating connection. I will then explain how this approach permits us to construct, study and classify representations of Leavitt pathalgebras by solely working with representations of quiver algebras. The methods are both explicit and functorial. These are joint works with Vaughan Jones and with Dilshan Wijesena.

Murad ÖZAYDIN

University of Oklahoma

Tuesday 16:30 - 17:30, June 17, 2025

Leavitt path algebras of polynomial growth up to isomorphism and Morita equivalence

Abstract: Let Γ be a finite digraph whose cycles are pairwise disjoint (equivalently the Leavitt path algebra L(Γ) has polynomial growth). How much of Γ can we recover from (the isomorphism type of) L(Γ) or the module category of L(Γ) (that is, the Morita type of L(Γ))? Satisfactory answers were known for finite dimensional Leavitt path algebras and those of linear growth (that is, for Gelfand-Kirillov dimensions 0 and 1).

In joint work with Ayten Koç we define a Morita invariant polynomial whose degree is the Gelfand-Kirillov dimension of L(Γ) and whose coefficients count the number of sinks and cycles of Γ. More generally the marked weighted Hasse diagram of the poset of sinks and cycles of Γ turns out to be a Morita invariant. We can classify all L(Γ) of quadratic and cubic growth up to Morita equivalence and all algebras Morita equivalent to even dimensional quantum disks and odd dimensional quantum spheres up to isomorphism.

*This study was supported by Scientific and Technological Research Council of Türkiye (TÜBİTAK) under the grant 122F414.

Kulumani M. RANGASWAMY

University of Colorado, Colorado Springs

Tuesday 17:30 - 18:30, June 17, 2025

On Naimark's problem for C*-algebras and Leavitt path algebras

Abstract: This is a joint work with Mark Tomforde. A positive solution to the Naimark problem for graph C*-algebras is obtained by using graphical techniques. Specifically, it is shown that a graph C*-algebra C*(E) of a directed graph E has exactly one irreducible representation up to unitary equivalence if and only if C*(E) is isomorphic to the C*-algebra of compact operators on a suitable Hilbert space. In this case, the graph E is shown to be acyclic, downward directed, and the vertex set E0 is the hereditary saturated closure of a line point. The algebraic version of Naimark's problem for a Leavitt path algebra LK(E) over a field K is considered. Here, the unitary equivalence of irreducible representations translates into isomorphisms of simple left LK(E)-modules. It is shown that a Leavitt path algebra LK(E) will have exactly one isomorphism class of simple left modules if and only if the graph E satisfies the same conditions as noted earlier for the case of C*(E). Under these conditions on E, it is shown that LK(E) is isomorphic to the algebra of finite rank endomorphisms of a suitable K-vector space V with a special decomposition property in terms of a specified basis. Equivalently, LK(E) is isomorphic to the algebra MX(K) of infinite matrices over the field K with finitely many non-zero entries whose rows and columns are indexed by an infinite set X. Further generalizations of Naimark's problem to other algebraic systems are also considered.

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