Graph Algebras Research Articles

The covariant functoriality of graph algebras

AbstractIn the standard category of directed graphs, graph morphisms map edges to edges. By allowing graph morphisms to map edges to finite paths (path homomorphisms of graphs), we obtain an ambient category in which we determine subcategories enjoying covariant functors to categories of algebras given by constructions of path algebras, Cohn path algebras, and Leavitt path algebras, respectively. Thus, we obtain new tools to unravel homomorphisms between Leavitt path algebras and between graph C*‐algebras. In particular, a graph‐algebraic presentation of the inclusion of the C*‐algebra of a quantum real projective plane into the Toeplitz algebra allows us to determine a quantum CW‐complex structure of the former. It comes as a mixed‐pullback theorem where two ‐homomorphisms are covariantly induced from path homomorphisms of graphs and the remaining two are contravariantly induced by admissible inclusions of graphs. As a main result and an application of new covariant‐induction tools, we prove such a mixed‐pullback theorem for arbitrary graphs whose all vertex‐simple loops have exits, which substantially enlarges the scope of examples coming from noncommutative topology.

On General Wide Graph Algebras and Orbit Equivalence

On General Wide Graph Algebras and Orbit Equivalence

Identification of maximal C*-covers of some operator algebras

We use results on inclusions of free products and extensions of completely positive maps to determine the maximal C ∗ ^* -envelope for upper triangular 3 × 3 3 \times 3 matrices. We consider these same results in the context of larger upper triangular matrices and graph algebras associated to cycle graphs.

Brauer Analysis of Some Cayley and Nilpotent Graphs and Its Application in Quantum Entanglement Theory

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages.

Measuring associativity: graph algebras of undirected graphs

We study two measures of associativity for graph algebras of finite undirected graphs: the index of nonassociativity and (a variant of) the semigroup distance. We determine “almost associative” and “antiassociative” graphs with respect to both measures. It turns out that the antiassociative graphs are exactly the balanced complete bipartite graphs, no matter which of the two measures we consider. In the class of connected graphs the two notions of almost associativity are also equivalent.

Hybrid algebras

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely the blocks of idempotent algebras of weighted surface algebras, up to socle deformations. More generally, for tame symmetric algebras whose Gabriel quiver is 2-regular, we show that the tree class of an arbitrary Auslander–Reiten component is Dynkin or Euclidean or one of the infinite trees A∞,A∞∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_{\\infty }, A_{\\infty }^{\\infty }$$\\end{document} or D∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{\\infty }$$\\end{document}.

The stable exotic Cuntz algebras are higher-rank graph algebras

For each odd integer n ≥ 3 n \geq 3 , we construct a rank-3 graph Λ n \Lambda _n with involution γ n \gamma _n whose real C ∗ C^* -algebra C R ∗ ( Λ n , γ n ) C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n) is stably isomorphic to the exotic Cuntz algebra E n \mathcal E_n . This construction is optimal, as we prove that a rank-2 graph with involution ( Λ , γ ) (\Lambda ,\gamma ) can never satisfy C R ∗ ( Λ , γ ) ∼ M E E n C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n , and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math. 10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution ( Λ , γ ) (\Lambda , \gamma ) whose real C ∗ C^* -algebra C R ∗ ( Λ , γ ) C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma ) is stably isomorphic to the suspension S R S \mathbb {R} . In the Appendix, we show that the i i -fold suspension S i R S^i \mathbb {R} is stably isomorphic to a graph algebra iff − 2 ≤ i ≤ 1 -2 \leq i \leq 1 .

Tracial weights on topological graph algebras

Abstract We describe two kinds of regular invariant measures on the boundary path space $\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C $^{*}$ -algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on $C^{*}(E)$ are gauge-invariant when $C^{*}(E)$ is simple and separable.

From Quantum Automorphism of (Directed) Graphs to the Associated Multiplier Hopf Algebras

This is a noticeably short biography and introductory paper on multiplier Hopf algebras. It delves into questions regarding the significance of this abstract construction and the motivation behind its creation. It also concerns quantum linear groups, especially the coordinate ring of Mq(n) and the observation that K [Mq(n)] is a quadratic algebra, and can be equipped with a multiplier Hopf ∗-algebra structure in the sense of quantum permutation groups developed byWang and an observation by Rollier–Vaes. In our next paper, we will propose the study of multiplier Hopf graph algebras. The current paper can be viewed as a precursor to this upcoming work, serving as a crucial intermediary bridging the gap between the abstract concept of multiplier Hopf algebras and the well-developed field of graph theory, thereby establishing connections between them! This survey review paper is dedicated to the 78th birthday anniversary of Professor Alfons Van Daele.

On simple-minded systems over domestic Brauer graph algebras

On simple-minded systems over domestic Brauer graph algebras

Graph Algebras and Derived Graph Operations

We revise our former definition of graph operations and correspondingly adapt the construction of graph term algebras. As a first contribution to a prospective research field, Universal Graph Algebra, we generalize some basic concepts and results from algebras to graph algebras. To tackle this generalization task, we revise and reformulate traditional set-theoretic definitions, constructions and proofs in Universal Algebra by means of more category-theoretic concepts and constructions. In particular, we generalize the concept of generated subalgebra and prove that all monomorphic homomorphisms between graph algebras are regular. Derived graph operations are the other main topic. After an in-depth analysis of terms as representations of derived operations in traditional algebras, we identify three basic mechanisms to construct new graph operations out of given ones: parallel composition, instantiation, and sequential composition. As a counterpart of terms, we introduce graph operation expressions with a structure as close as possible to the structure of terms. We show that the three mechanisms allow us to construct, for any graph operation expression, a corresponding derived graph operation in any graph algebra.

Finite Dimensional Labeled Graph Algebras

Given a directed graph E and a labeling L, one forms the labeled graph algebra by taking a weakly left-resolving labeled space (E, L, B) and considering a generating family of partial isometries and projections. In this paper, we discuss details in the formulation of the algebras, provide examples, and formulate a process that describes the algebra given the graph and a labelling.

Dessins D’Enfants, Brauer Graph Algebras and Galois Invariants

In this paper, we associate a finite dimensional algebra, called a Brauer graph algebra, to every clean dessin d’enfant by constructing a quiver based on the monodromy of the dessin. We show that Galois conjugate dessins d’enfants give rise to derived equivalent Brauer graph algebras and that the stable Auslander-Reiten quiver and the dimension of the Brauer graph algebra are invariant under the induced action of the absolute Galois group.

Crystal limits of compact semisimple quantum groups as higher-rank graph algebras

Abstract Let O q ⁢ [ K ] \mathcal{O}_{q}[K] be the quantized coordinate ring over the field C ⁢ ( q ) \mathbb{C}(q) of rational functions corresponding to a compact semisimple Lie group 𝐾, equipped with its ∗-structure. Let A 0 ⊂ C ⁢ ( q ) {\mathbf{A}_{0}}\subset\mathbb{C}(q) denote the subring of regular functions at q = 0 q=0 . We introduce an A 0 \mathbf{A}_{0} -subalgebra O q A 0 ⁢ [ K ] ⊂ O q ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]\subset\mathcal{O}_{q}[K] which is stable with respect to the ∗-structure and which has the following properties with respect to the crystal limit q → 0 q\to 0 . The specialization of O q ⁢ [ K ] \mathcal{O}_{q}[K] at each q ∈ ( 0 , ∞ ) ∖ { 1 } q\in(0,\infty)\setminus\{1\} admits a faithful ∗-representation π q \pi_{q} on a fixed Hilbert space, a result due to Soibelman. We show that, for every element a ∈ O q A 0 ⁢ [ K ] a\in\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] , the family of operators π q ⁢ ( a ) \pi_{q}(a) admits a norm limit as q → 0 q\to 0 . These limits define a ∗-representation π 0 \pi_{0} of O q A 0 ⁢ [ K ] \mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K] . We show that the resulting ∗-algebra O ⁢ [ K 0 ] = π 0 ⁢ ( O q A 0 ⁢ [ K ] ) \mathcal{O}[K_{0}]=\pi_{0}(\mathcal{O}_{q}^{{\mathbf{A}_{0}}}[K]) is a Kumjian–Pask algebra, in the sense of Aranda Pino, Clark, an Huef and Raeburn. We give an explicit description of the underlying higher-rank graph in terms of crystal basis theory. As a consequence, we obtain a continuous field of C * C^{*} -algebras ( C ⁢ ( K q ) ) q ∈ [ 0 , ∞ ] (C(K_{q}))_{q\in[0,\infty]} , where the fibres at q = 0 q=0 and ∞ are explicitly defined higher-rank graph algebras.

Stable Isomorphisms of Operator Algebras

Abstract Let ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ be operator algebras with $c_{0}$-isomorphic diagonals and let ${{\mathcal{K}}}$ denote the compact operators. We show that if ${{\mathcal{A}}}\otimes{{\mathcal{K}}}$ and ${{\mathcal{B}}}\otimes{{\mathcal{K}}}$ are isometrically isomorphic, then ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are isometrically isomorphic. If the algebras ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ satisfy an extra analyticity condition a similar result holds with ${{\mathcal{K}}}$ being replaced by any operator algebra containing the compact operators. For nonselfadjoint graph algebras this implies that the graph is a complete invariant for various types of isomorphisms, including stable isomorphisms, thus strengthening a recent result of Dor-On, Eilers, and Geffen. Similar results are proven for algebras whose diagonals satisfy cancellation and have $K_{0}$-groups isomorphic to ${{\mathbb{Z}}}$. This has implications in the study of stable isomorphisms between various semicrossed products.

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let k be an arbitrary field, be a k-algebra and V be a -module. When it exists, the universal deformation ring of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let be any 1-domestic, symmetric special biserial algebra. By viewing as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring .

On the classification and description of quantum lens spaces as graph algebras

AbstractWe investigate quantum lens spaces, $C(L_q^{2n+1}(r;\underline {m}))$ , introduced by Brzeziński and Szymański as graph $C^*$ -algebras. We give a new description of $C(L_q^{2n+1}(r;\underline {m}))$ as graph $C^*$ -algebras amending an error in the original paper by Brzeziński and Szymański. Furthermore, for $n\leq 3$ , we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz, and Sørensen.

Graded K-theory, filtered K-theory and the classification of graph algebras

Graded K-theory, filtered K-theory and the classification of graph algebras

Annihilator ideals of graph algebras

If I is a (two-sided) ideal of a ring R, we let $${\text {ann}}_l(I)=\{r\in R\mid rI=0\},$$ $${\text {ann}}_r(I)=\{r\in R\mid Ir=0\},$$ and $${\text {ann}}(I)={\text {ann}}_l(I)\cap {\text {ann}}_r(I)$$ be the left, the right and the double annihilators. An ideal I is said to be an annihilator ideal if $$I={\text {ann}}(J)$$ for some ideal J (equivalently, $${\text {ann}}({\text {ann}}(I))=I$$ ). We study annihilator ideals of Leavitt path algebras and graph $$C^*$$ -algebras. Let $$L_K(E)$$ be the Leavitt path algebra of a graph E over a field K. If I is an ideal of $$L_K(E),$$ it has recently been shown that $${\text {ann}}(I)$$ is a graded ideal (with respect to the natural grading of $$L_K(E)$$ by $$\mathbb Z$$ ). We note that $${\text {ann}}_l(I)$$ and $${\text {ann}}_r(I)$$ are also graded. If I is graded, we show that $${\text {ann}}_l(I)={\text {ann}}_r(I)={\text {ann}}(I)$$ and describe $${\text {ann}}(I)$$ in terms of the properties of a pair of sets of vertices of E, known as an admissible pair, which naturally corresponds to I. Using such a description, we present properties of E which are equivalent with the requirement that each graded ideal of $$L_K(E)$$ is an annihilator ideal. We show that the same properties of E are also equivalent with each of the following conditions: (1) the lattice of graded ideals of $$L_K(E)$$ is a Boolean algebra; (2) each closed gauge-invariant ideal of $$C^*(E)$$ is an annihilator ideal; (3) the lattice of closed gauge-invariant ideals of $$C^*(E)$$ is a Boolean algebra. In addition, we present properties of E which are equivalent with each of the following conditions: (1) each ideal of $$L_K(E)$$ is an annihilator ideal; (2) the lattice of ideals of $$L_K(E)$$ is a Boolean algebra; (3) each closed ideal of $$C^*(E)$$ is an annihilator ideal; (4) the lattice of closed ideals of $$C^*(E)$$ is a Boolean algebra.

Compact quantum metric spaces from free graph algebras

Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra [Formula: see text] defined in Hartglass–Penneys’ article on canonical [Formula: see text]-algebras associated to a planar algebra. Under mild conditions, [Formula: see text] is a non-nuclear simple [Formula: see text]-algebra with unique tracial state. There is a canonical polynomial subalgebra [Formula: see text] together with a Dirac number operator [Formula: see text] such that [Formula: see text] is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify [Formula: see text] yields a compact quantum metric space in the sense of Rieffel.We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our [Formula: see text]-algebras are non-nuclear, we adjust the Lip-norm coming from [Formula: see text] to utilize the finite dimensional filtration of [Formula: see text]. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces.As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) [Formula: see text]-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS [Formula: see text]-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance.