Some generalizations of the variety of transposed Poisson algebras (Q6591993)

The paper discusses relations between transposed Poisson algebras, \(F\)-manifold algebras, and Gelfand-Dorfman algebras. Namely, let us consider a vector space \(V\) with two multiplications: an associative commutative multiplication \(\cdot\) and a Lie multiplication \([\cdot,\cdot].\) Then,\N\N1. If \((V, \cdot, [\cdot,\cdot])\) satisfies \N\[\N[x_1, x_2 x_3] - [x_3, x_2 x_1] + [x_2, x_1] x_3 - [x_2, x_3] x_1 - x_2 [x_1, x_3] = 0,\N\]\Nwe have a commutative Gelfand-Dorfman algebra.\N\N2. If \((V, \cdot, [\cdot,\cdot])\) satisfies \N\[\N2 x_1 [x_2, x_3] = [x_1 x_2, x_3] + [x_2, x_1 x_3],\N\]\Nwe have a transposed Poisson algebra.\N\N3. If \((V, \cdot, [\cdot,\cdot])\) satisfies \N\[\begin{multlined}\N[x_1 x_2, x_3 x_4] = [x_1 x_2, x_3] x_4 + [x_1 x_2, x_4] x_3 + x_1 [x_2, x_3 x_4] + x_2 [x_1, x_3 x_4] \\ -\N\big( x_1 x_3 [x_2, x_4] + x_2 x_3 [x_1, x_4] + x_2 x_4 [x_1, x_3] + x_1 x_4 [x_2, x_3]\big),\N\end{multlined}\]\Nwe have a \(F\)-manifold algebra.\N\NThe author proves that every transposed Poisson algebra is a \(F\)-manifold algebra (Theorem 3.2.), and the variety of transposed Poisson algebras and commutative Gelfand-Dorfman algebras coincide. Moreover, every transposed Poisson algebra satisfies two special identities of Gelfand-Dorfman algebras (Theorem 4.1). The Gröbner-Shirshov basis for the transposed Poisson operad is calculated up to degree 4.