A controlling cohomology of the deformation theory of Lie triple systems
This article develops the deformation theory of Lie triple systems using the cohomology of such systems developed by \textit{K. Yamaguti} [Kumamoto J. Sci. Soc. A 5, 44--52 (1960; Zbl 0123.01105)]. For \(T\) a Lie triple system with ternary composition \(\alpha\) and deformations of the form \(\alpha_t (a,b,c) = \alpha (a,b,c) + t\alpha_1 (a,b,c) + t^2\alpha_2 (a,b,c) + \cdots\) the authors prove that the first-order terms of equivalent deformations belong to the same cohomology class in the third Yamaguti cohomology group \(H^3 (T,T)\). If \(H^3 (T,T) = 0\), then every deformation is equivalent to the null deformation. They also show that if \(H^5 (T,T) = 0\), then every 3-cocycle \(\alpha_1 \in Z^3 (T,T)\) is integrable, i.e. there exists a one parameter family \(\alpha_t\) such that \(\alpha_t = \alpha + t \alpha_1 + t^2 \alpha_2 + \cdots\).
scientific article; zbMATH DE number 2102499
scientific article