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Transposed Poisson structures on the Lie algebra of upper triangular matrices

Ivan Kaygorodov, Mykola Khrypchenko

Abstract

We describe transposed Poisson structures on the upper triangular matrix Lie algebra $T_n(F)$, $n>1$, over a field $F$ of characteristic zero. We prove that, for $n>2$, any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for $n=2$, there is one more class of transposed Poisson structures on $T_n(F)$. We also show that, up to isomorphism, the full matrix Lie algebra $M_n(F)$ admits only one non-trivial transposed Poisson structure, and it is of Poisson type.

Transposed Poisson structures on the Lie algebra of upper triangular matrices

Abstract

We describe transposed Poisson structures on the upper triangular matrix Lie algebra , , over a field of characteristic zero. We prove that, for , any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for , there is one more class of transposed Poisson structures on . We also show that, up to isomorphism, the full matrix Lie algebra admits only one non-trivial transposed Poisson structure, and it is of Poisson type.
Paper Structure (5 sections, 11 theorems, 63 equations)

This paper contains 5 sections, 11 theorems, 63 equations.

Table of Contents

  1. Definitions and preliminaries
  2. Transposed Poisson structures on the upper triangular matrix Lie algebra
  3. Upper triangular matrix algebra
  4. $\frac{1}{2}$-derivations of the upper triangular matrix Lie algebra
  5. Transposed Poisson structures on the full matrix Lie algebra

Key Result

Lemma 4

Let $({\mathfrak L},[\cdot,\cdot])$ be a Lie algebra and $\cdot$ a new binary (bilinear) operation on ${\mathfrak L}$. Then $({\mathfrak L},\cdot,[\cdot,\cdot])$ is a transposed Poisson algebra if and only if $\cdot$ is commutative and associative and for every $z\in{\mathfrak L}$ the multiplication

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 13 more