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Prolongation rigidity of sub-free Lie algebras

Boris Kruglikov

Abstract

We prove that if the 0-th Tanaka prolongation $\mathfrak{g}_0=\mathfrak{der}_0(\mathfrak{m})$ of a fundamental graded nilpotent Lie algebra $\mathfrak{m}=\mathfrak{g}_{-s}\oplus\dots\oplus\mathfrak{g}_{-1}$ is irreducible on $\mathfrak{g}_{-1}$, then $\mathfrak{m}$ is prolongation rigid: $\text{pr}_+(\mathfrak{m})=0$. The only exceptions are given by negative gradations of maximal parabolic subalgebras of a simple Lie algebra.

Prolongation rigidity of sub-free Lie algebras

Abstract

We prove that if the 0-th Tanaka prolongation of a fundamental graded nilpotent Lie algebra is irreducible on , then is prolongation rigid: . The only exceptions are given by negative gradations of maximal parabolic subalgebras of a simple Lie algebra.
Paper Structure (3 sections, 5 theorems, 10 equations, 1 table)

This paper contains 3 sections, 5 theorems, 10 equations, 1 table.

Table of Contents

  1. Formulation of the Problem and the Results
  2. Examples of sub-free Lie algebras
  3. Prolongation rigidity and generalization

Key Result

Theorem 1

Let ${\frak m}$ be a fundamental GNLA different from ${\frak m}_I$ in the case of parabolic ${\mathfrak p}_1$ for $G_2$ or ${\mathfrak p}_r$ for $B_r$. If ${\frak m}$ is sub-free, then it is prolongation rigid: $\mathop{\rm pr}\nolimits_+({\frak m})=0$.

Theorems & Definitions (16)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['Th1']}
  • Remark 1
  • proof : Proof of Corollary \ref{['Cor1']}
  • Definition 2
  • Theorem 2
  • ...and 6 more