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Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras

Nobuyoshi Takahashi

TL;DR

This work establishes a tight link between representations of Lie-Yamaguti algebras and representations of their enveloping Lie algebras under semisimplicity assumptions. By introducing tight representations and effective minimal representations of enveloping triples, it proves categorical equivalences: when $T$ is $L$-semisimple and $[T,T,T]=T$, Rep$(T)$ is equivalent to Rep$^{em}(L(T),T,D(T))$, and similarly for infinitesimal $s$-manifolds via $(L(T),L(\sigma))$. The method relies on reductive triples and local regular $s$-pairs, constructing functors that translate LY representations into enveloping Lie algebra data and back. The results clarify how semisimple enveloping algebras control LY representations and provide a practical pathway to classify representations through well understood Lie algebra theory. The paper also discusses limitations by presenting cases where the enveloping Lie algebra is not semisimple, illustrating the boundary of the equivalence.

Abstract

Let $T$ be a Lie-Yamaguti algebra such that its standard enveloping Lie algebra $L(T)$ is semisimple and $[T, T, T]=T$. Then we give a description of representations of $T$ in terms of representations of $L(T)$ with certain additional data. Similarly, if $(T, σ)$ is an infinitesimal $s$-manifold such that $L(T)$ is semisimple, then any representation of $(T, σ)$ comes from a representation of $L(T)$.

Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras

TL;DR

This work establishes a tight link between representations of Lie-Yamaguti algebras and representations of their enveloping Lie algebras under semisimplicity assumptions. By introducing tight representations and effective minimal representations of enveloping triples, it proves categorical equivalences: when is -semisimple and , Rep is equivalent to Rep, and similarly for infinitesimal -manifolds via . The method relies on reductive triples and local regular -pairs, constructing functors that translate LY representations into enveloping Lie algebra data and back. The results clarify how semisimple enveloping algebras control LY representations and provide a practical pathway to classify representations through well understood Lie algebra theory. The paper also discusses limitations by presenting cases where the enveloping Lie algebra is not semisimple, illustrating the boundary of the equivalence.

Abstract

Let be a Lie-Yamaguti algebra such that its standard enveloping Lie algebra is semisimple and . Then we give a description of representations of in terms of representations of with certain additional data. Similarly, if is an infinitesimal -manifold such that is semisimple, then any representation of comes from a representation of .
Paper Structure (8 sections, 27 theorems, 60 equations)

This paper contains 8 sections, 27 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Lie-Yamaguti algebras and infinitesimal $s$-manifolds
  3. Lie-Yamaguti algebras
  4. Infinitesimal $s$-manifolds
  5. $L$-Semisimplicity
  6. Representations
  7. Tight representations
  8. Correspondence of representations

Key Result

Proposition 2.3

Let $(\mathfrak{g}, \mathfrak{m}, \mathfrak{h})$ be a reductive triple. Then $\mathfrak{m}$ forms a Lie-Yamaguti algebra with the operations If $f: (\mathfrak{g}, \mathfrak{m}, \mathfrak{h})\to (\mathfrak{g}', \mathfrak{m}', \mathfrak{h}')$ is a homomorphism of reductive triples, then the induced map $\mathfrak{m}\to\mathfrak{m}'$ is a homomorphism of Lie-Yamaguti algebras. In other words, there

Theorems & Definitions (79)

  • Definition 2.1: Yamaguti1969
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • Definition 2.8: Yamaguti1969
  • Definition 2.9
  • ...and 69 more