Matrix representation of λ−oscillator algebras and their corresponding Leibniz algebras
Chao Deng , Xiaomin Tang
Electronic Research Archive ›› 2026, Vol. 34 ›› Issue (5) : 3079 -3092.
In this paper, we systematically characterize the types of oscillator algebras, ranging from the simplest case $\mathfrak{D}_{4}$ to the most general $\mathfrak{D}_{2 m+2}^{\lambda}$. Subsequently, we construct multiple faithful matrix representations and vector field representations for $\mathfrak{D}_{2 m+2}^{\lambda}(\mathbb{R})$, and give a non-go theorem for a certain representation of $\mathfrak{D}_{2 m+2}^{\lambda}(\mathbb{C})$. Furthermore, leveraging the faithful representation of $\mathfrak{D}_{2 m+2}^{\lambda}(\mathbb{R})$ via the special symplectic Lie algebra $\mathfrak{s p}_{2 m+2}(\mathbb{R})$, we establish a Leibniz algebra associated with $\mathfrak{D}_{2 m+2}^{\lambda}(\mathbb{R})$.
oscillator algebra / Lie algebra / matrix representation / algebra representation / Leibniz algebra
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