Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transposed Poisson structures

Patrícia Damas Beites, Bruno Leonardo Macedo Ferreira, Ivan Kaygorodov

Abstract

To present a survey on known results from the theory of transposed Poisson algebras, as well as to establish new results on this subject, are the main aims of the present paper. Furthermore, a list of open questions for future research is given.

Transposed Poisson structures

Abstract

To present a survey on known results from the theory of transposed Poisson algebras, as well as to establish new results on this subject, are the main aims of the present paper. Furthermore, a list of open questions for future research is given.
Paper Structure (15 sections, 20 theorems, 12 equations)

This paper contains 15 sections, 20 theorems, 12 equations.

Table of Contents

  1. Transposed Poisson algebras and structures
  2. Unital transposed Poisson algebras
  3. Transposed Poisson structures on nilpotent $n$-Lie algebras
  4. Transposed Poisson structures and quasi-automorphisms
  5. Transposed Poisson structures and ${\rm Hom}$-Lie algebra structures
  6. Transposed Poisson structures on simple graded Lie algebras of finite growth
  7. Relations with other algebras
  8. Quasi-Poisson algebras
  9. Generalized Poisson algebras and algebras of Jordan brackets
  10. Algebras of Jordan brackets
  11. Gelfand-Dorfman algebras (GD-algebras) and $F$-manifold algebras
  12. Transposed Poisson PI algebras
  13. Transposed Poisson fields
  14. Transposed Poisson $n$-Lie algebras and Poisson $n$-Lie algebras
  15. Open questions

Key Result

Lemma 4

Let $({\mathfrak L},\cdot,[\cdot,\cdot])$ be a transposed Poisson algebra, and let $z$ be an arbitrary element from ${\mathfrak L}.$ Then the right multiplication $R_z$ in the commutative associative algebra $({\mathfrak L},\cdot)$ gives a $\frac{1}{2}$-derivation of the Lie algebra $({\mathfrak L},

Theorems & Definitions (39)

  • Definition 1: see bai20
  • Definition 2
  • Definition 3
  • Lemma 4
  • Theorem 5: see FKL
  • Theorem 6: see, bai20
  • Definition 7
  • Definition 8: see, bai20
  • Theorem 9
  • proof
  • ...and 29 more