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Homogeneous spaces and invariant tensors. (English. Russian original) Zbl 0717.53030

J. Sov. Math. 42, No. 5, 1944-1973 (1988); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 18, 105-142 (1986).
The survey presents the results of the author and his collaborators on the construction and classification of tensors which are invariant with respect to a semisimple Lie algebra H. The main tool for the construction of such tensors is the following “inclusion principle” proposed by the author. Let \(\phi\) be a linear representation of a Lie algebra H in a vector space V. Assume that H is a subalgebra of a Lie algebra G with a reductive decomposition \(G=H+B\) such that the H-module V is isomorphic to a submodule \(B_ 1\) of the H-module B. Then for any linear representation \(\Phi\) of the Lie algebra G, the polynomial functions \[ F(X_ 1,...,X_ p)=Tr \Phi (X_ 1)\cdot...\cdot \Phi (X_ p),\quad X_ 1,...,X_ p\in B_ 1, \] define H-invariant tensors on the space \(B_ 1\approx V\). The author also states some results about classification of invariant tensors of order \(\leq 6\) on Riemannian symmetric spaces and about algebras with irreducible groups of automorphisms.
Reviewer: D.V.Alekseevski

MSC:

53C30 Differential geometry of homogeneous manifolds
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
22E46 Semisimple Lie groups and their representations
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