We introduce a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We show that the transposed Poisson algebra thus defined not only shares common properties of the Poisson algebra, including the closure under taking tensor products and the Koszul self-duality as an operad, but also admits a rich class of identities. More significantly, a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Consequently, the classic construction of a Poisson algebra from a commutative associative algebra with a pair of commuting derivations has a similar construction of a transposed Poisson algebra when there is one derivation. More broadly, the transposed Poisson algebra also captures the algebraic structures when the commutator is taken in pre-Lie Poisson algebras and two other Poisson type algebras. Furthermore, the transposed Poisson algebra improves two processes in~[17] that produce 3-Lie algebras from Poisson algebras with a strongness condition. When transposed Poisson algebras are used in one process, the strongness condition is no longer needed and the resulting 3-Lie algebra gives a transposed Poisson 3-Lie algebra. In the other process, the resulting 3-Lie algebra is shown to again give a transposed Poisson 3-Lie algebra.

Transposed poisson algebras, novikov-poisson algebras and 3-lie algebras

In this paper, we define a class of 3-algebras which are called 3-Lie-Rinehart algebras. A 3-Lie-Rinehart algebra is a triple $(L, A, \rho)$, where $A$ is a commutative associative algebra, $L$ is an $A$-module, $(A, \rho)$ is a 3-Lie algebra $L$-module and $\rho(L, L)\subseteq Der(A)$. We discuss the basic structures, actions and crossed modules of 3-Lie-Rinehart algebras and construct 3-Lie-Rinehart algebras from given algebras, we also study the derivations from 3-Lie-Rinehart algebras to 3-Lie $A$-algebras. From the study, we see that there is much difference between 3-Lie algebras and 3-Lie-Rinehart algebras.

3-Lie-Rinehart Algebras

We introduce the concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via the cohomology theory.

Structure and cohomology of 3-Lie-Rinehart superalgebras

We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to two Hom-Lie algebras constructed on an associative, commutative algebra using 
$$\sigma $$

-derivation and involution, and we obtain two 3-Hom-Lie algebras.

3-Hom-Lie Algebras Based on $$\sigma $$σ -Derivation and Involution

Abstract In this paper, we introduce the concept of transposed BiHom-Poisson (abbr. TBP) algebras which can be constructed by BiHom-Novikov-Poisson algebras. Several useful identities for TBP algebras are provided. We also prove that the tensor products of two (T)BP algebras are closed. The notions of BP 3-Lie algebras and TBP 3-Lie algebras are presented and TBP algebras can induce TBP 3-Lie algebras by two approaches. Finally, we give some examples for the TBP algebras of dimension 2.

Transposed BiHom-Poisson algebras

3-Lie algebras are constructed by commutative associative algebras, involutions and derivations. Then the 3-Lie algebras are obtained from Lie algebras and linear functions, and from group algebras F[G] of an abelian group G and homomorphisms .  At the end of the paper, the 3-Lie algebras are obtained from Laurent polynomials , and whose structures are studied.

Yong Wu

Papers

Constructions of 3-Lie algebras