The Classification of the Simple Modular Lie Algebras: I. Determination of the Two-Sections

1989 ◽  
Vol 130 (3) ◽  
pp. 643 ◽  
Author(s):  
Helmut Strade
Keyword(s):  
Lie Algebras ◽  
Modular Lie Algebras

scholarly journals Contributions to the classification of simple modular Lie algebras

1994 ◽  
Vol 341 (1) ◽  
pp. 227-252 ◽  
Author(s):  
Georgia Benkart ◽  
J. Marshall Osborn ◽  
Helmut Strade
Keyword(s):  
Lie Algebras ◽  
Modular Lie Algebras

scholarly journals Μελέτη ιδεωδών σε ειδικές lie άλγεβρες

2002 ◽  
Author(s):  
Θεόδουλος Ταπανίδης
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Open Problems ◽  
Nilpotent Lie Algebras ◽  
Central Sequence ◽  
Special Values ◽  
The Third

In this paper we study special properties of Nilpotent Lie Algebras of dimension eight over the field K of characteristic zero. The complete classification of these Lie Algebras has been done recently and there exist a great number of open problems. The problems, which have been solved in the thesis, are the following: i. There is not an Algebra of this category, which has two maximum abellian ideals of different dimension. ii. Extension of a Nilpotent Lie Algebra to others of bigger dimension. iii. Determination of Nilpotent Lie Algebras from another category iv. Determination of characteristic Nilpotent Lie Algebras from this category of Nilpotent Lie Algebras of dimensions eight. This thesis has three chapters. Each of them is analyzed as follows. The first chapter contains basic elements of the theory of Nilpotent Lie Algebras. This has eleven paragraphs; each of them consists of the following. The first paragraph has a general theory of algebra. Basic elements about Lie Algebras are given in the second paragraph. The structure constants of a Lie algebra are also given in this paragraph and also some relations between them. Finally it contains the determination of a Lie Algebra by constant structure and conversely. The third paragraph includes mappings between Lie Algebras. The notions of homomorphic and isomorphic Lie Algebras are defined by these mappings. The definitions of subalgebras and ideals of Lie Algebras are given in the fourth paragraph. It also contains some of their properties. Finally it has the notion of quotient Lie Algebra. The derivations of a Lie Algebra are contain in the fifth paragraph. It also contains some of their properties. The sixth paragraph includes some basic subsets of Lie Algebra. These basic sets play an important role in the theory of Lie Algebras. From a Lie Algebra g we can form sequences of ideals of g. Two basic ideals are the central sequence and the derived sequence. These are in the seventh paragraph. The eighth paragraph contains some elements of solvable Lie Algebras. Some elements of Nilpotent Lie Algebras are included in the ninth paragraph. The tenth paragraph contains basic elements of simple and semi-simple Lie Algebras. Finally the problem of classification of Lie Algebras is included in the last paragraph. The purpose of the second chapter is to study some properties of Nilpotent Lie Algebras of dimension eight. The whole chapter contains three paragraphs; each of them is analyzed as follows. The first paragraph describes the maximum abelian ideals of a Nilpotent Lie Algebra. The Nilpotent Lie Algebras of dimension eight are studied in the second paragraph. It is given their separation in categories according to the number of parameters, which have the none zero Lie brackets. Special categories of Nilpotent Lie Algebras of dimension eight are determined in the third paragraph. Furthermore some basic problems are studied for which we have some solutions. One of them is to determine a Nilpotent Lie Algebra of dimension eight which has two maximum abelian ideals of different dimension. The answer to this problem is negative, that mean there exists no such Lie Algebra of dimension eight, which has two maximum abelian ideals of different dimension. In this paragraph is also given the theory of extension of a Nilpotent Lie Algebra of bigger dimensions. The third chapter contains the study of Nilpotent Lie Algebras of dimension eight which are characteristically Nilpotent for all the parameters. Another category of Nilpotent Lie Algebras is determined which is characteristically Nilpotent for special values of parameters. The chapter has two paragraphs. The first paragraph gives special elements for characteristically Nilpotent Lie Algebras, which are necessary for the next paragraph. In the second paragraph we determine the category of Nilpotent Lie Algebras of dimension eight which are characteristically Nilpotent. We also determine other such Nilpotent Lie Algebras of dimension eight for special values of the parameters.

scholarly journals Classification of 5-dimentional subalgebras for 6-dimentional nilpotent Lie algebras

2020 ◽  
Vol 56 (2) ◽  
pp. 175-188
Author(s):  
U. L. Shtukar
Keyword(s):  
Lie Algebras ◽  
Standard Method ◽  
Classical Problem ◽  
Vector Spaces ◽  
Nilpotent Lie Algebras ◽  
New Approach ◽  
Zero Characteristic ◽  
Higher Dimensional

In this paper, we consider the classical problem of the classification of subalgebras of small dimensional Lie algebras. We found all 5-dimentional subalgebras of 6-dimentional nilpotent Lie algebras under the field with the zero characteristic. As is known, up to isomorphism all 6-dimensional nilpotent Lie algebras (their number is 32) were received by V. V. Morosov. However, the standard method based on the Campbell – Hausdorf formula is not effective for the determination of subalgebras of Lie 5- or higher dimensional algebras. In our research, we use a new approach to the solution of the problem of the determination of 5-dimensional subalgeras of indicated 6-dimensional nilpotent Lie algerbas, namely, the application of canonical bases for subspaces of vector spaces.

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