Finiteness conditions for graph algebras over tropical semirings

Connection matrices for graph parameters with values in a field have been introduced by M. Freedman, L. Lov{\'a}sz and A. Schrijver (2007). Graph parameters with connection matrices of finite rank can be computed in polynomial time on graph classes of bounded tree-width. We introduce join matrices, a generalization of connection matrices, and allow graph parameters to take values in the tropical rings (max-plus algebras) over the real numbers. We show that rank-finiteness of join matrices implies that these graph parameters can be computed in polynomial time on graph classes of bounded clique-width. In the case of graph parameters with values in arbitrary commutative semirings, this remains true for graph classes of bounded linear clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that definability of a graph parameter in Monadic Second Order Logic implies rank finiteness. We also show that there are uncountably many integer valued graph parameters with connection matrices or join matrices of fixed finite rank. This shows that rank finiteness is a much weaker assumption than any definability assumption.Comment: 12 pages, accepted for presentation at FPSAC 2014 (Chicago, June 29 -July 3, 2014), to appear in Discrete Mathematics and Theoretical Computer Scienc

Algebras of quantum monodromy data and decorated character varieties

The Riemann-Hilbert correspondence is an isomorphism between the de Rham moduli space and the Betti moduli space, defined by associating to each Fuchsian system its monodromy representation class. In 1997 Hitchin proved that this map is a symplectomorphism. In this paper, we address the question of what happens to this theory if we extend the de Rham moduli space by allowing connections with higher order poles. In our previous paper arXiv:1511.03851, based on the idea of interpreting higher order poles in the connection as boundary components with bordered cusps (vertices of ideal triangles in the Poincar\'e metric) on the Riemann surface, we introduced the notion of decorated character variety to generalize the Betti moduli space. This decorated character variety is the quotient of the space of representations of the fundamental groupid of arcs by a product of unipotent Borel sub-groups (one per bordered cusp). Here we prove that this representation space is endowed with a Poisson structure induced by the Fock--Rosly bracket and show that the quotient by unipotent Borel subgroups giving rise to the decorated character variety is a Poisson reduction. We deal with the Poisson bracket and its quantization simultaneously, thus providing a quantisation of the decorated character variety. In the case of dimension 2, we also endow the representation space with explicit Darboux coordinates. We conclude with a conjecture on the extended Riemann-Hilbert correspondence in the case of higher order poles.Comment: Dedicated to Nigel Hitchin for his 70th birthday. 22 pages, 6 figure

Leavitt path algebras: the first decade

The algebraic structures known as {\it Leavitt path algebras} were initially developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a different approach) by the author and Aranda Pino. During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in C$^*$-algebras, group theorists, and symbolic dynamicists as well. The goal of this article is threefold: to introduce the notion of Leavitt path algebras to the general mathematical community; to present some of the important results in the subject; and to describe some of the field's currently unresolved questions.Comment: 53 pages. To appear, Bulletin of Mathematical Sciences. (page numbering in arXiv version will differ from page numbering in BMS published version; numbering of Theorems, etc ... will be the same in both versions

Generalized double affine Hecke algebras of higher rank

We define generalized double affine Hecke algebras (GDAHA) of higher rank, attached to a non-Dynkin star-like graph D. This generalizes GDAHA of rank 1 defined in math.QA/0406480 and math.QA/0409261. If the graph is extended D4, then GDAHA is the algebra defined by Sahi in q-alg/9710032, which is a generalization of the Cherednik algebra of type BCn. We prove the formal PBW theorem for GDAHA, and parametrize its irreducible representations in the case when D is affine (i.e. extended D4, E6, E7, E8) and q=1. We formulate a series of conjectures regarding algebraic properties of GDAHA. We expect that, similarly to how GDAHA of rank 1 provide quantizations of del Pezzo surfaces (as shown in math.QA/0406480), GDAHA of higher rank provide quantizations of deformations of Hilbert schemes of these surfaces. The proofs are based on the study of the rational version of GDAHA (which is closely related to the algebras studied in math.QA/0401038), and differential equations of Knizhnik-Zamolodchikov type.Comment: 25 pages, latex; minor corrections are made, some proofs were expande

On the periodicity of Coxeter transformations and the non-negativity of their Euler forms

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for triangular algebras, as demonstrated by incidence algebras of posets. We also give a simple, direct proof, that certain products of reflections, defined for any square matrix A with 2 on its main diagonal, and in particular the Coxeter transformation corresponding to a generalized Cartan matrix, can be expressed as $-A_{+}^{-1} A_{-}^t$, where A_{+}, A_{-} are closely associated with the upper and lower triangular parts of A.Comment: 12 pages, (v2) revision, to appear in Linear Algebra and its Application

Wavelets and graph C∗C^*-algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs

From conformal embeddings to quantum symmetries: an exceptional SU(4) example

We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a quantum graph. For known examples, the corresponding modular invariant partition function, which is sometimes associated with a conformal embedding, provides enough information to recover the whole structure. We illustrate these notions with the example of the conformal embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures (ps files) in such a way that they are still understood when converted to gray levels. Version 3: Several references have been adde

The Classification of the Simply Laced Berger Graphs from Calabi-Yau CY3CY_3 spaces

The algebraic approach to the construction of the reflexive polyhedra that yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres reveals graphs that include and generalize the Dynkin diagrams associated with gauge symmetries. In this work we continue to study the structure of graphs obtained from $CY_3$ reflexive polyhedra. The objective is to describe the ``simply laced'' cases, those graphs obtained from three dimensional spaces with K3 fibers which lead to symmetric matrices. We study both the affine and, derived from them, non-affine cases. We present root and weight structurea for them. We study in particular those graphs leading to generalizations of the exceptional simply laced cases $E_{6,7,8}$ and $E_{6,7,8}^{(1)}$. We show how these integral matrices can be assigned: they may be obtained by relaxing the restrictions on the individual entries of the generalized Cartan matrices associated with the Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras. These graphs keep, however, the affine structure present in Kac-Moody Dynkin diagrams. We conjecture that these generalized simply laced graphs and associated link matrices may characterize generalizations of Cartan-Lie and affine Kac-Moody algebras