The notion of a transposed Poisson $n$-Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson $n$-Lie algebra with a derivation gives rise to a transposed Poisson $(n+1)$-Lie algebra. In this paper, we focus on transposed Poisson $n$-Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of $(n+1)$-Lie algebras from transposed Poisson $n$-Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases.

The notion of generalized quantum cluster algebras was introduced as a natural generalization of Berenstein and Zelevinsky's quantum cluster algebras as well as Chekhov and Shapiro's generalized cluster algebras. In this paper, we focus on a generalized quantum cluster algebra of Kronecker type which possesses infinitely many cluster variables. We obtain the cluster multiplication formulas for this algebra. As an application of these formulas, a positive bar-invariant basis is explicitly constructed. Both results generalize those known for the Kronecker cluster algebra and quantum cluster algebra.