Prolongations of Lie Algebra Representations

An Introduction to Lie Groups and Lie Algebras

This paper studies the relationship between representations of a Lie group and representations of its Lie algebra. We will make the correspondence in two steps: First we shall prove that a given representation of a Lie group will provide us with a corresponding representation of its Lie algebra. Second, we shall go backwards and see whether a given representation of a Lie algebra will have a corresponding representation of its Lie group. 1. Introduction to Lie groups and Lie algebras 1 2. From Representations of Lie groups to Lie algebras 8 3. From Representations of Lie algebras to Lie groups 11 Acknowledgements 14 References 14 Date: October 1st 2013. 1 Both GL(n,R) and GL(n,C) are equipped with a standard smooth structure. For details, please refer to chapter 1 of Lee's Introduction to Smooth Manifolds.

2008

We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out.

2000

We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. 2000 Mathematics Subject Classification. Primary 17B05, 17B56. Key words and phrases. Extensions of Lie algebras, cohomology of Lie algebras. P.W.M. was supported by 'Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P 14195 MAT'. Typeset by A M S-T E X

This text presents basic results from a projected monograph on "Lie Theory and the Structure of Pro-Lie groups and Locally Compact Groups" which may be consid-ered a sequel to our book "The Structure of Compact Groups" [De Gruyter, Berlin, 1998]. In focus are the categories of pro-jective limits of finite dimensional Lie groups and of projec-tive limits of finite dimensional Lie algebras, their functorial relationship, and their intrinsic Lie theory. Explicit informa-tion on pro-Lie algebras, simply connected pro-Lie groups and abelian pro-Lie groups is given.

2017

2020

In this research article, Lie Groups and Lie Algebras are projected in a distinct direction and with innovative proofs. We develop the necessary and sufficient condition for a topological group to be Hausdorff. The criteria of topological group and Haussdorff to be connected is also derived. This research article mainly explores the concept of left in variant field and presents an important hypothesis namely ―any Lie group is parallelizable‖ with a very simple accurate logical and analytical reasoning. Moreover this paper covers the impervious of another property namely every one parameter subgroup is an integral curve of some left-invariant vector field.

arXiv: Differential Geometry, 2020

We first define the concept of Lie algebroid in the convenient setting. In reference to the finite dimensional context, we adapt the notion of prolongation of a Lie algebroid over a fibred manifold to a convenient Lie algebroid over a fibred manifold. Then we show that this construction is stable under projective and direct limits under adequate assumptions.

A Lie group G is a space which possesses two structures: 1) structure of a group; 2) structure of a smooth manifold. These structures are compatible in the sense that the group operations (" multiplication " a, b → a · b and " taking the inverse element " a → a −1) are both smooth. Main examples of Lie groups are matrix groups:

2020

Lie groups are important to describe symmetries, both in mathematics and in applications (physics, chemistry, engineering,. . . ). The classical Lie groups are for example the orthogonal groups O(n), the unitary groups U(n), but mathematicians and physicists are also fascinated by more exotic examples such as the symmetry group of the octonions which is discussed a lot in modern mathematical physics. Many of these Lie groups can be represented as subgroups of Gl(k,R) for some sufficiently large k, but there are also Lie groups which cannot. Lie groups are manifolds G together with a multiplication μ : G×G→ G which is a smooth map, such that (G,μ) is a group.1 Lie groups and their representation is a mighty theory which allows effect calculations both for problems inside mathematics and also for applications outside.

Algorithmic Lie Theory for Solving Ordinary Differential Equations, 2007

Suppose G is a Lie group and M is a manifold (G and M are not necessarily finite dimensional). Let D(M) denote the group of diffeomorphisms on M and V(M) denote the Lie algebra of vector fields on M. If X: isv a. complete-vector: field-then; Exp tX will denote the one-parameter group of X. A local action <£ of G oh M gives rise to a Lie algebra homomorphism <J> + from L(G) into V(M). In particular if G is a subgroup-of D(M> and <|> : G x M-> M is the natural global action (g»p)-> g(p). then G is called a Lie transformation group of M. If M is a Hausdorff manifold and G is a Lie transformation group of M we show that <j> is an isomorphism of L(G) onto <f> (L(G)) and L = <j> + (L(G)) satisfies the following conditions : (A) L consists of complete vector fields. (B) L has a Banach Lie algebra structure satisfying the following two conditions : (BI) the evaluation map ev : (X,p)-> X(p) is a vector bundle morphism from the trivial bundle L x M into T(M), (B2) there exists an open ball B r (0) of radius r at 0 such that Exp : L-> D(M) is infective on B r (0). Conversely, if L is a suba-lgebra of V(M) (M Hausdorff) satisfying conditions (A) and (B) we show there exists a unique connected Lie transfor-+ mation group with natural action <j> : G x M-> M such that tj) is a Banach Lie algebra isomorphism of L(G) onto L .