A linear space \(V\) with two bilinear operations \(\circ\) and \([\cdot, \cdot]\) is called a Gelfand-Dorfman algebra if \((V, \circ)\) is a Novikov algebra, \((V,[\cdot, \cdot])\) is a Lie algebra, and the following additional identity holds: \[ b \circ [a,c]=[a, b \circ c]-[c, b \circ a]+[b, a] \circ c - [b,c] \circ a. \] Recall that the variety of Novikov algebras is defined by the following identities: \[ (a \circ b) \circ c - a \circ (b \circ c) = (b \circ a) \circ c - b \circ (a \circ c),\qquad (a \circ b) \circ c = (a \circ c) \circ b. \]
It is known that any Poisson algebra \((P, \circ, \{\cdot,\cdot\})\) with a derivation \(d\) equipped with operations \[ x \circ y = xy,\qquad [x, y]=\{x, y\}, \] is a Gelfand-Dorfman algebra denoted \(P^{(d)}\) [X. Xu, Southeast Asian Bull. Math. 27, No. 3, 561–574 (2003; Zbl 1160.17301)]. A Gelfand-Dorfman algebra \(V\) is said to be special if it can be embedded into a Gelfand-Dorfman algebra \(P^{(d)}\) for an appropriate differential Poisson algebra \(P\).
The main results of the paper are the following.
Corollary 1. The class of special Gelfand-Dorfman algebras forms a variety.
Theorem 2. Let \(V\) be a two-dimensional Gelfand-Dorfman algebra. Then \(V\) is special.
Corollary 3. All special identities of Gelfand-Dorfman algebras of degree \(<6\) are consequences of two following special identities \[ [c, a \circ d] \circ b + ([a,c] \circ d) \circ b = [c,(a \circ b) \circ d]-[c, a \circ b] \circ d \] and \[ 2([a, b] \circ c) \circ d = [b \circ c, a \circ d]-[a \circ c, b \circ d] + ([a, b \circ c]-[b, a \circ c]) \circ d + ([a, b \circ d]-[b, a \circ d]) \circ c. \]
Later, Sartayev proved that each transposed Poisson algebra satisfies these special identities [B. Sartayev, Commun. Math. 32, No. 2, Paper No. 3, 8 p. (2024; Zbl 07900710)].