Let $ \mathfrak{L} $ and $ A $ be Lie triple systems, and let $ \theta_A $ be a representation of $ \mathfrak{L} $ on $ A. $ We first construct the third-order cohomology classes by derivations of $ A $ and $ \mathfrak{L}, $ then obtain a Lie algebra $ G_{\theta_A} $ with a representation $ \Phi $ on $ H^3(\mathfrak{L}, A), $ where $ \theta_A $ is given by an abelian extension
$ 0\longrightarrow A\longrightarrow {\tilde {\mathfrak{L}}} \xrightarrow{\pi} \mathfrak{L}\longrightarrow 0. $
We study obstruction classes for extensibility of derivations of $ A $ and $ \mathfrak{L} $ to those of $ \tilde{\mathfrak{L}}. $ An application of $ \Phi $ is discussed.
Citation: Xueru Wu, Yao Ma, Liangyun Chen. Abelian extensions of Lie triple systems with derivations[J]. Electronic Research Archive, 2022, 30(3): 1087-1103. doi: 10.3934/era.2022058
Let $ \mathfrak{L} $ and $ A $ be Lie triple systems, and let $ \theta_A $ be a representation of $ \mathfrak{L} $ on $ A. $ We first construct the third-order cohomology classes by derivations of $ A $ and $ \mathfrak{L}, $ then obtain a Lie algebra $ G_{\theta_A} $ with a representation $ \Phi $ on $ H^3(\mathfrak{L}, A), $ where $ \theta_A $ is given by an abelian extension
$ 0\longrightarrow A\longrightarrow {\tilde {\mathfrak{L}}} \xrightarrow{\pi} \mathfrak{L}\longrightarrow 0. $
We study obstruction classes for extensibility of derivations of $ A $ and $ \mathfrak{L} $ to those of $ \tilde{\mathfrak{L}}. $ An application of $ \Phi $ is discussed.
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Xueru Wu, Yao Ma, Liangyun Chen. Abelian extensions of Lie triple systems with derivations[J]. Electronic Research Archive, 2022, 30(3): 1087-1103. doi: 10.3934/era.2022058
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