Classification of 5-DIMENSIONAL MD-Algebras Having Commutative Derived Ideals (original) (raw)
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The paper presents a subclass of the class of MD-algebras, i.e., solvable real Lie algebras such that their orbits in the co-adjoint representation (say by K-orbits) of corresponding connected and simply connected Lie groups are orbit of zero or maximal dimension. The main result of the paper is the classification up to an isomorphism of all 5-dimensional MD-algebras which have commutative derived ideal.
Classification of 5Dimensional MD-Algebras Having Non-Commutative Derived Ideals
2011
The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e. five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbits) are orbits of zero or maximal dimension. The main result of the paper is the classification up to an isomorphism of all MD5-algebras with the non-commutative derived ideal. With this result, we have the complete classification of 5-dimensional solvable Lie algebras.
We study a special subclass of real solvable Lie algebras having small dimensional or small codimensional derived ideals. It is well-known that the derived ideal of any Heisenberg Lie algebra is 1-dimensional and the derived ideal of the 4-dimensional real Diamond algebra is 1-codimensional. Moreover, all the coadjoint orbits of any Heisenberg Lie group as well as 4-dimensional real Diamond group are orbits of dimension zero or maximal dimension. In general, a (finite dimensional) real solvable Lie group is called an MD-group if its coadjoint orbits are zero-dimensional or maximal dimensional. The Lie algebra of an MD-group is called an MD-algebra and the class of all MD-algebras is called MD-class. Simulating the mentioned above characteristic of Heisenberg Lie algebras and 4-dimensional real Diamond algebra , we give a complete classification of MD-algebras having 1-dimensional or 1-codimensional derived ideals.
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In this paper we study a special subclass of real solvable Lie algebras having small dimensional or small codimensional derived ideal. It is wellknown that the derived ideal of any Heisenberg Lie algebra is 1-dimensional and the derived ideal of the 4-dimensional real Diamond algebra is 1-codimensional. Moreover, all the coadjoint orbits of any Heisenberg Lie group as well as 4dimensional real Diamond group are orbits of dimension zero or maximal dimension. In general, a (finite dimensional) real solvable Lie group is called an M D-group if its coadjoint orbits are zero-dimensional or maximal dimensional. The Lie algebra of an M D-group is called an M D-algebra and the class of all M D-algebras is called M D-class. Simulating the mentioned above characteristic of Heisenberg Lie algebras and 4-dimensional real Diamond algebra, we give a complete classification of M D-algebras having 1-dimensional or 1codimensional derived ideals.
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We give a complete classification of the class of Lie algebras of simply connected real Lie groups whose nontrivial coadjoint orbits are of codimension 1. Such a Lie group belongs to a well-known class, called the class of MD-groups. The Lie algebra of an MD-group is called an MD-algebra. Some interest properties of MD-algebras will be investigated as well. The techniques used in this paper is elementary techniques in matrix theory and available to apply to more general cases.
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Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject.
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The literature contains two different classifications of solvable Lie algebras of dimensions up to and including 4. This paper is devoted to comparing the two classifications and translating each into the other. In particular, we exhibit an isomorphism between each solvable Lie algebra of one classification and the corresponding algebra of the second. The first classification is provided by de Graaf, and the second classification is from a recent book by Snobl and Winternitz.
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We give a complete classification of the class of simply connected real Lie groups whose nontrivial coadjoint orbits are of codimension 1. Such a Lie group belongs to a well-known class, called the class of MD-groups. By definition, an MD-group is a solvable real Lie group such that its nontrivial coadjoint orbits have the same dimension. The Lie algebra of an MD-group is called an MD-algebra. Some interest properties of MD-algebras will be investigated as well.
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The problem of Lie algebras' classification, in their different varieties, has been dealt with by theory researchers since the early 20 th century. This problem has an intrinsically infinite nature since it can be inferred from the results obtained that there are features specific to each field and dimension. Despite the hundreds of attempts published, there are currently fields and dimensions in which only partial classifications of some families of algebras of low dimensions have been obtained. This article intends to bring some order to the achievements of this prolific line of research so far, in order to facilitate future research.