Sabinin L.V. Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

New York: Kluwer Academic Publishers. – 2004. – 330 p. As K. Nomizu has justly noted, Differential Geometry ever will be initiating newer and newer aspects of the theory of Lie groups. This monograph is devoted to just some such aspects of Lie groups and Lie algebras. The detailed consideration of involutive sums of Lie algebras has shown, however, that their role is more significant than the role of the only convenient auxiliary apparatus for solving some differential-geometric problems. We may talk about the theory of independent interest and it is natural to call it ‘Mirror geometry of Lie algebras’, or ‘Mirror calculus’; the role and significance of which is comparable with the role and significance of the well known ‘Root Method’ in the theory of Lie algebras. Part I and II of this treatise are devoted to the presentation of Mirror Geometry over the reals. The suggested theory is a theory of structures of a new type for compact real Lie algebras and is related to discrete involutive groups of automorphisms and the corresponding involutive decompositions. Let us now turn to possible geometric applications, which, in particular, maybe found in Part III and IV. Some applications are concentrated at the beginning of Part III after some necessary definitions. Despite that here many results have been obtained simply as a reformulation of theorems of Part I and II from the language of Lie algebras into the language of Lie groups, homogeneous spaces, and mirrors, those are very interesting. (For example, the characterization of symmetric spaces of rank 1 by the properties of geodesic mirrors). In addition, the theory of Part I and II implies two new interesting types of symmetric spaces—principal and special—and allows us to explore geometric properties of their mirrors. We now intend to compare the well known ‘Root Method’ with our new theory, which we briefly call ‘Mirror Geometry’. First of all, Mirror Geometry deals with new types of structures (involutive sums) and is introduced independently of the Root Method. Thus these two theories seem to be different. But since Mirror Geometry leads us to the classification of simple compact Lie algebras (through the classification of principal unitary involutive automorphisms) we need some comparisons.On the artistic and poetic fragments of the book.Preliminaries.
Curvature tensor of involutive pair. Classical involutive pairs of index 1.
Iso-involutive sums of Lie algebras.
Iso-involutive base and structure equations.
Iso-involutive sums of types 1 and 2.
Iso-inolutive sums of lower index 1.
Principal central involutive automorphism of type U.
Principal unitary involutive automorphism of index 1.
Hyper-involutive decomposition of a simple compact Lie algebra.
Some auxiliary results.
Principal involutive automorphisms of type O.
Fundamental theorem.
Principal di-unitary involutive automorphism.
Singular principal di-unitary involutive automorphism.
Mono-unitary non-central principal involutive automorphism.
Principal involutive automorphism of types f and e.
Classification of simple special unitary subalgebras.
Hyper-involutive reconstruction of basic decompositions.
Special hyper-involutive sums.
Notations, definitions and some preliminaries.
Symmetric spaces of rank 1.
Principal symmetric spaces.
Essentially special symmetric spaces.
Some theorems on simple compact Lie groups.
Tri-symmetric and hyper-tri-symmetric spaces.
Tri-symmetric spaces with exceptional compact groups.
Tri-symmetric spaces with groups of motions SO(n), Sp(n), SU(n).
Subsymmetric Riemannian homogeneous spaces.
Subsymmetric homogeneous spaces and Lie algebras.
Mirror Subsymmetric Lie triplets of Riemannian type.
Mobile mirrors. Iso-involutive decompositions.
Homogeneous Riemannian spaces with two-dimensional mirrors.
Homogeneous Riemannian space with groups G SO(n), SU(3) and two-dimensional mirrors.
Homogeneous Riemannian spaces with simple compact Lie groups SU(3) and two-dimensional mirrors.
Homogeneous Riemannian spaces with simple compact Lie group of motions and two-dimensional immobile mirrors.
On the structure of T, U, V-isospins in the theory of higher symmetry.
Description of contents.
Definitions.
Theorems.