Abstract
We propose a general procedure to construct noncommutative deformations of an algebraic submanifold M of mathbb {R}^{n}, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds ofmathbb {R}^{n}, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883–1911, 2006), whereby the commutative pointwise product is replaced by the ⋆-product determined by a Drinfel’d twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds Mc that are level sets of the fa(x), where fa(x) = 0 are the polynomial equations solved by the points of M, employing twists based on the Lie algebra Ξt of vector fields that are tangent to all the Mc. The twisted Cartan calculus is automatically equivariant under twisted Ξt. If we endow mathbb {R}^{n} with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted M is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and ⋆-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in mathbb {R}^{3} except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean mathbb {R}^{3} and twisted hyperboloids embedded in twisted Minkowski mathbb {R}^{3} [the latter are twisted (anti-)de Sitter spaces dS2, AdS2].
Highlights
The concept of a submanifold of a manifold plays a fundamental role in mathematics and physics
The question whether, and to what extent, a notion of a submanifold is possible in Noncommutative Geometry (NCG) has received little systematic attention
-equivariant noncommutative algebra “of functions on the quantum Euclidean space R ”, which is generated by non-commuting coordinates, one can obtain the one A on the quantum Euclidean sphere 1 by imposing that the [central and
Summary
The concept of a submanifold of a manifold plays a fundamental role in mathematics and physics. Algebraic submanifolds of affine spaces such as R or. In the last few decades the program of generalizing differential geometry into so-called Noncommutative Geometry (NCG) has made a remarkable progress [14, 35, 41,42,43]; NCG might provide a suitable framework for a theory of quantum spacetime allowing the quantization of gravity [1, 20]) or for unifying fundamental interactions On several noncommutative (NC) spaces one can make sense of special classes of NC submanifolds, but some aspects of the latter may depart from their commutative counterparts. -equivariant noncommutative algebra “of functions on the quantum Euclidean space R ”, which is generated by non-commuting coordinates , one can obtain the one A on the quantum Euclidean sphere 1 by imposing that the [central and
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