Popovych D. Generalizations of Inönü–Wigner contractions and Lie-orthogonal operators

In the thesis, the main attention is paid to problems related to contractions between finite-dimensional real and complex Lie algebras, special kinds of contractions such as Saletan contractions, contractions with necessarily unbounded matrices and generalized Inönü–Wigner contractions. Also studied are Lie-orthogonal operators on such algebras. We describe the behavior of subalgebra and subspace flags of Lie algebras under contractions, which gives new criteria for non-existence of contractions. For each dimension not less than five, we construct a contraction between solvable Lie algebras that can be realized only with matrices whose Euclidean norms approach infinity at the limit value of the contraction parameter. We find the canonical form for Saletan (linear) contractions and introduce the notion of signature of a Saletan contraction. An algorithm of finding generalized IW-contractions or proving their nonexistence for a pair of Lie algebras is suggested. Using this algorithm, we optimize the known description of the generalized IW-contractions of three- and four-dimensional real and complex Lie algebras. We demonstrate that generalized IW-contractions are not universal in the dimension four. Any diagonal contraction (e.g., a generalized Inönü–Wigner contraction) is proved to be equivalent to a generalized Inönü–Wigner contraction with integer parameter powers. We introduce the equivalence relation of Lie-orthogonal operators on Lie algebras. It is proved that the center, the radical and the components of the ascending central series are invariant with respect to any Lie-orthogonal operator. Lie algebras that admit Lie-orthogonal operators whose all eigenvalues differ from 1 and –1 are described. We obtain a representation for Lie-orthogonal automorphisms. Lie-orthogonal operators on metric Lie algebras are completely described. The sets of Lie-orthogonal operators of some classes of Lie algebras are directly computed.