arXiv subjects

Ayten Koç

Publications and source records attributed to Ayten Koç.

7 recordsLinked to original sources

arXiv2026

A Noncommutative Nullstellensatz for Leavitt Path Algebras

Hilbert's Nullstellensatz states that a quotient of the algebra of polynomial functions on an algebraic variety over an algebraically closed field is the algebra of polynomial functions on a (sub)variety if and only if its radical is trivial. We prove a noncommutative analog: A quotient of a Leavitt path algebra over an algebraically closed field is isomorphic to a Leavitt path algebra if and only if its radical is trivial. This suggests that a Leavitt path algebra behaves like a noncommutative algebra of polynomial functions on a directed graph. Along the way we provide a complete answer to the question "When is a quotient of a Leavitt path algebra isomorphic to a Leavitt path algebra?". We also define a Morita invariant stratification and a parametrization of the ideal space of a Leavitt path algebra and show that a generic quotient of a Leavitt path algebra is a Leavitt path algebra. We end this article by pointing out a connection with quantum spaces strengthening the analogy with function algebras and several examples involving algebraic quantum spaces illustrating our results.

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arXiv2025

Classification of Leavitt Path Algebras with Gelfand-Kirillov Dimension <4 up to Morita Equivalence

Leavitt path algebras are associated to di(rected )graphs and there is a combinatorial procedure (the reduction algorithm) making the digraph smaller while preserving the Morita type. We can recover the vertices and most of the arrows of the completely reduced digraph from the module category of a Leavitt path algebra of polynomial growth. We give an explicit classification of all irreducible representations of when the coefficients are a commutative ring with 1. We define a Morita invariant filtration of the module category by Serre subcategories and as a consequence we obtain a Morita invariant (the weighted Hasse diagram of the digraph) which captures the poset of the sinks and the cycles of $Γ$, the Gelfand-Kirillov dimension and more. When the Gelfand-Kirillov dimension of the Leavitt path algebra is less than 4, the weighted Hasse diagram (equivalently, the complete reduction of the digraph) is a complete Morita invariant.

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arXiv2023

Leavitt Path Algebras with Coefficients in a Commutative Unital Ring

In addition to extending some facts from field coefficients to commutative ring coefficients for Leavitt path algebras with new shorter proofs, we also prove some results that are new even for field coefficients. In particular, we show that the ideal lattice of a Leavitt path algebra embeds into the ideal lattice of the path algebra of the same digraph, we construct a new basis for a Leavitt path algebra of polynomial growth and give a formula for the Gelfand-Kirillov dimension of a Leavitt path algebra in terms of its digraph.

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arXiv2019

Representations of Leavitt Path Algebras

We study representations of a Leavitt path algebra $L$ of a finitely separated digraph $Γ$ over a field. We show that the category of $L$-modules is equivalent to a full subcategory of quiver representations. When $Γ$ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of $L$ after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to $L$ having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of $L$-modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, $\mathbb{F}Γ$-modules) via a new colimit model for $M\otimes_{\mathbb{F}Γ} L$.

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arXiv2018

On Prüfer-Like Properties of Leavitt Path Algebras

Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra $L$, in spite of being non-commutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [8] it was shown that the ideals of $L$ satisfy the distributive law, a property of Prüfer domains and that $L$ is a multiplication ring, a property of Dedekind domains. In this paper, we first show that $L$ satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers $a,b,c$, $\gcd(a,b)\cdot\operatorname{lcm}(a,b)=a\cdot b$ and $a\cdot \operatorname{gcd}(b,c)=\operatorname{gcd}(ab,ac)$. We also show that $L$ satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which $L$ satisfies another important characterizing property of almost Dedekind domains, namely the cancellative property of its non-zero ideals.

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arXiv2017

Finite Dimensional Representations of Leavitt Path Algebras

When $Γ$ is a row-finite di(rected )graph we classify all finite dimensional modules of the Leavitt path algebra $L(Γ)$ via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph $Γ$. The category of (unital) $L(Γ)$-modules is equivalent to a subcategory of quiver representations of $Γ$. However the category of finite dimensional representations of $L(Γ)$ is tame in contrast to the finite dimensional quiver representations of $Γ$ which are almost always wild.

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arXiv2012

A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph - which we name as a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant κ(A) for A and count the number of isomorphism classes of Leavitt path algebras with κ(A)=n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.

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