On free Gelfand–Dorfman–Novikov–Poisson algebras and a PBW theorem

Transposed Hom-Poisson and Hom-pre-Lie Poisson algebras and bialgebras

“…Poisson algebras appear in many areas of mathematics and physics being important for integrable systems, Poisson geometry, Poisson brackets and Poisson manifolds, geometry of smooth loops, vertex algebras, quantization theory, deformation quantization, classical mechanics, Hamiltonian mechanics, quantum mechanics, quantum field theory, Lie theory and operads [2,3,13,17,21,25-27,29,30,33-36,38,44,45,48,57,58,74-76, 80, 90-92]. Poisson algebras led to development of other fundamental structures such as noncommutative Poisson algebras [92], Jacobi algebras (generalized Poisson algebras) [2,20,27,46,58], Gerstenhaber algebras and Lie-Rinehart algebras [32,49,60,79] and Novikov-Poisson algebras [18,93] arising from Novikov algebra in connection with the Poisson brackets of hydrodynamic type [12] and Hamiltonian operators in the formal variational calculus [31].…”

On the Lie-solvability of Novikov algebras

“…(x, y, z) = (y, x, z), (1) (xy)z = (xz)y, (2) where (x, y, z) = (xy)z − x(yz) is the associator of elements x, y, z.…”

“…The construction described above is called the Gelfand-Dorfman construction for Novikov algebras. Recently, L.A. Bokut, Y. Chen, and Z. Zhang [2] proved that any Novikov algebra over a field of characteristic zero is a subalgebra of a Novikov algebra obtained from some differential algebra by the Gelfand-Dorfman construction.…”

One-generated nilpotent Novikov algebras

“…: e 1 e 1 = e 2 e 1 e 2 = e 3 e 1 e 3 = −e 5 + e 6 e 1 e 5 = −3e 6 e 2 e 1 = 3e 3 + e 4 e 2 e 2 = 3e 5 e 2 e 3 = −3e 6 e 3 e 1 = 3e 5 e 3 e 2 = 3e 6 e 4 e 1 = −3e 6 e 5 e 1 = 3e 6 N 6 16 (λ) λ / ∈{0,3} : e 1 e 1 = e 2 e 1 e 2 = e 3 e 1 e 3 = (2 − λ)e 5 e 1 e 4 = e 6 e 1 e 5 = (3 − 2λ)e 6 e 2 e 1 = λe 3 + e 4 e 2 e 2 = λe 5 e 2 e 3 = λ(2 − λ)e 6 e 3 e 1 = λe 5 e 3 e 2 = λe 6 e 4 e 1 = −e 6 e 5 e 1 = λe 6 3.3. Central extensions of N 5 02 (0).…”

“…e 1 e 1 = e 2 e 1 e 2 = e 3 e 1 e 3 = e 4 e 1 e 4 = e 5 e 1 e 5 = e 6 e 2 e 1 = e 3 + e 6 e 2 e 2 = e 4 e 2 e 3 = e 5 e 2 e 4 = e 6 e 3 e 1 = e 4 e 3 e 2 = e 5 e 3 e 3 = e 6 e 4 e 1 = e 5 e 4 e 2 = e 6 e 5 e 1 = e 6 N 6 34 (λ) : e 1 e 1 = e 2 e 1 e 2 = e 3 e 1 e 3 = e 4 e 1 e 4 = e 5 e 1 e 5 = 2e 6 e 2 e 1 = e 5 + λe 6 e 2 e 2 = e 6 e 3 e 1 = e 6 N 6 35 (λ) λ =0 : e 1 e 1 = e 2 e 1 e 2 = e 3 e 1 e 3 = e 4 e 1 e 4 = e 5 e 1 e 5 = 2e 6 e 2 e 1 = e 5 e 2 e 2 = e 6 e 2 e 3 = λe 6 e 3 e 1 = e 6 e 3 e 2 = −λe 6 e 4 e 1 = λe 6 N 6…”