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{sp}_{2m+2}(\mathbb{R}) $, we establish a Leibniz algebra associated with $ \mathfrak{D}_{2m+2}^{\lambda}(\mathbb{R}) $.
Citation: Chao Deng, Xiaomin Tang. Matrix representation of $ \lambda- $oscillator algebras and their corresponding Leibniz algebras[J]. Electronic Research Archive, 2026, 34(5): 3079-3092. doi: 10.3934/era.2026139
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{sp}_{2m+2}(\mathbb{R}) $, we establish a Leibniz algebra associated with $ \mathfrak{D}_{2m+2}^{\lambda}(\mathbb{R}) $.
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Chao Deng, Xiaomin Tang. Matrix representation of $ \lambda- $oscillator algebras and their corresponding Leibniz algebras[J]. Electronic Research Archive, 2026, 34(5): 3079-3092. doi: 10.3934/era.2026139
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