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Infinitesimal deformations of $\mathfrak{sl}_2$ with a twisted Jacobi identity

Haoran Zhu

Abstract

We show that whenever \[ [\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad α_t = \mathrm{id} + tα_1 \] define an infinitesimal Hom--Lie deformation of $\mathfrak{sl}_2(\mathbb K)$ over $\mathbb K[t]/(t^2)$ and $(\mathfrak{sl}_2(\mathbb K),[\,\cdot,\cdot]_0,α_1)$ is a Hom--Lie algebra, then the deformed bracket $[\,\cdot,\cdot]_t$ satisfies the ordinary Jacobi identity over $\mathbb K[t]$. This solves a conjecture of Makhlouf and Silvestrov from 2010.

Infinitesimal deformations of $\mathfrak{sl}_2$ with a twisted Jacobi identity

Abstract

We show that whenever \[ [\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad α_t = \mathrm{id} + tα_1 \] define an infinitesimal Hom--Lie deformation of over and is a Hom--Lie algebra, then the deformed bracket satisfies the ordinary Jacobi identity over . This solves a conjecture of Makhlouf and Silvestrov from 2010.
Paper Structure (5 sections, 5 theorems, 31 equations)

This paper contains 5 sections, 5 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Proof of the Makhlouf--Silvestrov conjecture
  3. The constraint that $(V,[\cdot,\cdot]_0,\alpha_1)$ is Hom--Lie
  4. First--order Hom--Lie deformations
  5. The ordinary Jacobi identity for $[\cdot,\cdot]_t$

Key Result

Theorem 1.1

Let $(V,[\cdot,\cdot]_0)\cong\mathfrak{sl}_2(\mathbb K)$. Fix a basis $x_1,x_2,x_3$ of $V$ and define $[\cdot,\cdot]_0$ by eq:sl2-bracket. Consider where $[\cdot,\cdot]_1\colon V\times V\to V$ is bilinear and skew--symmetric and $\alpha_1\colon V\to V$ is linear. Assume that $([\cdot,\cdot]_t,\alpha_t)$ defines an infinitesimal Hom--Lie deformation of $(V,[\cdot,\cdot]_0)$ over $\mathbb K[t]/(t^2

Theorems & Definitions (11)

  • Conjecture 1: Makhlouf--Silvestrov
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Corollary 2.4: Makhlouf-Silvestrov conjecture
  • ...and 1 more