Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0002215

Geometrical techniques for the N-dimensional Quantum Euclidean Spaces

B. L. Cerchiai·G. Fiore·J. Madore
View original source ↗Read source document ↗Export metadata ↓

Abstract

We briefly report our application of a version of noncommutative geometry to the quantum Euclidean space $R^N_q$, for any $N \ge 3$; this space is covariant under the action of the quantum group $SO_q(N)$, and two covariant differential calculi are known on it. More precisely, we describe how to construct in a Cartan `moving-frame formalism' the metric, two covariant derivatives, the Dirac operator, the frame, the inner derivations dual to the frame elements, for both of these calculi. The components of the frame elements in the basis of differentials provide a `local realization' of the Faddeev-Reshetikhin-Takhtadjan generators of $U_q^{\pm}(so(N))$.

Explore related subjects

math.QAmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

B. L. Cerchiai, G. Fiore, J. Madore. 2000-02-25. Geometrical techniques for the N-dimensional Quantum Euclidean Spaces. https://arxiv.org/abs/math/0002215

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

KEEP EXPLORING

Related papers

arXiv2026

On singular vectors and associated Weyl elements for simply-laced universal affine vertex operator algebras

Given a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ and a complex number $κ$, let $V^κ(\mathfrak{g})$ be the associated universal affine vertex algebra. Gorelik and Kac [GK07] gave a sufficient and necessary condition for $V^κ(\mathfrak{g})$ to be simple. In this paper, for simply-laced $\mathfrak{g}$ and non-critical $κ$, we determine the weights of singular vectors of $V^κ(\mathfrak{g})$ with minimal conformal weights, when $V^κ(\mathfrak{g})$ is not simple. We further determine all the longest Weyl elements in the Kashiwara-Tanisaki character theorem [KT00] which correspond to the weights of the singular vectors.

math.QA↗
arXiv2026

Modular invariants and NIM-reps

Given a pivotal module category over a spherical fusion category, we introduce the encircling module, a module over the fusion algebra defined using the pivotal structure, and prove that it is isomorphic to the NIM-rep as a fusion algebra module. When applied to the $\mathcal{TM}$ realisation of the modular invariant partition function (arXiv:1911.09024), this identifies the NIM-rep multiplicities with the entries $Z_{I,I^\vee}$ of the modular invariant (its diagonal relative to the charge-conjugation invariant, which is the one given by the regular module category), providing a categorical generalisation of Böckenhauer, Evans and Kawahigashi's results (arXiv:math/9907149). We also show that for indecomposable module categories the dimension condition on $\mathcal{TM}$ required for modular invariance is automatically satisfied, and that $\mathcal{TM}$ is, as an algebra, the full centre of the module category in the sense of Davydov, Kong and Runkel (arXiv:1307.5956), which for module categories of algebras is the full centre of Fjelstad, Fuchs, Runkel and Schweigert (arXiv:hep-th/0612306). Consequently $\mathcal{TM}$ gives rise to Cardy algebras in the sense of Kong and Runkel (arXiv:0807.3356).

math.QA↗
arXiv2026

A diagrammatic presentation for every pivotal pointed fusion category

We provide a generators and relations presentation of pivotal pointed fusion categories, $Vec(G,ω,π)$. Unlike the well-known skeletal model, our presentation is strict and allows multiple isomorphic objects. Our main tool is skein theory, which allows us to apply topological tools to understand the relations of morphisms in the category. This expository paper provides details of skein-theoretic work in the setup that allows multiple isomorphic labels in the generators, which didn't previously exist in the literature.

math.QA↗