Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.11888

On two notions of torsion and metric compatibility of connections in noncommutative geometry

Jyotishman Bhowmick·Bappa Ghosh·Satyajit Guin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We compare the notions of metric-compatibility and torsion of a connection in the frameworks of Beggs-Majid and Mesland-Rennie. It follows that for $\ast$-preserving connections, compatibility with a real metric in the sense of Beggs-Majid corresponds to Hermitian connections in the sense of Mesland-Rennie. If the calculus is quasi-tame, the torsion zero conditions are equivalent. A combination of these results proves the existence and uniqueness of Levi-Civita connections in the sense of Mesland-Rennie for unitary cocycle deformations of a large class of Riemannian manifolds as well as the Heckenberger-Kolb calculi on all quantized irreducible flag manifolds.

Explore related subjects

math.QAmath-phmath.MPmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jyotishman Bhowmick, Bappa Ghosh, Satyajit Guin. 2025-09-15. On two notions of torsion and metric compatibility of connections in noncommutative geometry. https://arxiv.org/abs/2509.11888

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

A $q$-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension

Let $H$ be a pointed Hopf algebra of square-free dimension over an algebraically closed field of characteristic $p>0$. We prove that either $H$ is a group algebra or $\dim H/|\G(H)|=p$, and that in the latter case $H$ belongs to exactly one of two explicit families of rank-one pointed Hopf algebras. We develop a truncated $q$-Weyl freeness principle for finite-dimensional Nichols algebras of quandle type. If $V=\bigoplus_{x\in X}\K e_x$, $X'\subsetneq X$ is a nonempty subquandle, $V'=\bigoplus_{x\in X'}\K e_x$, and $s\in X\setminus X'$, then $\mathcal B(V)\simeq\K[e_s]/(e_s^{m_s})\otimes C_{s,X'}\otimes\mathcal B(V')$ for some graded vector space $C_{s,X'}$, where $m_s$ is the nilpotency order of $e_s$; in particular, $(m_s)_z\,\mathcal H_{\mathcal B(V')}(z)\mid\mathcal H_{\mathcal B(V)}(z)$. In the non-group case, this yields a $p^2$-divisibility obstruction that rules out noncentral support for the infinitesimal braiding. Together with a graded-dual argument, the resulting rank-one reduction forces the diagram of $H$ to have dimension $p$.

math.QA↗