On free Gelfand-Dorfman-Novikov-Poisson algebras and a PBW theorem (Q1702694)

Gelfand-Dorfman-Novikov algebras (GDN algebras, known also as Novikov algebras) were introduced independently in [\textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman}, Funkts. Anal. Prilozh. 13, No. 4, 13--30 (1979; Zbl 0428.58009); translation in Funct. Anal. Appl. 13, 248--262 (1980; Zbl 0437.58009)] in connection with Hamiltonian operators in the formal calculus of variations and in [\textit{A. A. Balinskii} and \textit{S. P. Novikov}, Sov. Math., Dokl. 32, 228--231 (1985; Zbl 0606.58018); translation from Dokl. Akad. Nauk SSSR 283, 1036--1039 (1985)] in connection with linear Poisson brackets of hydrodynamic type. These algebras satisfy the identities of left symmetry \[ x\circ(y\circ z)-(x\circ y)\circ z = y\circ (x\circ z)-(y\circ x)\circ z\] and right commutativity \[ (x\circ y)\circ z = (x\circ z)\circ y. \] GDN-Poisson algebras were introduced in [\textit{X. Xu}, J. Algebra 185, No. 3, 905--934 (1996; Zbl 0863.17003); J. Algebra 190, No. 2, 253--279 (1997; Zbl 0872.17030)]. These are GDN algebras with an additional operation \(\cdot\) which equips the algebra with the structure of a commutative associative algebra with the compatibility conditions \[ (x\cdot y)\circ z = x\cdot (y\circ z)\quad\text{and}\quad (x\circ y)\cdot z-x\circ (y\cdot z) = (y\circ x)\cdot z-y\circ (x\cdot z). \] In the paper under review the authors construct a linear basis of the free GDN-Poisson algebra. Then they define the notion of a special GDN-Poisson admissible algebra. This is a differential algebra with two commutative associative products and some extra identities.The authors prove that any GDN-Poisson algebra can be embedded into its universal enveloping special GDN-Poisson admissible algebra. Finally, they establish that any GDN-Poisson algebra with the identity \[ x\circ(y\cdot z)=(x\circ y)\cdot z+(x\circ z)\cdot y\] is isomorphic to a commutative associative differential algebra both as GDN-Poisson algebra and as a commutative associative differential algebra.