Twisted Quadrics and Algebraic Submanifolds in $\mathbb {R}^{n}$

Abstract

We propose a general procedure to construct noncommutative deformations of an algebraic submanifold M of $\mathbb {R}^{n}$ ℝ n , specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of$\mathbb {R}^{n}$ 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 $\mathbb {R}^{n}$ 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 $\mathbb {R}^{3}$ ℝ 3 except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $\mathbb {R}^{3}$ ℝ 3 and twisted hyperboloids embedded in twisted Minkowski $\mathbb {R}^{3}$ ℝ 3 [the latter are twisted (anti-)de Sitter spaces dS2, AdS2].

BMS algebras in 4 and 3 dimensions, their quantum deformations and duals

“…In the present section we employ the quantum Lie algebra formalism. Following [37] (for recent applications see also [38,40,41]) we define the twisted (quantum Lie algebra) generators of U B 3 n as…”

Twisted submanifolds of $${\mathbb {R}}^n$$

“…If f a (x) are polynomial functions fulfilling suitable irreducibility conditions and we set X = Pol • (R n ), the * -algebra of complex-valued polynomial functions on R n (instead of X = C ∞ (D f )), then again the * -algebra X M of complex-valued polynomial functions on M can be expressed as the quotient X M = X /C, where C ⊂ X is the ideal of polynomial functions vanishing on M, := {X = X i ∂ i | X i ∈ X } is the Lie algebra of polynomial vector fields X on R n , etc. C can be decomposed again in the form (3), with X = Pol • (R n ) [38].…”

“…In particular, one can apply [38] this procedure to algebraic submanifolds M ⊂ R n , e.g., quadrics (i.e., level sets of a polynomial function f (x) = 0 of degree 2); for the latter there exists a Lie subalgebra g (of dimension at least 2) of both t and the Lie algebra a f f (n) of the affine group…”

“…The same occurs with the X -bimodules and algebras • of differential forms, that of differential operators, etc. In [38] the authors discuss in detail deformations of all families of quadric surfaces embedded in R 3 that are induced by twists of the abelian [54] or Jordanian [52,53] type. In Sect.…”

On the Notion of Noncommutative Submanifold

“…Submanifold algebras have been recently studied from the point of view of Drinfel'd twists in [20,21,39].…”