Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1706.10168

A noncommutative catenoid

Joakim Arnlind·Christoffer Holm
View original source ↗Read source document ↗Export metadata ↓

Abstract

A noncommutative algebra corresponding to the classical catenoid is introduced together with a differential calculus of derivations. We prove that there exists a unique metric and torsion-free connection that is compatible with the complex structure, and the curvature is explicitly calculated. A noncommutative analogue of the fact that the catenoid is a minimal surface is studied by constructing a Laplace operator from the connection and showing that the embedding coordinates are harmonic. Furthermore, an integral is defined and the total curvature is computed. Finally, classes of left and right modules are introduced together with constant curvature connections, and bimodule compatibility conditions are discussed in detail.

Explore related subjects

math.QAmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joakim Arnlind, Christoffer Holm. 2017-06-30. A noncommutative catenoid. https://doi.org/10.1007/s11005-017-1042-z

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

Schneps \cite{Schneps2012,Schneps2025} and Enriquez-Furusho \cite{EF4} proved that the double shuffle Lie algebra $\mathfrak{dmr}_0$ embeds into the Kashiwara--Vergne Lie algebra $\mathfrak{krv}_2$. We give a Brunnian braid interpretation of a related embedding into the symmetric Kashiwara--Vergne Lie algebra $\mathfrak{krv}_2^{\mathrm{sym}}$. More precisely, the map \[ φ\longmapsto \bigl(φ(-x_0-x_1,x_0),φ(-x_0-x_1,x_1)\bigr) \] defines an injective Lie algebra homomorphism from the subalgebra of $\mathfrak{dmr}_0$ satisfying the condition \[ [x_0,φ(-x_0-x_1,x_0)] +[x_1,φ(-x_0-x_1,x_1)]=0 \] into $\mathfrak{krv}_2^{\mathrm{sym}}$. The proof reformulate the double shuffle and symmetric Kashiwara--Vergne relations through abelianizations of Brunnian Lie algebras associated with the disk and punctured disks. We generalize this inclusion in two directions. First, replacing these abelianizations by higher lower central series quotients yields generalizations of relations and implications among them. Second, we establish explicit identities relating the linear pentagon defect to the stuffle coproduct, the divergence map, and the necklace cobracket.

math.QA↗
arXiv2026

Birational Equivalences for Kac--Moody Borel Enveloping Algebras

A Coxeter ordering of the simple roots of a finite-rank Kac--Moody algebra determines a finite family of commuting real-root vectors. We prove that $U^{\geq0}(\mathfrak g)$ is birationally equivalent to $Z\otimes\mathbb A_n$, where $Z$ is the residual Coxeter centralizer, by identifying the Coxeter localization $U^{\geq0}(\mathfrak g)[\mathbf X^{-1}]$ with $Z\otimes\mathbb A_n[\mathbf x^{-1}]$. For symmetrizable Cartan matrices the residual algebra is generated by finite Coxeter windows and is finitely presented. For the generic quantum Borel with torus dual to the root lattice, we prove the analogous birational equivalence.

math.QA↗
arXiv2026

Kontsevich's Cocycle Construction and Quantization of the Loday-Quillen-Tsygan Theorem

We relate graph complexes, Calabi-Yau $A_\infty$-categories and Kontsevich's cocycle construction. Our main result produces a commutative square of shifted Poisson algebras; one of its edges is the Loday-Quillen-Tsygan map, generalized to $A_\infty$-categories. We describe a quantized version via Beilinson-Drinfeld algebras. The larger context is to provide categorical methods which relate enumerative geometry (as in mirror symmetry) and large $N$ gauge theories.

math.QA↗