Transposed δ-Poisson (super)algebra Structures on the Virasoro-like algebra and its Kantor Lie-double
Jie Lin, Chengyu Liu, Jingjing Jiang
TL;DR
This work investigates transposed δ-Poisson (super)algebra structures on the Virasoro-like algebra $V$ and its Kantor Lie-double $L(V)$. By analyzing δ-derivations and employing the left-multiplication/derivation correspondence, the authors show that nontrivial δ-derivations occur only at δ = 1, and in all tested cases (δ ∈ {−1,0,1}) the transposed δ-Poisson (super)algebra structures are necessarily trivial. Consequently, both $V$ and $L(V)$ admit no nontrivial transposed δ-Poisson (super)algebra structures, although their derivation algebras are explicitly characterized (with $Der(V) = ad(V) ty ty \\) and ${ m Der}(L(V)) = { m ad}(L(V)) ty ty \\ + \\mathbb{C}d_1 \\ + \\mathbb{C}d_2$). The results refine understanding of transposed δ-Poisson theory in the setting of infinite-dimensional Lie algebras and their Kantor doubles, informing related structural and representation-theoretic questions.
Abstract
We undertake a study of transposed δ-Poisson (super)algebra structures on the Virasoro-like algebra and its Kantor Lie-double -- the latter being constructed via Kantor's procedure. This work leads to the finding that, whereas non-trivial δ-derivations exist solely at δ=1, non-trivial transposed δ-Poisson (super)algebra structures are entirely absent.