[Paper Review] $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras

Bruno Leonardo Macedo Ferreira, Ivan Kaygorodov+1 more|arXiv (Cornell University)|Oct 1, 2020

Advanced Topics in Algebra47 references26 citations

TL;DR

This paper establishes a correspondence between 1/2-derivations of Lie algebras and transposed Poisson algebra structures, demonstrating that such algebras arise precisely when the associative part's right multiplications are 1/2-derivations of the Lie part. The key contribution is a complete classification of transposed Poisson structures on specific Lie algebras: non-trivial structures exist on the Witt algebra, W(a, -1), the thin Lie algebra, and solvable Lie algebras with abelian nilpotent radical of codimension 1, while no non-trivial structures exist on semisimple finite-dimensional Lie algebras, the Virasoro algebra, or N=1 and N=2 superconformal algebras.

ABSTRACT

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras was established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, algebra $\mathcal{W}(a,-1)$, thin Lie algebra and solvable Lie algebra with abelian nilpotent radical) were constructed. In particular, we constructed an example of the transposed Poisson algebra with associative and Lie parts isomorphic to the Laurent polynomials and the Witt algebra. On the other side, it was proven that there are no non-trivial transposed Poisson algebras with Lie algebra part isomorphic to a semisimple finite-dimensional algebra, a simple finite-dimensional superalgebra, the Virasoro algebra, $N=1$ and $N=2$ superconformal algebras, or a semisimple finite-dimensional $n$-Lie algebra.

Motivation & Objective

• . The paper aims to classify all transposed Poisson algebra structures on specific Lie algebras by linking them to 1/2-derivations.
• It investigates the existence of non-trivial transposed Poisson algebras on various Lie algebras, including infinite-dimensional and finite-dimensional ones.
• The research seeks to determine which Lie algebras admit non-trivial transposed Poisson structures by analyzing the space of 1/2-derivations.
• It provides a systematic method to construct transposed Poisson algebras from 1/2-derivations of the Lie part and right multiplications of the associative part.
• The objective includes proving the non-existence of such structures on semisimple finite-dimensional Lie algebras and related algebras like the Virasoro and superconformal algebras.

Proposed method

• . The main method is establishing a one-to-one correspondence between right multiplications in the associative part of a transposed Poisson algebra and 1/2-derivations of its Lie part.
• The paper uses the known classification of 1/2-derivations of semisimple finite-dimensional Lie algebras to prove non-existence of non-trivial transposed Poisson structures on such algebras.
• For infinite-dimensional Lie algebras like the Witt algebra and W(a, b), the authors compute the space of 1/2-derivations explicitly using recursive formulas derived from the 1/2-derivation condition.
• The construction of transposed Poisson algebras is achieved by defining the associative multiplication via right multiplication operators that are 1/2-derivations.
• The method involves solving systems of linear equations arising from the 1/2-derivation condition and the associativity of the multiplication.
• For the thin Lie algebra and solvable Lie algebras with abelian nilpotent radical of codimension 1, the authors derive explicit formulas for 1/2-derivations and classify all possible associative multiplications satisfying the transposed Poisson identity.

Experimental results

Research questions

• RQ1 . Does every transposed Poisson algebra structure on a Lie algebra arise from a 1/2-derivation of its Lie part via right multiplication in the associative part?
• RQ2 . Which Lie algebras admit non-trivial transposed Poisson algebra structures?
• RQ3 . Are there non-trivial transposed Poisson algebras with the Witt algebra as the Lie part?
• RQ4 . Does the algebra W(a, -1) admit non-trivial transposed Poisson structures, and if so, how can they be classified?
• RQ5 . What is the structure of transposed Poisson algebras on the Virasoro algebra, N=1 and N=2 superconformal algebras, and solvable Lie algebras with abelian nilpotent radical of codimension 1?

Key findings

• . The Witt algebra admits infinitely many non-trivial transposed Poisson algebra structures, and all such structures on W(a, -1) are completely classified.
• . The algebra W(a, b) admits no non-trivial transposed Poisson structures if and only if b ≠ -1; for b = -1, all such structures are explicitly described.
• . The thin Lie algebra admits non-trivial transposed Poisson algebra structures, and all such structures are isomorphic to one where e1 ∗ e1 = ek for some k ≥ 2.
• . The solvable Lie algebra with abelian nilpotent radical of codimension 1 admits three distinct isomorphism classes of non-trivial transposed Poisson algebra structures, depending on the parameters of the right multiplication.
• . There are no non-trivial transposed Poisson algebras with a semisimple finite-dimensional Lie algebra as the Lie part, as such algebras have no non-trivial 1/2-derivations.
• . The Virasoro algebra, N=1 and N=2 superconformal algebras admit no non-trivial transposed Poisson algebra structures, as they are not compatible with the existence of non-trivial 1/2-derivations.