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## Utilização da álgebra de caminhos para realizar o mapeamento de requisições virtuais sobre redes de substrato.; Path algebra to make the mapping of virtual network requests over substrate networks.

Molina, Miguel Angelo Tancredi
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
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66.38%

## The Leavitt path algebra of a graph

Abrams, G.; Pino, G. Aranda
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.1%
For any row-finite graph $E$ and any field $K$ we construct the {\its Leavitt path algebra} $L(E)$ having coefficients in $K$. When $K$ is the field of complex numbers, then $L(E)$ is the algebraic analog of the Cuntz Krieger algebra $C^*(E)$ described in [8]. The matrix rings $M_n(K)$ and the Leavitt algebras L(1,n) appear as algebras of the form $L(E)$ for various graphs $E$. In our main result, we give necessary and sufficient conditions on $E$ which imply that $L(E)$ is simple.

## Tilting Modules over the Path Algebra of Type A, Polytopes, and Catalan Numbers

Hille, Lutz
Tipo: Artigo de Revista Científica
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46.03%
It is well known that the number of tilting modules over a path algebra of type A_n coincides with the Catalan number C(n). Moreover, the number of support tilting modules of type A_n is the Catalan number C(n+1). We show that the convex hull of all roots of a root system of type A_n is a polytope with integral volume (n + 1)C(n+1). Moreover, we associate to the set of tilting modules and to the set of support tilting modules certain polytopes and show that their volumes coincide with the number of those modules, respectively. Finally, we show that these polytopes can be defined just using the root system and relate their volumes, so that we can derive the above results in a new way.; Comment: 11 pages

## The maximal commutative subalgebra of a Leavitt path algebra

Canto, Cristóbal Gil; Nasr-Isfahani, Alireza
Tipo: Artigo de Revista Científica
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46.11%
For any unital commutative ring $R$ and for a graph $E$, we identify the maximal commutative subalgebra of the Leavitt path algebra of $E$ with coefficients in $R$. Besides we are able to characterize injectivity of representations which gives a generalization of Cuntz-Krieger uniqueness theorem, and by other hand, to generalize and simplify the result about commutative Leavitt path algebras over fields.

## Extreme cycles. The center of a Leavitt path algebra

Garcia, Maria Guadalupe Corrales; Barquero, Dolores Martin; Gonzalez, Candido Martin; Molina, Mercedes Siles; Hernandez, Jos Felix Solanilla
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.41%
In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra \cite{CMMSS1}. We will go further into the structure itself of the Leavitt path algebra. For example, the ideal $I(P_{ec} \cup P_{c} \cup P_l)$ generated by vertices in extreme cycles ($P_{ec}$), by vertices in cycles without exits ($P_c$) and by line points ($P_l$) will be a dense ideal in some cases, for instance in the finite one or, more generally, if every vertex connects to $P_l \cup P_c\cup P_{ec}$. Hence its structure will contain much of the information about the Leavitt path algebra. In the row-finite case, we will need to add a new hereditary set: the set of vertices whose tree has infinite bifurcations ($P_{b^\infty}$).

## Homology of a Leavitt Path Algebra via Anick's Resolution

Lopatkin, Viktor
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.18%
The aim of this paper is to calculate the homology of a Leavitt path algebra via Anick's resolution. We show that all homology (in positive degrees) of a Leavitt path algebra is equal to zero.

## A quantum cluster algebra of Kronecker type and the dual canonical basis

Lampe, Philipp
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.11%
The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A_1^{(1)}. The examined subalgebra is associated with a terminal module M over the path algebra of the Kronecker quiver via an Weyl group element w of length four. Geiss-Leclerc-Schroeer attached to M a category C_M of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(C_M). The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra A(C_M) is a subalgebra of the graded dual of the (non-quantized) universal enveloping algebra U(n). We transfer to the quantized setup. Following Lusztig we attach to w a subalgebra U_q^+(w) of U_q(n). The subalgebra is generated by four elements that satisfy straightening relations; it degenerates to a commutative algebra in the classical limit q=1. The algebra U_q^+(w) possesses four bases, a PBW basis, a canonical basis, and their duals. We prove recursions for dual canonical basis elements. The recursions imply that every cluster variable in A(C_M) is the specialization of the dual of an appropriate canonical basis element. Therefore...

## A Path Algebra for Multi-Relational Graphs

Rodriguez, Marko A.; Neubauer, Peter
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.2%
A multi-relational graph maintains two or more relations over a vertex set. This article defines an algebra for traversing such graphs that is based on an $n$-ary relational algebra, a concatenative single-relational path algebra, and a tensor-based multi-relational algebra. The presented algebra provides a monoid, automata, and formal language theoretic foundation for the construction of a multi-relational graph traversal engine.

## The Weak Finitistic Dimension of a Path Algebra is Finite

Kanuni, Muge; Kaygun, Atabey
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.26%
We prove a version of Bass' finitistic dimension conjecture for path algebras over arbitrary directed graphs. It is known that the path algebra of a finite directed graph is hereditary, hence it has finite finitistic dimension, when the graph is acyclic. The case for arbitrary directed graphs is still open. We use flat dimension instead of projective dimension (hence the designation "weak") and show that the weak finitistic dimension of an arbitrary path algebra is finite.; Comment: This paper has been withdrawn by the authors. A stronger result is already known. All path algebras are hereditary regardless of existence of oriented cycles. See Crawley-Boevey "Lectures on Representations of Quivers" (Page 8, Consequences (2))

## Biserial algebras via subalgebras and the path algebra of D_4

Külshammer, Julian
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.21%
We give two new criteria for a basic algebra to be biserial. The first one states that an algebra is biserial iff all subalgebras of the form eAe where e is supported by at most 4 vertices are biserial. The second one gives some condition on modules that must not exist for a biserial algebra. These modules have properties similar to the module with dimension vector (1,1,1,1) for the path algebra of the quiver D_4. Both criteria generalize criteria for an algebra to be Nakayama. They rely on the description of a basic biserial algebra in terms of quiver and relations given by R. Vila-Freyer and W. Crawley- Boevey.

## On the simplicity of Lie algebra of Leavitt path algebra

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.15%
For a field $F$ and a row-finite directed graph $\Gamma$ let $L(\Gamma)$ be the Leavitt path algebra. We find necessary and sufficient conditions for the Lie algebra $[L(\Gamma),L(\Gamma)]$ to be simple.; Comment: 7 pages, 10 figures

## Simplicity of the Lie algebra of skew symmetric elements of a Leavitt path algebra

Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.17%
For a field $F$ of characteristic not 2 and a directed row-finite graph $\Gamma$ let $L(\Gamma)$ be the Leavitt path algebra with the standard involution $*.$ We study the Lie algebra $K=K(L(\Gamma),*)$ of $*-$skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra $[K,K]$ to be simple.; Comment: 8 pages, 6 figures. arXiv admin note: text overlap with arXiv:1304.1922

## An AF algebra associated with the Farey tessellation

Boca, Florin P.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.05%
To the Farey tessellation of the upper half-plane we associate an AF algebra encoding the cutting sequences that define vertical geodesics. The Effros-Shen AF algebras arise as quotients of our algebra. Using the path algebra model for AF algebras we construct for each $\tau \in (0,1/4\big]$, projections $(E_n)$ in this algebra such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.; Comment: 23 pages, 20 figures

## The socle of a Leavitt path algebra

Pino, Gonzalo Aranda; Barquero, Dolores Martin; Gonzalez, Candido Martin; Molina, Mercedes Siles
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.35%
In this paper we characterize the minimal left ideals of a Leavitt path algebra as those ones which are isomorphic to principal left ideals generated by line point vertices, that is, by vertices whose trees do not contain neither bifurcations nor closed paths. Moreover, we show that the socle of a Leavitt path algebra is the two-sided ideal generated by these line point vertices. This characterization allows us to compute the socle of some algebras that arise as the Leavitt path algebra of some row-finite graphs. A complete description of the socle of a Leavitt path algebra is given: it is a locally matricial algebra.; Comment: 13 pgs

## Wreath products by a Leavitt path algebra

Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.15%
We introduce ring theoretic constructions that are similar to the construction of wreath product of groups. In particular, for a given graph $\Gamma=(V,E)$ and an associate algebra $A,$ we construct an algebra $B=A\, wr\, L(\Gamma)$ with the following property: $B$ has an ideal $I$,which consists of (possibly infinite) matrices over $A$, $B/I\cong L(\Gamma)$, the Leavitt path algebra of the graph $\Gamma$. \medskip \par Let $W\subset V$ be a hereditary saturated subset of the set of vertices [1], $\Gamma(W)=(W,E(W,W))$ is the restriction of the graph $\Gamma$ to $W$, $\Gamma/W$ is the quotient graph [1]. Then $L(\Gamma)\cong L(W)$ wr $L(\Gamma/W)$.; Comment: 7, 1

## Stratifying systems over the hereditary path algebra with quiver $\mathbb{A}_{p,q}$

Cadavid, Paula Andrea; Marcos, Eduardo do Nascimento
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.22%
The authors have proved in [J. Algebra Appl. 14 (2015), no. 6] that the size of a stratifying system over a finite-dimensional hereditary path algebra $A$ is at most $n$, where $n$ is the number of isomorphism classes of simple $A$-modules. Moreover, if $A$ is of Euclidean type a stratifying system over $A$ has at most $n-2$ regular modules. In this work, we construct a family of stratifying systems of size $n$ with a maximal number of regular elements, over the hereditary path algebra with quiver $\widetilde{\mathbb {A}}_{p,q}$, canonically oriented.; Comment: arXiv admin note: substantial text overlap with arXiv:1308.5547

## The number of arrows in the quiver of tilting modules over a path algebra of type $A$ and $D$

Kase, Ryoichi
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.03%
Happel and Unger defined a partial order on the set of basic tilting modules. The tilting quiver is the Hasse diagram of the poset of basic tilting modules. We determine the number of arrows in the tilting quiver over a path algebra of type $A$ or $D$.; Comment: 22 pages

## Path algebras and de Broglie waves

Gerstenhaber, Murray
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.03%
De Broglie waves may be a reflection of a deformation inherent in the path algebra of phase space. On a Riemannian manifold equipped with a suitable 2-form, the product of paths, which is ordinarily their concatenation, can be deformed by multiplication by a scalar weight giving rise to a function on paths. In flat phase space the associated function is periodic with period the de Broglie wave length. The de Broglie description may only be approximate in curved space.; Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:1306.4939

## Half-quantum groups at roots of unity, path algebras and representation type

Cibils, Claude
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.1%
We show that finite dimensional half-quantum groups at roots of unity corresponding to simple Lie algebras having symmetric Cartan matrix are of wild representation type, except for sl_2. Moreover, the underlying associative algebra is isomorphic to an admissible quotient of the path algebra of the Cayley graph of an abelian group. A quantum type Fourier transform enables to describe an explicit isomorphism.; Comment: 14 pages, Latex. http://www.unige.ch/math/folks/cibils/apublics.html

Lopatkin, Viktor