Results 321 to 330 of about 15,318,133 (354)

Groups, Lie Groups, and Lie Algebras

2011
This 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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Lie groups as spin groups [PDF]

Journal of Mathematical Physics, 1993
It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras.
N. Van Acker, David Hestenes, Chris Doran, Frank Sommen   +3 more
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Lie Groups and Lie Algebras

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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On a Compact Lie Group Acting on a Manifold

, 1957
Let M be a manifold (= connected, separable, locally euclidean space) of dimension n + 1, and G a compact connected Lie group acting on M in such a way that there is at least one n-dimensional orbit.2 In this paper, we show that the space of orbits M/G ...
P. Mostert
semanticscholar   +1 more source

On the tangent Lie group of a symplectic Lie group

Ricerche di Matematica, 2019
Motivated by the recent work of Asgari and Salimi Moghaddam (Rend Circ Mat Palermo II Ser 67:185–195, 2018) on the Riemannian geometry of tangent Lie groups, we prove that the tangent Lie group $${ TG}$$ of a symplectic Lie group $$(G,\omega )$$ admits the structure of a symplectic Lie group. On $${ TG}$$, we construct a left invariant symplectic form $
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Characterization of Lie groups on the cotangent bundle of a Lie group

Letters in Mathematical Physics, 1986
The author gives a simple characterization, in differential geometric terms, of the groups K of dimension 2n which can be interpreted as semidirect products of an n-dimensional Lie group G by the group of translations of the dual space of its Lie algebra \(g^*\).
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Integrability of Poisson–Lie Group Actions

, 2009
We establish a 1:1 correspondence between Poisson–Lie group actions on integrable Poisson manifolds and twisted multiplicative Hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson–Lie group G on a Poisson manifold M,
R. Fernandes, David Iglesias Ponte
semanticscholar   +1 more source

Quantization of Lie Groups and Lie Algebras

1988
Publisher Summary This chapter focuses on the quantization of lie groups and lie algebras. The Algebraic Bethe Ansatz—the quantum inverse scattering method—emerges as a natural development of the various directions in mathematical physics: the inverse scattering method for solving nonlinear equations of evolution, the quantum theory of magnets, the ...
L. D. Faddeev, L. A. Takhtajan, N. Yu. Reshetikhin   +2 more
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The heat equation on compact Lie group

, 1975
McKean and Singer [9] posed the problem of the existence of an analogue of the Poisson's summation formula for manifolds other than flat tori. Y. Colin de Verdiere [3] gave an answer to it in the case of a 2-dimensional compact Riemannian manifold with ...
H. Urakawa
semanticscholar   +1 more source

Lie Algebras and Lie Groups

2004
In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion ...
Joe Harris, William Fulton
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