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On conformal pseudo-subriemannian fundamental graded Lie algebras associated with pseudo $H$-type Lie algebras

Tomoaki Yatsui

Abstract

A pseudo $H$-type Lie algebra naturally gives rise to a conformal pseudo-subriemannian fundamental graded Lie algebras. In this paper we investigate the prolongations of the associated fundamental graded Lie algebra and the associated conformal pseudo-subriemannian fundamental graded Lie algebra. In particular, we show that the prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under some assumptions.

On conformal pseudo-subriemannian fundamental graded Lie algebras associated with pseudo $H$-type Lie algebras

Abstract

A pseudo -type Lie algebra naturally gives rise to a conformal pseudo-subriemannian fundamental graded Lie algebras. In this paper we investigate the prolongations of the associated fundamental graded Lie algebra and the associated conformal pseudo-subriemannian fundamental graded Lie algebra. In particular, we show that the prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under some assumptions.

Paper Structure

This paper contains 22 sections, 23 theorems, 90 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Pseudo H-type Lie algebras
  3. ComH-type Lie algebras
  4. Cayley algebras
  5. ComH-type Lie algebras of the first class
  6. ComH-type Lie algebras of the second and the third classes
  7. Prolongations of FGLAs associated with comH-type Lie algebras
  8. General results
  9. Pseudo H-type Lie algebras with $\dim\mathfrak n_{-2}\leqq2$
  10. Matricial models of comH-type Lie algebras of the first class
  11. Matricial Models of comH-type Lie algebras of the second class
  12. Matricial models of comH-type Lie algebras of the third class
  13. The case of signature $(1,3)$
  14. The case of signature $(3,1)$
  15. Summary
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $(\mathfrak n=\mathfrak n_{-1}\oplus \mathfrak n_{-2},\langle\cdot\,|\,\cdot\rangle)$ be a pseudo H-type Lie algebra. We define a new scalar product $\langle\cdot\,|\,\cdot\rangle'$ on $\mathfrak n$ as follows: where $\alpha,\beta$ are nonzero real numbers. The pair $(\mathfrak n=\mathfrak n_{-1}\oplus \mathfrak n_{-2},\langle\cdot\,|\,\cdot\rangle')$ also becomes a pseudo H-type Lie algebra

Theorems & Definitions (31)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Proposition 2.1
  • Lemma 3.1
  • Remark 3.1
  • Lemma 3.2
  • Remark 3.2
  • Remark 3.3
  • Proposition 4.1: AS14:2*Theorem 2.3
  • ...and 21 more