Solvability and nilpotency of transposed Novikov-Poisson algebras

In this paper, we develop the theory of nilpotency and solvability for transposed Novikov-Poisson algebras. We first establish several equivalent conditions for a dialgebra to be nilpotent, and show that the lower central series of a transposed Novikov-Poisson algebra $P$ admits a simplified form. We then prove that $P$ is solvable if and only if it is right nilpotent, and also if and only if $P^2$ is nilpotent. Moreover, we show that nilpotency (respectively, solvability) of a transposed Novikov-Poisson algebra is equivalent to nilpotency (respectively, solvability) of both its underlying commutative associative algebra and its underlying Novikov algebra. Finally, we prove that It\^{o}'s theorem holds for transposed Novikov-Poisson algebras.