Transposed Poisson structures on Block Lie algebras and superalgebras

Abstract

We describe transposed Poisson algebra structures on Block Lie algebras $\mathcal{B}(q)$ and Block Lie superalgebras $\mathcal{S}(q)$, where $q$ is an arbitrary complex number. Specifically, we show that the transposed Poisson structures on $\mathcal{B}(q)$ are trivial whenever $q\notin \mathbb{Z}$, and for each $q\in \mathbb{Z}$ there is only one (up to an isomorphism) non-trivial transposed Poisson structure on $\mathcal{B}(q)$. The superalgebra $\mathcal{S}(q)$ admits only trivial transposed Poisson superalgebra structures for $q\ne 0$ and two non-isomorphic non-trivial transposed Poisson superalgebra structures for $q=0$. As a consequence, new Lie algebras and superalgebras that admit non-trivial $\mathrm{Hom}$-Lie algebra structures are found.

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• Transposed Poisson structures on generalized Witt algebras and Block Lie algebras
• Transposed Poisson structures
• Some generalizations of the variety of transposed Poisson algebras
• Transposed Poisson structures on solvable and perfect Lie algebras
• Transposed Poisson structures on the Lie algebra of upper triangular matrices
• Transposed Poisson structures on Lie incidence algebras