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This paper studies the problem of graph-level clustering, which is a novel yet challenging task. This problem is critical in a variety of real-world applications such as protein clustering and genome analysis in bioinformatics. Recent years…

Machine Learning · Computer Science 2023-03-09 Wei Ju , Yiyang Gu , Binqi Chen , Gongbo Sun , Yifang Qin , Xingyuming Liu , Xiao Luo , Ming Zhang

The main purpose of this work is to develop the basic notions of the Lie theory for commutative algebras. We introduce a class of $\mathbbZ_2$-graded commutative but not associative algebras that we call ``Lie antialgebras''. These algebras…

Mathematical Physics · Physics 2010-10-18 Valentin Ovsienko

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and…

Differential Geometry · Mathematics 2018-11-09 Camilo Angulo

The $\Upsilon$ invariant is a concordance invariant defined by using knot Floer homology. F\"{o}ldv\'{a}ri gives a combinatorial restructure of it using grid homology. We extend the combinatorial $\Upsilon$ invariant for balanced spatial…

Geometric Topology · Mathematics 2024-06-10 Hajime Kubota

We propose a new nonlinear factorization model for graphs that are with topological structures, and optionally, node attributes. This model is based on a pseudometric called Gromov-Wasserstein (GW) discrepancy, which compares graphs in a…

Machine Learning · Computer Science 2019-11-21 Hongteng Xu

We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy…

Algebraic Topology · Mathematics 2013-08-21 Jim Conant , Martin Kassabov , Karen Vogtmann

Homography has an essential relationship with the special linear group and the embedding Lie algebra structure. Although the Lie algebra representation is elegant, few researchers have established the connection between homography and…

Computer Vision and Pattern Recognition · Computer Science 2022-11-15 Xinrui Zhan , Yang Li , Wenyu Liu , Jianke Zhu

Let N be a regular branched cover of a homology 3-sphere M with deck group G isomorphic to Z_2^d and branch set a trivalent graph Gamma; such a cover is determined by a coloring of the edges of Gamma with elements of G. For each index-2…

Geometric Topology · Mathematics 2009-09-25 R. A. Litherland

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a "bridge" between GLMY-theory and group homology theory, which helps to reduce path homology…

Algebraic Topology · Mathematics 2024-04-02 Shaobo Di , Sergei O. Ivanov , Lev Mukoseev , Mengmeng Zhang

Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways…

Algebraic Topology · Mathematics 2007-05-23 Dmitry N. Kozlov

We explain how to calculate link homology for a Lie algebra $\mathfrak{g}$ using the Fukaya category associated to a 2d A-model. Links are represented as configurations of particular A-branes and link homology is given by Homs between these…

High Energy Physics - Theory · Physics 2023-07-17 Elise LePage

We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\times$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a…

Operator Algebras · Mathematics 2008-02-22 Karl-Hermann Neeb

It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a…

Combinatorics · Mathematics 2021-10-28 Tobias Fritz

We give a presentation in terms of generators and relations of the cohomology in degree zero of the Campos-Willwacher graph complexes associated to compact orientable surfaces of genus $g$. The results carry a natural Lie algebra structure,…

Quantum Algebra · Mathematics 2021-05-06 Matteo Felder

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some…

Algebraic Topology · Mathematics 2015-04-14 Shigeyuki Morita , Takuya Sakasai , Masaaki Suzuki

We describe a correspondence between GL_n-invariant tensors and graphs, and show how this correspondence accomodates various types of symmetries and orientations.

Representation Theory · Mathematics 2009-08-12 Martin Markl

We consider homological finiteness properties $FP_n$ of certain $\mathbb{N}$-graded Lie algebras. After proving some general results, see Theorem A, Corollary B and Corollary C, we concentrate on a family that can be considered as the Lie…

Group Theory · Mathematics 2023-01-02 Dessislava H. Kochloukova , Conchita Martínez-Pérez

We prove that the inclusion from oriented graph complex into graph complex with at least one source is a quasi-isomorphism, showing that homology of the "sourced" graph complex is also equal to the homology of standard Kontsevich's graph…

Quantum Algebra · Mathematics 2018-02-14 Marko Živković

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$…

Differential Geometry · Mathematics 2016-05-24 Sergio Console , Anna Fino , Hisashi Kasuya

E. Sk\"oldberg's Morse Theory from an Algebraic Viewpoint and M. J\"ollenbeck's Algebraic Discrete Morse Theory and Applications to Commutative Algebra, which is the algebraic generalization of R. Forman's discrete Morse Theory for Cell…

Algebraic Topology · Mathematics 2019-08-08 Leon Lampret , Aleš Vavpetič