Topics in Hidden Symmetries

This note being devoted to some aspects of the inverse problem of representation theory contains a new insight into it illustrated by two topics. The attention is concentrated on the manner of representation of abstract objects by the concrete ones as well as on the abstract objects themselves. This paper being a continuation of the previous three parts [1] explicates novel features of the general ideology presented in the review [2] (see also the research article [3], where a wide interpretation of the inverse problem of representation theory was originally proposed). The attention is concentrated on the manner of concrete representations of abstract objects as well as on the least themselves. The concrete lines of topics have their origins in the author’s papers [4,5:App.A], where some objects, which gave start to the constructions below, appeared. 1. Topic Nine: Lie composites and their representation. Composed representations of Lie algebras This topic may be considered as a variation of a theme briefly discussed at the end of the article [6] (the so–called isotopic composites and their representations). Though the particular case considered below looses some interesting combinatorial features of the general one, e.g. the graph–representations, it is nevertheless rather interesting. Definition 1. A. A linear space v is called a Lie composite iff there are fixed its subspaces v1, . . .vn (dim vi > 1) supplied by the compatible structures of Lie algebras. Compatibility means that the structures of the Lie algebras induced in vi ∩ vj from vi and vj are the same. The Lie composite is called dense iff v1 ∪ . . . ∪ vn = v. The Lie composite is called connected iff for all i and j there exists a sequence k1, . . . km (k1 = i, km = j) such that vkl ∩ vkl+1 6= ∅. B. A representation of the Lie composite v in the space H is the linear mapping T : v 7→ End(H) such that T |vi is a representation of the Lie algebra vi for all i.