Construction of canonical coordinates for exponential Lie groups

Texts in Applied Mathematics, 2001

Annales De L Institut Henri Poincare-physique Theorique, 1970

Some facts concerning symplectic vector spaces, their automorphism groups Spl(2n, R, E), and derivation Lie algebras spl(2n, R, E) are given. For every element R of these Lie algebras a solvable Lie group exp(RR) x E x R is constructed, which is nilpotent iff R is nilpotent. We calculate the Lie algebras f~R (B E of these groups, all of which contain the Heisenberg Lie algebra. Automorphism groups and derivation Lie algebras of RR fl3 E fl3 R, and faithful finite dimensional representations of them together with the corresponding representations of exp(RR) x E x R are given. In Part II a modification weyl(E, cr) of the universal enveloping algebra of the Heisenberg Lie algebra is defined. We realize the Lie algebras E E9 R in this algebra. Finally some automorphisms and derivations of are constructed by means of the adjoint representation of weyl(E, ~). Attention is given to the case of the harmonic oscillator and especially to the free nonrelativistic particle whose group is nilpot...

Symmetries in Science X, 1998

2012

In this note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair (L, A) of algebroids. In particular, we prove that the quotient L/A of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid A, which we call Kapranov module.

Functional Analysis and Its Applications, 1993

2014

In this paper we prove that matrix groups are manifolds and use them as a special case to introduce the concepts of Lie groups, Lie algebras, and the exponential map.

2013

Abstract. In this note we envisage the relation existing between the Lie Groups and the Theory of Complex Variables. In particular, it is shown that the dimensions of the irreducibles representations of SUpNq may be written in terms of the Eisenstein integers and an identity is built up between the imaginary parts of the dimensions of the irreducible representations of the Lie Groups SUp3q and Spp4q.

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.

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.

euler.slu.edu

We show that when the methods of are combined with the explicit stratification and orbital parameters of [9] and [10], the result is a construction of explicit analytic canonical coordinates for any coadjoint orbit O of a completely solvable Lie group. For each layer in the stratification, the canonical coordinates and the orbital cross-section together constitute an analytic parametrization for the layer.