Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Product Lie Algebroids, and Collective Motion

Begüm Ateşli, Oğul Esen, Serkan Sütlü

TL;DR

The paper addresses how to model collective motion of two mutually interacting systems within the Lie algebroid setting by constructing algebraic extensions that couple two bundles. It introduces the bicocycle double cross product (BDCP) Lie algebroids as the most general extension, establishing compatibility conditions that subsume unified products, double cross products, and cocycle extensions, and then develops reversible and irreversible dynamics (Euler-Poincaré-Herglotz, Lie-Poisson-Herglotz) on both the algebroid and its dual, including their BDCP decompositions. The key contributions are explicit BDCP bracket formulas, representation/cocycle data, and a full dynamical treatment—both in quasisymmetric (reversible) and dissipative (Herglotz/ Jacobi) contexts—for Lagrangian and Hamiltonian formalisms, with special cases recapitulating known constructions. This BDCP framework offers a versatile toolkit for modeling coupled physical systems (e.g., magnetohydrodynamics, plasma-fluid interactions) and for extending to Dirac, Jacobi, and higher-order algebroid structures in future work.

Abstract

This work explores the geometrical/algebraic framework of Lie algebroids, with a specific focus on the decoupling and coupling phenomena within the bicocycle double cross product realization. The bicocycle double cross product theory serves as the most general method for (de)coupling an algebroid into the direct sum of two vector bundles in the presence of mutual \textit{representations}, along with two twisted cocycle terms. Consequently, it encompasses unified product, double cross product (matched pairs), semi-direct product, and cocycle extension frameworks as particular instances. In addition to algebraic constructions, the research extends to both reversible and irreversible Lagrangian and Hamiltonian dynamics on (de)coupled Lie algebroids, as well as Euler-Poincaré-(Herglotz) and Lie-Poisson-(Herglotz) dynamics on (de)coupled Lie algebras, providing insights into potential physical applications.

On Product Lie Algebroids, and Collective Motion

TL;DR

The paper addresses how to model collective motion of two mutually interacting systems within the Lie algebroid setting by constructing algebraic extensions that couple two bundles. It introduces the bicocycle double cross product (BDCP) Lie algebroids as the most general extension, establishing compatibility conditions that subsume unified products, double cross products, and cocycle extensions, and then develops reversible and irreversible dynamics (Euler-Poincaré-Herglotz, Lie-Poisson-Herglotz) on both the algebroid and its dual, including their BDCP decompositions. The key contributions are explicit BDCP bracket formulas, representation/cocycle data, and a full dynamical treatment—both in quasisymmetric (reversible) and dissipative (Herglotz/ Jacobi) contexts—for Lagrangian and Hamiltonian formalisms, with special cases recapitulating known constructions. This BDCP framework offers a versatile toolkit for modeling coupled physical systems (e.g., magnetohydrodynamics, plasma-fluid interactions) and for extending to Dirac, Jacobi, and higher-order algebroid structures in future work.

Abstract

This work explores the geometrical/algebraic framework of Lie algebroids, with a specific focus on the decoupling and coupling phenomena within the bicocycle double cross product realization. The bicocycle double cross product theory serves as the most general method for (de)coupling an algebroid into the direct sum of two vector bundles in the presence of mutual \textit{representations}, along with two twisted cocycle terms. Consequently, it encompasses unified product, double cross product (matched pairs), semi-direct product, and cocycle extension frameworks as particular instances. In addition to algebraic constructions, the research extends to both reversible and irreversible Lagrangian and Hamiltonian dynamics on (de)coupled Lie algebroids, as well as Euler-Poincaré-(Herglotz) and Lie-Poisson-(Herglotz) dynamics on (de)coupled Lie algebras, providing insights into potential physical applications.

Paper Structure

This paper contains 13 sections, 6 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Product Lie Algebroids
  3. Unified Product Lie Algebroids
  4. Bicocycle Double Cross Product Lie Algebroids
  5. Twisted Poisson Algebroids and Dirac Structures
  6. Decomposition of Jacobi Lie Algebroid
  7. Dynamics on Product Lie Algebroids
  8. Reversible Dynamics on Lie Algebroids
  9. Reversible Dynamics on BDCP Lie Algebroids
  10. Irreversible Dynamics via Real Line Extension
  11. Irreversible Dynamics on Lie Algebroids
  12. Irreversible Dynamics on Bicocycle Double Cross Product Lie Algebroids
  13. Conclusion

Key Result

Theorem 2.1

Let $(\mathcal{A},\tau,M,\mathfrak{a}_{\mathcal{A}},[\bullet,\bullet]_{\mathcal{A}})$ be a Lie algebroid, and $(\mathcal{B},\kappa,M)$ a vector bundle. Equipped with the maps muratcan and rolar, the product bundle is a Lie algebroid along with the Lie bracket hohoho and the anchor map bobobo if and only if the conditions gabon-B-gabon3 are satisfied.

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6