Matrix representation of $\lambda -$oscillator algebras and their corresponding Leibniz algebras

In this paper, we systematically characterize the types of oscillator algebras, ranging from the simplest case ${\mathfrak{D}}_4$ to the most general ${\mathfrak{D}}_{2m+2}^{\lambda }$. Subsequently, we construct multiple faithful matrix representations and vector field representations for ${\mathfrak{D}}_{2m+2}^{\lambda }(\mathbb{R})$, and give a non-go theorem for a certain representation of ${\mathfrak{D}}_{2m+2}^{\lambda }(\mathbb{C})$. Furthermore, leveraging the faithful representation of ${\mathfrak{D}}_{2m+2}^{\lambda }(\mathbb{R})$ via the special symplectic Lie algebra ${\mathfrak{s}\mathfrak{p}}_{2m+2}(\mathbb{R})$, we establish a Leibniz algebra associated with ${\mathfrak{D}}_{2m+2}^{\lambda }(\mathbb{R})$.