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Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei

Tomasz Trześniewski

Abstract

There is a surge of research devoted to the formalism and physical manifestations of non-Lorentzian kinematical symmetries, which focuses especially on the ones associated with the Galilei and Carroll relativistic limits (the speed of light taken to infinity or to zero, respectively). The investigations have also been extended to quantum deformations of the Carrollian and Galilean symmetries, in the sense of (quantum) Hopf algebras. The case of 2+1 dimensions is particularly worth to study due to both the mathematical nature of the corresponding (classical) theory of gravity, and the recently finalized classification of all quantum-deformed algebras of spacetime isometries. Consequently, the list of all quantum deformations of (anti-)de Sitter-Carroll algebra is immediately provided by its well-known isomorphism with either Poincaré or Euclidean algebra. Quantum contractions from the (anti-)de Sitter to (anti-)de Sitter-Carroll classification allow to almost completely recover the latter. One may therefore conjecture that the analogous contractions from the (anti-)de Sitter to (anti-)de Sitter-Galilei $r$-matrices provide (almost) all coboundary deformations of (anti-)de Sitter-Galilei algebra. This scheme is complemented by deriving (Carrollian and Galilean) quantum contractions of deformations of Poincaré algebra, leading to coboundary deformations of Carroll and Galilei algebras.

Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei

Abstract

There is a surge of research devoted to the formalism and physical manifestations of non-Lorentzian kinematical symmetries, which focuses especially on the ones associated with the Galilei and Carroll relativistic limits (the speed of light taken to infinity or to zero, respectively). The investigations have also been extended to quantum deformations of the Carrollian and Galilean symmetries, in the sense of (quantum) Hopf algebras. The case of 2+1 dimensions is particularly worth to study due to both the mathematical nature of the corresponding (classical) theory of gravity, and the recently finalized classification of all quantum-deformed algebras of spacetime isometries. Consequently, the list of all quantum deformations of (anti-)de Sitter-Carroll algebra is immediately provided by its well-known isomorphism with either Poincaré or Euclidean algebra. Quantum contractions from the (anti-)de Sitter to (anti-)de Sitter-Carroll classification allow to almost completely recover the latter. One may therefore conjecture that the analogous contractions from the (anti-)de Sitter to (anti-)de Sitter-Galilei -matrices provide (almost) all coboundary deformations of (anti-)de Sitter-Galilei algebra. This scheme is complemented by deriving (Carrollian and Galilean) quantum contractions of deformations of Poincaré algebra, leading to coboundary deformations of Carroll and Galilei algebras.
Paper Structure (16 sections, 69 equations, 4 figures, 1 table)

This paper contains 16 sections, 69 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classical $r$-matrices (in 2+1 dimensions)
  3. The classification for Poincaré algebra
  4. The classification for (anti-)de Sitter algebra
  5. Carrollian and Galilean (quantum) contractions
  6. Contractions of Poincaré and (a)dS algebras
  7. Quantum contractions of $r$-matrices for Poincaré algebra
  8. (a)dS-Carroll algebras and their quantum deformations
  9. The complete classification of adSC and dSC $r$-matrices
  10. Carrollian quantum contractions in the de Sitter case
  11. Carrollian quantum contractions in the anti-de Sitter case
  12. (A)dS-Galilei $r$-matrices from quantum contractions
  13. The de Sitter case
  14. The anti-de Sitter case
  15. Conclusions
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Quantum ($\Lambda \to 0$) contractions relating all $r$-matrix classes for (anti-)de Sitter and Poincaré algebras; a two-headed arrow means that a given contraction recovers the full class; double arrows denote automorphisms of a given algebra; arrows leading to $r_8$ are lightened to make the diagram more legible.
  • Figure 2: Quantum ($c \to 0$ and $c \to \infty$) contractions relating all $r$-matrix classes for Poincaré algebra, and those obtained for Carroll and Galilei algebras; a dashed line means that a given contraction leads to a subclass of a larger class.
  • Figure 3: Quantum ($c \to 0$ and $\Lambda \to 0$) contractions relating all $r$-matrix classes for de Sitter and anti-de Sitter algebras, all of those for dS-Carroll and adS-Carroll algebras, and those obtained for Carroll algebra; a two-headed arrow means that a given $c \to 0$ contraction recovers the full class (i.e., it is surjective).
  • Figure 4: Quantum ($c \to \infty$ and $\Lambda \to 0$) contractions relating all $r$-matrix classes for de Sitter and anti-de Sitter algebras, those obtained for dS-Galilei and adS-Galilei algebras, and those obtained for Galilei algebra; a dashed line means that a given $c \to \infty$ contraction leads to a subclass of a larger class.