Results 201 to 210 of about 107,381 (262)
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2009
We restrict ourselves to the study of linear Lie groups, that is, to closed subgroups of GL(n,ℝ), for an integer n, in other words, to groups of real matrices. We adopt the convention, introduced in Chapter 1, of calling such a group simply a Lie group. We shall show that to each Lie group there corresponds a Lie algebra.
Stephanie Frank Singer +1 more
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We restrict ourselves to the study of linear Lie groups, that is, to closed subgroups of GL(n,ℝ), for an integer n, in other words, to groups of real matrices. We adopt the convention, introduced in Chapter 1, of calling such a group simply a Lie group. We shall show that to each Lie group there corresponds a Lie algebra.
Stephanie Frank Singer +1 more
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1999
The theory of differential equations had flourished to such a level by the 1860s that a systematic study of their solutions became possible. Sophus Lie, a Norwegian mathematician, undertook such a study using the same tool that was developed by Galois and others to study algebraic equations: group theory.
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The theory of differential equations had flourished to such a level by the 1860s that a systematic study of their solutions became possible. Sophus Lie, a Norwegian mathematician, undertook such a study using the same tool that was developed by Galois and others to study algebraic equations: group theory.
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Sobolev algebras on nonunimodular Lie groups
Calculus of Variations and Partial Differential Equations, 2017Let G be a noncompact connected Lie group and $$\rho $$ρ be the right Haar measure of G. Let $$\mathbf{{X}}=\{X_1,\dots ,X_q\}$$X={X1,⋯,Xq} be a family of left invariant vector fields which satisfy Hörmander’s condition, and let $$\Delta =-\sum _{i=1 ...
M. Peloso, M. Vallarino
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Groups, Lie Groups, and Lie Algebras
2011This chapter introduces abstract groups and Lie groups, which are a formalization of the notion of a physical transformation. The chapter begins with a heuristic introduction that motivates the definition of a group and gives an intuitive sense for what an “infinitesimal generator” is.
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2020
In this chapter, we recall some well-known results on Lie groups and Lie algebras. In particular, we discuss the third Lie theorem, the Ado theorem, and the Cartan semisimplicity criterion. Some important types of Lie algebras and Lie groups together with their important ideals and normal subgroups are discussed.
Valerii Berestovskii, Yurii Nikonorov
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In this chapter, we recall some well-known results on Lie groups and Lie algebras. In particular, we discuss the third Lie theorem, the Ado theorem, and the Cartan semisimplicity criterion. Some important types of Lie algebras and Lie groups together with their important ideals and normal subgroups are discussed.
Valerii Berestovskii, Yurii Nikonorov
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1985
In this chapter we explain what a Lie group is and quickly review the basic concepts of the theory of differentiable manifolds. The first section illustrates the notion of a Lie group with classical examples of matrix groups from linear algebra. The spinor groups are treated in a separate section, §6, but the presentation of the general theory of ...
Theodor Bröcker, Tammo tom Dieck
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In this chapter we explain what a Lie group is and quickly review the basic concepts of the theory of differentiable manifolds. The first section illustrates the notion of a Lie group with classical examples of matrix groups from linear algebra. The spinor groups are treated in a separate section, §6, but the presentation of the general theory of ...
Theodor Bröcker, Tammo tom Dieck
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1994
In various fields of geometry and applications object that simultaneously carry the structure of a group and a structure of a smooth manifold occur. These objects are called Lie groups provided that the group operations are smooth. As a rule, Lie groups that occur in applications have nontrivial topological structure.
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In various fields of geometry and applications object that simultaneously carry the structure of a group and a structure of a smooth manifold occur. These objects are called Lie groups provided that the group operations are smooth. As a rule, Lie groups that occur in applications have nontrivial topological structure.
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Lie-Group and Lie-Algebra Inhomogenizations
Journal of Mathematical Physics, 1968A systematic formulation of the concept of inhomogenization is given both for Lie groups and for Lie algebras, and the connection between the two structures is clarified in terms of the notion of semidirect product. Special emphasis is devoted to the classification of the inhomogenizations of semisimple Lie algebras. As an application, a lemma due to O'
Vittorio Gorini, Vittorio Berzi
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1986
This chapter is a continuation of Chapter I on the basic elements of group theory, and is devoted to continuous groups whose elements depend on a given number of continuous parameters. We are already familiar with some aspects of continuous groups from our experience with the three-dimensional rotation group.
Young S. Kim, Marilyn E. Noz
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This chapter is a continuation of Chapter I on the basic elements of group theory, and is devoted to continuous groups whose elements depend on a given number of continuous parameters. We are already familiar with some aspects of continuous groups from our experience with the three-dimensional rotation group.
Young S. Kim, Marilyn E. Noz
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The Lie Algebra of a Lie Group
2017The Lie algebra of a Lie group is introduced via the tangent space and distributions and differential operators are discussed and used in this setting.
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