Correction and addenda to: On the surjectivity of the exponential map for certain lie groups

1997 ◽  
Vol 173 (1) ◽  
pp. 351-358 ◽  
Author(s):  
Martin Moskowitz
Keyword(s):  
Lie Groups ◽  
Exponential Map

New Parameterizations of \(\mathrm{SL}(2,\mathbb{R})\) and Some Explicit Formulas for Its Logarithm

2021 ◽  
Vol 60 ◽  
pp. 65-81
Author(s):  
Tihomir Valchev ◽  
◽  
Clementina Mladenova ◽  
Ivaïlo Mladenov
Keyword(s):  
Lie Groups ◽  
Analytic Description ◽  
Exponential Map ◽  
Explicit Formulas ◽  
Field Of Values

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\bbr)\cong\mathrm{SL}(2,\bbr)$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\bbr)\to\mathrm{SL}(2,\bbr)$, we have found a few explicit formulas for the logarithm of the matrices in these groups.\\ Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.

Harmonic analysis and the global exponential map for compact Lie groups

1993 ◽  
Vol 27 (1) ◽  
pp. 21-27 ◽  
Author(s):  
A. H. Dooley ◽  
N. J. Wildberger
Keyword(s):  
Harmonic Analysis ◽  
Lie Groups ◽  
Exponential Map ◽  
Compact Lie Groups

scholarly journals On the Dimension of Certain Graded Lie Algebras Arising in Geometric Integration of Differential Equations

2000 ◽  
Vol 3 ◽  
pp. 44-75
Author(s):  
Arieh Iserles ◽  
Antonella Zanna
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Algebra ◽  
Homogeneous Spaces ◽  
Practical Interest ◽  
Exponential Map ◽  
Graded Lie Algebras ◽  
Graded Algebras ◽  
Algebraic Setting ◽  
Cayley Transformation

AbstractMany discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

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