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Lie-Yamaguti Algebra Bundle

Saikat Goswami, Goutam Mukherjee

TL;DR

The paper introduces Lie-Yamaguti algebra bundles as a natural generalization of Lie algebra bundles, anchored in geometric considerations from Kikkawa. It develops a representation theory for these bundles and a fiberwise cohomology theory $H^{(2p,2p+1)}(\xi;\eta)$ by adapting KY-cohomology to the bundle setting, including a framework for (twisted) semidirect products. Key contributions include explicit constructions (product bundles, End$(\xi)$, reductive decompositions, and tangent-bundle examples from homogeneous Lie loops) and an existence clutching-type theorem for LY algebra bundles using a Lie group of symmetry. The work provides a rigorous geometric-algebraic framework for LY-symmetric structures on bundles and lays groundwork for abelian extensions and potential algebroid interpretations with broad differential-geometric and physical implications.

Abstract

We introduce the notion of Lie-Yamaguti algebra bundle, define its cohomology groups with coefficients in a representation and show that such bundles appeared naturally from geometric considerations in the work of M. Kikkawa, which motivates us to introduce this object in the proper mathematical framework. We also study abelian extensions of Lie-Yamaguti algebra bundles and investigate their relationship with suitable cohomology group.

Lie-Yamaguti Algebra Bundle

TL;DR

The paper introduces Lie-Yamaguti algebra bundles as a natural generalization of Lie algebra bundles, anchored in geometric considerations from Kikkawa. It develops a representation theory for these bundles and a fiberwise cohomology theory by adapting KY-cohomology to the bundle setting, including a framework for (twisted) semidirect products. Key contributions include explicit constructions (product bundles, End, reductive decompositions, and tangent-bundle examples from homogeneous Lie loops) and an existence clutching-type theorem for LY algebra bundles using a Lie group of symmetry. The work provides a rigorous geometric-algebraic framework for LY-symmetric structures on bundles and lays groundwork for abelian extensions and potential algebroid interpretations with broad differential-geometric and physical implications.

Abstract

We introduce the notion of Lie-Yamaguti algebra bundle, define its cohomology groups with coefficients in a representation and show that such bundles appeared naturally from geometric considerations in the work of M. Kikkawa, which motivates us to introduce this object in the proper mathematical framework. We also study abelian extensions of Lie-Yamaguti algebra bundles and investigate their relationship with suitable cohomology group.
Paper Structure (5 sections, 7 theorems, 98 equations)

This paper contains 5 sections, 7 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie-Yamaguti Algebra Bundle
  4. Representation of Lie-Yamaguti Algebra Bundles
  5. Cohomology of Lie-Yamaguti Algebra Bundle

Key Result

Proposition 3.10

Let $\xi = (L,p,M,[~,~],\{~,~,~\})$ be a locally trivial Lie-Yamaguti algebra bundle with fibers isomorphic to a given Lie-Yamguti algebra $\mathfrak{g}$. Let $\mathfrak{h}$ be a subalgebra of $\mathfrak{g}$ such that $\varphi(\mathfrak{h}) = \mathfrak{h}$ for all $\varphi \in \operatorname{Aut}_{\o

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • ...and 50 more