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# Lie group

## Definition of the noun Lie group

What does Lie group mean as a name of something?

noun

1. [topology] Any of many analytic groups that are also a smooth manifold; they arise as groups of rotational symmetries

## Explanation

Lie group: In mathematics, a Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, page 3.

## Printed encyclopedias and other books with definitions for Lie group

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### Theory of Group Representations and Applications (1986)

by Asim Orhan Barut, Ryszard R?czka

Any Lie group is a topological group with respect to the topology induced by its analytic structure. Indeed, a manifold is a Hausdorff space and the analytic mapping (x, y) -* xy'1 is continuous. Hence, by def. 2 (2.1) a Lie group is a topological ...

### Applied Differential Geometry (2007)

A Modern Introduction by Vladimir G. Ivancevic, Tijana T. Ivancevic

A Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group SO(3), which can be parametrized by the Euler angles. More formally, a Lie group is an analytic real or complex manifold ...

### Smooth Homogeneous Structures in Operator Theory (2005)

by Daniel Beltita

A Lie group is a group possessing a manifold structure that makes the group operations into smooth maps. The corresponding Lie algebra is just the tangent space at the unit element, with a bracket reflecting the group structure of the Lie ...

### Handbook of Mathematics (2016)

by Thierry Vialar

Lie group: A Lie group is a topological group which is also a differentiable manifold in such a way that the group operations are themselves analytic functions. But a more precise approach is required: Lie groups are sets endowed with two ...

### Geometric Control of Mechanical Systems (2004)

Modeling, Analysis, and Design for Simple Mechanical Control Systems by Francesco Bullo, Andrew D. Lewis

Roughly speaking, a Lie group is a smooth manifold equipped with a group operation and a Lie algebra is a vector space equipped with a bracket operation. Via the notion of a one-parameter subgroup generator, we present an explicit ...

### Geometric Methods and Applications (2012)

For Computer Science and Engineering by Jean Gallier

is an embedded submanifold, and thus, a Lie group (see Warner [176], Chapter 3, Theorem 3.42, page 110). Thus, a linear Lie group is a closed subgroup of ...

### Geometry and its Applications (2014)

It was proved that each minimal left-invariant unit vector field on three- dimensional unimodular Lie group is an eigenvector of the Ricci operator. A. Borisenko [1] was the first who asked on unit vector fields with totally geodesic image in the unit ...

### Introduction to Topological Manifolds (2010)

by John Lee

A Lie group is a group (in the algebraic sense) that is also a manifold, together with some technical conditions to ensure that the group structure and the manifold structure are compatible with each other. They play central roles in differential ...

### Encyclopaedia of Mathematics, Supplement III (2001)

by Michiel Hazewinkel

A Lie group G can act from the right on M by a: M x G — > M in a way which respects oj, so that one obtains a homomorphism a': g — } X{M,ui), where g is the Lie algebra of G. (For a left action one gets an anti-homomorphism of Lie algebras.) ...

### Encyclopedia of Nonlinear Science (2006)

by Alwyn Scott

A Lie group actson its Lie algebra g by the adjoint representation and on the dual space g* by the coadjoint representation. The coadjoint orbits are symplectic submanifolds withrespect tothenatural Lie¥Poisson structureon g*, andareof ...

### The Catholic Encyclopedia; (1909)

An International Work of Reference on the Constitution, Doctrine, Discipline, and History of the Catholic Church; by Charles George Herbermann

Political Groups, the Press, and Intellectual and Social Organizations, — Politically speaking, the Catho-7 lie group which receives the active sympathies of the) Catholic press is that known as the Action Liberate Populaire, founded bv M.

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Lie group Meaning

Video shows what Lie group means. Any of many analytic groups that are also a smooth manifold, they arise as groups of rotational symmetries. Lie group ...

## Anagrams of LIEGROUP

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