A root system is any collection of vectors that has properties that satisfy the roots of a semi simple Lie algebra. If g is semi simple, then the root system A, (Q) can be described as a system of vectors in a Euclide...A root system is any collection of vectors that has properties that satisfy the roots of a semi simple Lie algebra. If g is semi simple, then the root system A, (Q) can be described as a system of vectors in a Euclidean vector space that possesses some remarkable symmetries and completely defines the Lie algebra of g. The purpose of this paper is to show the essentiality of the root system on the Lie algebra. In addition, the paper will mention the connection between the root system and Ways chambers. In addition, we will show Dynkin diagrams, which are an integral part of the root system.展开更多
In this paper, the most important liner groups are classified. Those that we often have the opportunity to meet when studying linear groups as well as their application in left groups. In addition to the introductory ...In this paper, the most important liner groups are classified. Those that we often have the opportunity to meet when studying linear groups as well as their application in left groups. In addition to the introductory part, we have general linear groups, special linear groups, octagonal groups, symplicit groups, cyclic groups, dihedral groups: generators and relations. The paper is summarized with brief deficits, examples and evidence as well as several problems. When you ask why this paper, I will just say that it is one of the ways I contribute to the community and try to be a part of this little world of science.展开更多
This paper is made up of a desire for me to contribute to this beautiful field of mathematics that I have encountered in recent years. In addition, I would like to mention that I am not aware that there are papers in ...This paper is made up of a desire for me to contribute to this beautiful field of mathematics that I have encountered in recent years. In addition, I would like to mention that I am not aware that there are papers in our Balkans on Lie algebra, although this is only an introductory part for which in the near future in collaboration with several professors from abroad I will do a book in our mother tongue on Lie groups and algebras. The main content of this paper is similar to the books that have been published regarding Lie algebras, from basic definition and example, structure, killing form, classification to root system. In my opinion, this paper is important in relation to Lie algebras, because it will be helpful to all those who write papers on algebra, as well as the fact that the paper will be written in Montenegrin, which is understood by almost more than 70 percent of the population. For me, this work has the significance of being useful to all who need it.展开更多
This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was...This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was more superficial in that book with some remarks about the examples given in relation to the left group. I decided to try a little harder and collect as much literature as possible, both for the needs of me and the others who will take after me. Searching for literature in my mother tongue I could not find anything, in English as someone who comes from a small country like Montenegro, all I could find was through the internet. I decided to gather what I could find from the literature in my own way and to my observation and make this kind of work. The main content of this paper is to present the Lie group in the simplest way. Before and before I started writing or collecting about Lie groups, it was necessary to say something about groups and subgroups that are taught in basic studies in algebra. In them I cited several deficits and an example. The following content of the paper is related to Lie groups primarily concerning the definition of examples such as <i>The General Linear Group GL(n, R)</i>, The <i>Complex General Linear Group GL(n, C)</i>, <i>The Special Linear Group SL(n, R)=SL(V)</i>, <i>The Complex Special Linear Group SL(n, C)</i>, <i>Unitary and Orthogonal Groups</i>, <i>Symplectic Group</i>, <i>The groups R*, C*, S<sup>1</sup> and R<sup>n</sup></i> and others. In addition, invariant vector fields and the exponential map and the lie algebra of a lie group. For me, this work has the significance of being useful to all who need it.展开更多
文摘A root system is any collection of vectors that has properties that satisfy the roots of a semi simple Lie algebra. If g is semi simple, then the root system A, (Q) can be described as a system of vectors in a Euclidean vector space that possesses some remarkable symmetries and completely defines the Lie algebra of g. The purpose of this paper is to show the essentiality of the root system on the Lie algebra. In addition, the paper will mention the connection between the root system and Ways chambers. In addition, we will show Dynkin diagrams, which are an integral part of the root system.
文摘In this paper, the most important liner groups are classified. Those that we often have the opportunity to meet when studying linear groups as well as their application in left groups. In addition to the introductory part, we have general linear groups, special linear groups, octagonal groups, symplicit groups, cyclic groups, dihedral groups: generators and relations. The paper is summarized with brief deficits, examples and evidence as well as several problems. When you ask why this paper, I will just say that it is one of the ways I contribute to the community and try to be a part of this little world of science.
文摘This paper is made up of a desire for me to contribute to this beautiful field of mathematics that I have encountered in recent years. In addition, I would like to mention that I am not aware that there are papers in our Balkans on Lie algebra, although this is only an introductory part for which in the near future in collaboration with several professors from abroad I will do a book in our mother tongue on Lie groups and algebras. The main content of this paper is similar to the books that have been published regarding Lie algebras, from basic definition and example, structure, killing form, classification to root system. In my opinion, this paper is important in relation to Lie algebras, because it will be helpful to all those who write papers on algebra, as well as the fact that the paper will be written in Montenegrin, which is understood by almost more than 70 percent of the population. For me, this work has the significance of being useful to all who need it.
文摘This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was more superficial in that book with some remarks about the examples given in relation to the left group. I decided to try a little harder and collect as much literature as possible, both for the needs of me and the others who will take after me. Searching for literature in my mother tongue I could not find anything, in English as someone who comes from a small country like Montenegro, all I could find was through the internet. I decided to gather what I could find from the literature in my own way and to my observation and make this kind of work. The main content of this paper is to present the Lie group in the simplest way. Before and before I started writing or collecting about Lie groups, it was necessary to say something about groups and subgroups that are taught in basic studies in algebra. In them I cited several deficits and an example. The following content of the paper is related to Lie groups primarily concerning the definition of examples such as <i>The General Linear Group GL(n, R)</i>, The <i>Complex General Linear Group GL(n, C)</i>, <i>The Special Linear Group SL(n, R)=SL(V)</i>, <i>The Complex Special Linear Group SL(n, C)</i>, <i>Unitary and Orthogonal Groups</i>, <i>Symplectic Group</i>, <i>The groups R*, C*, S<sup>1</sup> and R<sup>n</sup></i> and others. In addition, invariant vector fields and the exponential map and the lie algebra of a lie group. For me, this work has the significance of being useful to all who need it.
