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Pluecker-type relations for orthogonal planes

José Figueroa-O'Farrill, George Papadopoulos

TL;DR

This work introduces and investigates a Plücker-type relation for middle-dimensional orthogonal planes in spaces with Euclidean or Lorentzian metrics, unifying the classical Plücker relations with Jacobi-type identities for metric Lie and n-Lie algebras. The authors formulate a conjecture: for F ∈ Λ^p V* and Ξ ∈ Λ^{p−2} V, the condition [ι_Ξ F, F] = 0 is satisfied precisely when F splits into a sum of two orthogonal simple p-forms (in d = 2p or 2p+1), while in smaller dimensions it forces simple F. They provide multiple low-dimensional verifications (notably p = 3 in d ≤ 7 and p = 4 in d = 8, as well as p = 5 in various signatures), using a combination of contraction analyses, Cartan subalgebras, and Lie-algebraic reinterpretations, including metric-n Lie algebra perspectives and Medina–Revoy-style decompositions. The results support a geometric interpretation of the Plücker-type relation and reveal a deep link between decomposability of p-forms and invariant metric structures, with implications for the geometry underlying maximally supersymmetric solutions in supergravity (e.g., type IIB and AdS backgrounds) and potential generalizations via metric n-Lie algebras. Overall, the paper provides explicit verifications that reinforce the conjecture in a broad range of low-dimensional cases and sketches a structural framework connecting classical Plücker geometry to modern algebraic structures and theoretical physics.

Abstract

We explore a Pluecker-type relation which occurs naturally in the study of maximally supersymmetric solutions of certain supergravity theories. This relation generalises at the same time the classical Pluecker relation and the Jacobi identity for a metric Lie algebra and coincides with the Jacobi identity of a metric n-Lie algebra. In low dimension we present evidence for a geometric characterisation of the relation in terms of middle-dimensional orthogonal planes in euclidean or lorentzian inner product spaces.

Pluecker-type relations for orthogonal planes

TL;DR

This work introduces and investigates a Plücker-type relation for middle-dimensional orthogonal planes in spaces with Euclidean or Lorentzian metrics, unifying the classical Plücker relations with Jacobi-type identities for metric Lie and n-Lie algebras. The authors formulate a conjecture: for F ∈ Λ^p V* and Ξ ∈ Λ^{p−2} V, the condition [ι_Ξ F, F] = 0 is satisfied precisely when F splits into a sum of two orthogonal simple p-forms (in d = 2p or 2p+1), while in smaller dimensions it forces simple F. They provide multiple low-dimensional verifications (notably p = 3 in d ≤ 7 and p = 4 in d = 8, as well as p = 5 in various signatures), using a combination of contraction analyses, Cartan subalgebras, and Lie-algebraic reinterpretations, including metric-n Lie algebra perspectives and Medina–Revoy-style decompositions. The results support a geometric interpretation of the Plücker-type relation and reveal a deep link between decomposability of p-forms and invariant metric structures, with implications for the geometry underlying maximally supersymmetric solutions in supergravity (e.g., type IIB and AdS backgrounds) and potential generalizations via metric n-Lie algebras. Overall, the paper provides explicit verifications that reinforce the conjecture in a broad range of low-dimensional cases and sketches a structural framework connecting classical Plücker geometry to modern algebraic structures and theoretical physics.

Abstract

We explore a Pluecker-type relation which occurs naturally in the study of maximally supersymmetric solutions of certain supergravity theories. This relation generalises at the same time the classical Pluecker relation and the Jacobi identity for a metric Lie algebra and coincides with the Jacobi identity of a metric n-Lie algebra. In low dimension we present evidence for a geometric characterisation of the relation in terms of middle-dimensional orthogonal planes in euclidean or lorentzian inner product spaces.

Paper Structure

This paper contains 48 sections, 1 theorem, 188 equations.

Table of Contents

  1. Introduction and main result
  2. The classical Plücker relations
  3. Orthogonal Plücker-type relations
  4. The case $p{=}2$
  5. The case $p{=}3$
  6. Metric $n$-Lie algebras
  7. Verifications in low dimension
  8. Proof for $F \in \Lambda^3\mathbb{E}^6$
  9. $\mathfrak{so}(4)$
  10. $\mathfrak{su}(2)$
  11. $\mathfrak{so}(2)$
  12. Proof for $F \in \Lambda^3\mathbb{E}^{1,5}$
  13. Proof for $F \in \Lambda^3\mathbb{E}^7$
  14. $\mathfrak{so}(6)$
  15. $\mathfrak{su}(2)\times \mathfrak{u}(1)$
  16. ...and 33 more sections

Key Result

Theorem 1

A $p$-form $F \in \Lambda^p\mathbb{V}^*$ is simple if and only if for every ($p{-}1$)-polyvector $\Xi \in \Lambda^{p-1}\mathbb{V}$, where $\iota_\Xi F$ denotes the one-form obtained by contracting $F$ with $\Xi$.

Theorems & Definitions (2)

  • Theorem 1
  • Conjecture 1