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Perturbative Expansion of Yang-Baxter Operators

Emanuele Zappala

Abstract

We study the deformations of a wide class of Yang-Baxter (YB) operators arising from Lie algebras. We relate the higher order deformations of YB operators to Lie algebra deformations. We show that the obstruction to integrating deformations of self-distributive (SD) objects lie in the corresponding Lie algebra third cohomology group, and interpret this result in terms of YB deformations. We show that there are YB operators that admit integrable deformations (i.e. that can be deformed infinitely many times), and that therefore give rise to a full perturbative series in the deformation parameter $\hbar$. We consider the second cohomology group of YB operators corresponding to certain types of Lie algebras, and show that this can be nontrivial even if the Lie algebra is rigid, providing examples of nontrivial YB deformations that do not arise from SD deformations.

Perturbative Expansion of Yang-Baxter Operators

Abstract

We study the deformations of a wide class of Yang-Baxter (YB) operators arising from Lie algebras. We relate the higher order deformations of YB operators to Lie algebra deformations. We show that the obstruction to integrating deformations of self-distributive (SD) objects lie in the corresponding Lie algebra third cohomology group, and interpret this result in terms of YB deformations. We show that there are YB operators that admit integrable deformations (i.e. that can be deformed infinitely many times), and that therefore give rise to a full perturbative series in the deformation parameter . We consider the second cohomology group of YB operators corresponding to certain types of Lie algebras, and show that this can be nontrivial even if the Lie algebra is rigid, providing examples of nontrivial YB deformations that do not arise from SD deformations.
Paper Structure (8 sections, 8 theorems, 41 equations)

This paper contains 8 sections, 8 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Racks and quandles
  4. Yang-Baxter operators
  5. Higher deformations of rack objects
  6. Higher deformations of Yang-Baxter operators
  7. More on second cohomology
  8. Examples

Key Result

Theorem 3.1

Let $\frak g$ be an $n$-Lie algebra and let $(X,T,\Delta)$ denote its corresponding $n$-rack object. Assume that $\phi = \sum_{i=0}^m \hbar^i\phi_i$ is an order $m$ deformation of the bracket of $\frak g$. Then, the correspondence $\Theta^{m+1}$ gives an order $m+1$ deformation of $X$ if the obstruc

Theorems & Definitions (20)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Corollary 4.3
  • proof
  • Corollary 4.4
  • ...and 10 more