Contractions of low-dimensional Lie algebras

Abstract

Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semiinvariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras.An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for both the complex and real Lie algebras of dimensions not greater than four are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases.Levels and colevels of low-dimensional Lie algebras are discussed in detail. Properties of multi-parametric and repeated contractions are also investigated.

On Contractions of Three-Dimensional Complex Associative Algebras

“…The proof of 1-4 is the same to that of Lie algebras [10,11]. For the parts 5 and 6, the proof is obtained by the following significant fact: let N be a Zariski closed subset of A n () and A 1 ,A 2 in A n ().…”

Classification of homogeneous CR-manifolds in dimension 4

“…We use the classification of 4-dimensional real Lie algebras, given in [19] (the results of [19] are also described, for example, in [20]). There are 22 types of such algebras: 10 solvable decomposable ones, 10 solvable indecomposable ones, and 2 non-solvable decomposable ones.…”

Non-solvable contractions of semisimple Lie algebras in low dimension

“…Further, the problem is technically a formidable task, not only because of the large number of isomorphism classes, but also because solvable algebras can depend on many parameters, and therefore the deformations must be analyzed for all possibilities of these parameters separately. The recent work [7] shows the difficulties that appear even in dimension four. Another possibility that is conceivable is to compute all deformations and contractions among Lie algebras with nontrivial Levi decomposition.…”

“…‡ The introduction of further techniques like the cohomology of Lie algebras [5] allowed one to interpret contractions geometrically in the variety of Lie algebras having a fixed dimension. Once the most important groups intervening in applications were analyzed, like the Lie algebras in the classical and quantum relativistic kinematics, the attention of various authors was turned to obtain complete diagrams of contractions in low dimension [6], which have been enlarged and completed in order to cover all the special types of contractions considered earlier [7]. Such lists have been obtained up to dimension 4 over the field of real numbers.…”