Let $\mathrm{\Sigma }$ be a Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\mathrm{\Sigma })$, $\mathcal{U}(\mathrm{\Sigma })$, and $\mathcal{L}(\mathrm{\Sigma })$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $\mathcal{A}(\mathrm{\Sigma })$ is uniquely defined by its cluster and any two seeds of $\mathcal{A}(\mathrm{\Sigma })$ with $n-1$ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $\mathcal{U}(\mathrm{\Sigma })$ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $\mathrm{\Sigma }$. Besides, we obtain the standard monomial bases of $\mathcal{L}(\mathrm{\Sigma })$. We also prove that $\mathcal{U}(\mathrm{\Sigma })$ coincides with $\mathcal{L}(\mathrm{\Sigma })$ under certain conditions.

We give the relationship between indecomposable modules over the finite dimensional $k$-algebra $A$ and the smash product $A\mathrm{♯}G$ respectively, where $G$ is a finite abelian group satisfying $G\subseteq Aut(A)$, and $k$ is an algebraically closed field with the characteristic not dividing the order of $G$. More precisely, we construct all indecomposable $A\mathrm{♯}G$-modules from indecomposable $A$-modules and prove that an $A\mathrm{♯}G$-module is indecomposable if and only if it is an indecomposable $G$-stable module over $A$. Besides, we give the relationship between simple, projective and injective modules in $modA$ and those in $modA\mathrm{♯}G$.

In this paper, assume that $H$ is a Hopf algebra and $A/B$ is an $H$-Galois extension. Firstly, by introducing the concept of an $H$-stable tilting complex $T_{\bullet }$ over $B$, we show that $T_{\bullet }{\otimes }_{B}A$ is a tilting complex over $A$ and a derived equivalence between two $H$-module algebras can be extended to smash product algebras under some conditions. Then we observe that $0\to {\mathrm{E}\mathrm{n}\mathrm{d}}_{{\mathcal{D}}^b(B)}(T_{\bullet })\to {\mathrm{E}\mathrm{n}\mathrm{d}}_{{\mathcal{D}}^b(A)}(T_{\bullet }{\otimes }_{B}A)$ is an $H$-Galois Frobenius extension if $A/B$ is an $H$-Galois Frobenius extension. Finally, for any perfect recollement of derived categories of $H$-module algebras, we apply the above results to construct a perfect recollement of derived categories of their smash product algebras and generalize it to $n$-recollements.