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Algebra, Lie Group and Lie Algebra
2010Geometry, algebra, and analysis are usually called the three main branches of mathematics. This chapter introduces some fundamental results in algebra that are mostly useful in systems and control. In section 4.1 some basic concepts of group and three homomorphism theorems are discussed. Ring and algebra are introduced briefly in section 4.2. As a tool,
Daizhan Cheng, Xiaoming Hu, Tielong Shen
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Canadian Journal of Mathematics, 1972
In this paper we attempt to define the notion of normed Lie algebra by endowing the corresponding algebraic concept with topologicalmetric properties. More precisely, we define normed Lie algebras as being normed spaces possessing a Lie product, the latter satisfying a ...
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In this paper we attempt to define the notion of normed Lie algebra by endowing the corresponding algebraic concept with topologicalmetric properties. More precisely, we define normed Lie algebras as being normed spaces possessing a Lie product, the latter satisfying a ...
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The Quarterly Journal of Mathematics, 1993
The author provides detailed information for the nilpotency class of Lie algebras (over a field \(k\)) satisfying the Engel condition \((\operatorname {ad}b)^n= 0\) for all \(b\), \(n=3,4\). The author shows that for \(n=3\), \(\operatorname {char}k\neq 2,5\), the nilpotency class is \(\leq 4\) (which is the best possible result) and for \(n=4 ...
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The author provides detailed information for the nilpotency class of Lie algebras (over a field \(k\)) satisfying the Engel condition \((\operatorname {ad}b)^n= 0\) for all \(b\), \(n=3,4\). The author shows that for \(n=3\), \(\operatorname {char}k\neq 2,5\), the nilpotency class is \(\leq 4\) (which is the best possible result) and for \(n=4 ...
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Canadian Journal of Mathematics, 1969
Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras.
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Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras.
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Lie Algebras and Lie Algebra Representations
2017In this chapter, we will introduce Lie algebras and Lie algebra representations, which provide a tractable linear construction that captures much of the behavior of Lie groups and Lie group representations.
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Mathematics of the USSR-Sbornik, 1986
In this paper general problems of the deformations of Lie algebras over a field of characteristic zero and related problems of calculating the cohomologies with coefficients in the adjoint representation are considered. The author adapts and generalizes to the case of Lie algebras the general constructions of the cohomology theory of (local ...
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In this paper general problems of the deformations of Lie algebras over a field of characteristic zero and related problems of calculating the cohomologies with coefficients in the adjoint representation are considered. The author adapts and generalizes to the case of Lie algebras the general constructions of the cohomology theory of (local ...
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Mathematics of the USSR-Sbornik, 1981
The author investigates Lie algebras which contain elements the squares of whose adjoint endomorphisms are zero. The basic theory is used, in particular, to improve the proof of the local nilpotence of Engel Lie algebras. Bibliography: 3 titles.
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The author investigates Lie algebras which contain elements the squares of whose adjoint endomorphisms are zero. The basic theory is used, in particular, to improve the proof of the local nilpotence of Engel Lie algebras. Bibliography: 3 titles.
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Variational algorithms for linear algebra
Science Bulletin, 2021Xiaosi Xu, Jinzhao Sun, Suguru Endo
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