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Manin triples and bialgebras of Left-Alia algebras associated to invariant theory

Kang Chuangchuang, Liu Guilai, Wang Zhuo, Yu Shizhuo

Abstract

A left-Alia algebra is a vector space together with a bilinear map satisfying symmetric Jocobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notion of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras.

Manin triples and bialgebras of Left-Alia algebras associated to invariant theory

Abstract

A left-Alia algebra is a vector space together with a bilinear map satisfying symmetric Jocobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notion of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras.
Paper Structure (17 sections, 20 theorems, 83 equations)

This paper contains 17 sections, 20 theorems, 83 equations.

Table of Contents

  1. Introduction and Main Statements
  2. Introduction
  3. Left-Alia Algebras Associated with Invariant Theory
  4. Manin Triples and Bialgebras of Left-Alia Algebras
  5. Pseudo-Reflections and Twisted Deviations in Invariant Theory
  6. Preliminary on Invariant Theory
  7. Pseudo-Reflections Induced by Twisted Deviations
  8. Left-Alia Algebras and Their Representations
  9. Left-Alia Algebras and Twisted Derivations
  10. Examples of Left-Alia Algebras
  11. From Left-Alia Algebras to Anti-Pre-Lie Algebras
  12. From Left-Alia Algebras to Lie Triple Systems
  13. Representations and Matched Pairs of left-Alia Algebras
  14. Quadratic Left-Alia Algebras
  15. Manin Triples of Left-Alia Algebras and Left-Alia Bialgebras
  16. ...and 2 more sections

Key Result

Theorem 2.1

Theorems & Definitions (66)

  • Definition 1.1: Dzh09
  • Theorem 2.1: ITHil
  • Definition 2.2: IT
  • Theorem 2.3: CheST
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • proof
  • ...and 56 more