By Lev V. Sabinin

As okay. Nomizu has justly famous [K. Nomizu, 56], Differential Geometry ever may be beginning more recent and more moderen points of the speculation of Lie teams. This monograph is dedicated to simply a few such features of Lie teams and Lie algebras. New differential geometric difficulties got here into being in reference to so known as subsymmetric areas, subsymmetries, and mirrors brought in our works relationship again to 1957 [L.V. Sabinin, 58a,59a,59b]. furthermore, the exploration of mirrors and structures of mirrors is of curiosity on the subject of symmetric areas. Geometrically, the main wealthy in content material there seemed to be the homogeneous Riemannian areas with structures of mirrors generated by means of commuting subsymmetries, specifically, so known as tri-symmetric areas brought in [L.V. Sabinin, 61b]. As to the concrete geometric challenge which wishes be solved and that is solved during this monograph, we point out, for instance, the matter of the class of all tri-symmetric areas with easy compact teams of motions. Passing from teams and subgroups hooked up with mirrors and subsymmetries to the corresponding Lie algebras and subalgebras results in a major new notion of the involutive sum of Lie algebras [L.V. Sabinin, 65]. this idea is without delay focused on unitary symmetry of simple par- cles (see [L.V. Sabinin, 95,85] and Appendix 1). the 1st examples of involutive (even iso-involutive) sums seemed within the - ploration of homogeneous Riemannian areas with and axial symmetry. the honor of areas with mirrors [L.V. Sabinin, 59b] back resulted in iso-involutive sums.

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**Sample text**

16 (Theorem 10) we construct the iso-involutive sum of lower index 1. 4 (Theorem 14), and the assertion of the theorem is evident. 2 (Theorem 20). 1 (Theorem 19) of type 1, generate the iso-involutive sum of type 1 and lower index 1. Thus with the natural embedding. 8 (Theorem 26) the three-dimensional simple compact algebra Thus and are conjugated by and consequently in The last case to be considered is This case is reduced to the case just considered. 10. Theorem 28. Let be a simple compact Lie algebra, be an iso-involutive sum of lower index 1.

Whence it follows that with the natural embedding. 14. Theorem 18. Let be a compact Lie algebra, be an elementary involutive pair, and let be an involutive sum of type 2 and of lower index 1, and Then with the natural embeddings. 1. Theorem 19. Let be a simple compact Lie algebra, and be an iso-involutive sum of lower index 1 and not of type 1 generated by an iso-involutive group Then and its corresponding iso-involutive sum is of lower index 1 and of type 1 (with the conjugating isomorphism Proof.

Let be a simple compact Lie algebra, be an involutive sum of lower index 1 and not of type 1. Then with the natural embedding, and the involutive automorphism is the special unitary involutive subalgebra of Proof. Indeed, under our assumptions is of lower index 1 and then where is an elementary involutive pair of lower index 1. 4 (Theorem 14) we have LEV SABININ 42 with the natural embedding. 17 (Definition 8), that is the special involutive subalgebra of the involutive automorphism of type U. 7.