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Jiarou Jin

Publications and source records attributed to Jiarou Jin.

3 recordsLinked to original sources

arXiv2026

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.

math.RA↗

arXiv2026

Nilpotency and Frattini theory for transposed Poisson algebras

We develop the theory of nilpotency and the Frattini theory for transposed Poisson algebras. The lower central series is shown to admit a simplified form, and an analogue of Engel's theorem is established: a finite-dimensional transposed Poisson algebra is nilpotent precisely when the left multiplication operators in both the associative and the Lie structures are nilpotent. Constructions of nilpotent and solvable algebras via tensor products and derivations are given. For a finite-dimensional Lie-nilpotent transposed Poisson algebra, we prove that the derived Lie subalgebra is a nilpotent ideal, which implies that the nilpotent radical coincides with the associative radical. In the framework of Frattini theory, we show that the Frattini subalgebra is always contained in the derived algebra and the Frattini ideal is associative nilpotent. When the algebra is nilpotent, all maximal subalgebras are ideals and the Frattini subalgebra equals the derived algebra. Conversely, for a Lie-nilpotent transposed Poisson algebra, if all maximal subalgebras are ideals, the algebra either is nilpotent or decomposes as a direct sum of a one-dimensional algebra generated by an idempotent and the nilpotent radical; if the Frattini subalgebra equals the derived algebra, the algebra is necessarily nilpotent. We also prove that the zero socle coincides with the nilpotent radical, and when the Frattini ideal is zero, the algebra splits into a subalgebra and its zero socle; in the Lie-nilpotent case this subalgebra is abelian as a Lie algebra.

math.RA↗

arXiv2026

Transposed Novikov-Poisson algebras

In this paper, we introduce the definition of transposed Novikov-Poisson algebras, whose affinization are transposed Poisson algebras. Moreover, we show that there is a natural transposed Poisson algebra structure on the tensor product of a transposed Novikov-Poisson algebra and a right differential Novikov-Poisson algebra. A transposed Poisson algebra also naturally arises from a transposed Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. We show that the tensor products of two transposed Novikov-Poisson algebras are also transposed Novikov-Poisson algebras. Several constructions of transposed Novikov-Poisson algebras are presented. Moreover, transposed Novikov-Poisson algebras are closely related to $\frac{1}{2}$-derivations of the associated Novikov algebras. By using $\frac{1}{2}$-derivations, we show that there are non-trivial transposed Novikov-Poisson algebra structures on solvable Novikov algebras with some conditions. We also prove that if a non-trivial transposed Novikov-Poisson algebra is simple, then the associated Novikov algebra is simple. Therefore, if the base field is algebraically closed and of characteristic 0, then any simple transposed Novikov-Poisson is of dimension $1$. Transposed Novikov-Poisson algebra structures on some simple Novikov algebras are also characterized.

math.RA↗