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The Einstein condition for quantum irreducible flag manifolds

Marco Matassa

Abstract

We show that any quantum irreducible flag manifold satisfies an analogue of the Einstein condition, expressing proportionality between the Ricci tensor and the metric, at least in a small open interval around the classical value of the quantization parameter. This makes use of various canonical constructions associated to these algebras, such as differential calculi and bimodule connections, which were previously introduced by various authors.

The Einstein condition for quantum irreducible flag manifolds

Abstract

We show that any quantum irreducible flag manifold satisfies an analogue of the Einstein condition, expressing proportionality between the Ricci tensor and the metric, at least in a small open interval around the classical value of the quantization parameter. This makes use of various canonical constructions associated to these algebras, such as differential calculi and bimodule connections, which were previously introduced by various authors.
Paper Structure (16 sections, 15 theorems, 44 equations)

This paper contains 16 sections, 15 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum irreducible flag manifolds
  4. Differential calculi
  5. Takeuchi's equivalence
  6. Quantum metrics and connections
  7. Quantum metrics
  8. Connections
  9. Construction of lifting maps
  10. Lifting maps
  11. Heckenberger--Kolb calculus
  12. Ricci tensor and Einstein condition
  13. Definitions
  14. Results for the Ricci tensor
  15. The Einstein condition
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathcal{O}_q(U / K_S)$ be a quantum irreducible flag manifold with the quantum metric $g$ as described above. Then there exists a lifting map $\ell$ such that $\mathcal{O}_q(U / K_S)$ satisfies the Einstein condition in an open interval around the classical value $q = 1$.

Theorems & Definitions (40)

  • Theorem
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4: takeuchi
  • Remark 2.5
  • Corollary 2.6
  • Remark 2.7
  • Definition 3.1
  • Theorem 3.2
  • ...and 30 more