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Öğe CLASSIFICATION OF LEAVITT PATH ALGEBRAS WITH TWO VERTICES(Independent Univ Moscow-Ium, 2019) Kanuni, Müge; Barquero, Dolores Martin; Gonzalez, Candido Martin; Molina, Mercedes SilesWe classify row-finite Leavitt path algebras associated to graphs with no more than two vertices. For the discussion we use the following invariants: decomposability, the K-0 group, det(N-E') (included in the Franks invariants), the type, as well as the socle, the ideal generated by the vertices in cycles with no exits and the ideal generated by vertices in extreme cycles. The starting point is a simple linear algebraic result that determines when a Leavitt path algebra is IBN. An interesting result that we have found is that the ideal generated by extreme cycles is invariant under any isomorphism (for Leavitt path algebras whose associated graph is finite). We also give a more specific proof of the fact that the shift move produces an isomorphism when applied to any row-finite graph, independently of the field we are considering.Öğe Cohn-Leavitt path algebras and the invariant basis number property(World Scientific Publ Co Pte Ltd, 2019) Kanuni, Müge; Özaydın, MuradWe give the necessary and sufficient condition for a separated Cohn-Leavitt path algebra of a finite digraph to have Invariant Basis Number (IBN). As a consequence, separated Cohn path algebras have IBN. We determine the non-stable K-theory of a corner ring in terms of the non-stable K-theory of the ambient ring. We give a necessary condition for a corner algebra of a separated Cohn-Leavitt path algebra of a finite graph to have IBN. We provide Morita equivalent rings which are non-IBN, but are of different types.Öğe A combinatorial discussion on ?nit edimensional Leavitt path algebras(2014) Koç, Ayten; Esin, Songül; Güloğlu, İsmail Şuayip; Kanuni, MügeAny finite dimensional semisimple algebra A over a field K is isomorphicto a direct sum of finite dimensional full matrix rings over suitabledivision rings. We shall consider the direct sum of finite dimensionalfull matrix rings over a fieldK:All such finite dimensional semisimplealgebras arise as finite dimensional Leavitt path algebras. For thisspecific finite dimensional semisimple algebraAover a fieldK;we definea uniquely determined specific graph - called a truncated tree associatedwithA- whose Leavitt path algebra is isomorphic toA. We define analgebraic invariant (A)forAand count the number of isomorphismclasses of Leavitt path algebras with the same fixed value of (A).Moreover, we find the maximum and the minimumK-dimensions of theLeavitt path algebras of possible trees with a given number of verticesand we also determine the number of distinct Leavitt path algebras ofline graphs with a given number of vertices.Öğe A combinatorial discussion on finite dimensional Leavitt path algebras(Hacettepe Univ, Fac Sci, 2014) Koç, Ayten; Esin, Songül; Güloğlu, İsmail Şuayip; Kanuni, MügeAny 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. We shall consider the direct sum of finite dimensional full matrix rings over a field K. All such finite dimensional semisimple algebras arise as finite dimensional Leavitt path algebras. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely determined specific graph - called a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant kappa(A) for A and count the number of isomorphism classes of Leavitt path algebras with the same fixed value of kappa(A). 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 we also determine the number of distinct Leavitt path algebras of line graphs with a given number of vertices.Öğe Hochschild cohomology of reduced incidence algebras(World Scientific Publ Co Pte Ltd, 2017) Kanuni, Müge; Kaygun, Atabey; Sütlü, SerkanWe compute the continuous Hochschild cohomology of four reduced incidence algebras: the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation for the coalgebra Cotor-groups of their pre-dual coalgebras.
