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.

PERTURBATIVE SOLUTIONS OF DIFFERENTIAL EQUATIONS IN LIE GROUPS

2005 ◽  
Vol 02 (01) ◽  
pp. 111-125 ◽  
Author(s):  
PAOLO ANIELLO
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Similarity Transformation ◽  
Perturbative Expansion ◽  
Analytic Perturbation ◽  
The Matrix ◽  
Matrix Formula

We show that, given a matrix Lie group [Formula: see text] and its Lie algebra [Formula: see text], a 1-parameter subgroup of [Formula: see text] whose generator is the sum of an unperturbed matrix Â0 and an analytic perturbation Â♢(λ) can be mapped — under suitable conditions — by a similarity transformation depending analytically on the perturbative parameter λ, onto a 1-parameter subgroup of [Formula: see text] generated by a matrix [Formula: see text] belonging to the centralizer of Â0 in [Formula: see text]. Both the similarity transformation and the matrix [Formula: see text] can be determined perturbatively, hence allowing a very convenient perturbative expansion of the original 1-parameter subgroup.

scholarly journals Lie-group methods

Acta Numerica ◽  
2000 ◽  
Vol 9 ◽  
pp. 215-365 ◽  
Author(s):  
Arieh Iserles ◽  
Hans Z. Munthe-Kaas ◽  
Syvert P. Nørsett ◽  
Antonella Zanna
Keyword(s):  
Differential Geometry ◽  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Group Structure ◽  
Practical Interest ◽  
The Novel ◽  
Group Methods ◽  
Novel Theory ◽  
Numerical Integrators

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

scholarly journals Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces

2021 ◽  
Vol 9 (1) ◽  
pp. 119-148
Author(s):  
Thomas Ernst
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Homogeneous Spaces ◽  
Iwasawa Decomposition ◽  
Base Field ◽  
Killing Form ◽  
Exact Sequences

Abstract We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples S U q ( 1 , 1 ) S O q ( 2 ) {{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}} and S O q ( 3 ) S O q ( 2 ) {{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).

The Darboux coordinate system and Holstein–Primakoff representations on Kähler manifolds

2006 ◽  
Vol 84 (10) ◽  
pp. 891-904
Author(s):  
J R Schmidt
Keyword(s):  
Coordinate System ◽  
Lie Groups ◽  
Lie Algebra ◽  
Poisson Structure ◽  
Kähler Manifolds ◽  
Coadjoint Orbits ◽  
Kahler Manifolds ◽  
Canonical Transformations ◽  
Kahler Geometry

The Kahler geometry of minimal coadjoint orbits of classical Lie groups is exploited to construct Darboux coordinates, a symplectic two-form and a Lie–Poisson structure on the dual of the Lie algebra. Canonical transformations cast the generators of the dual into Dyson or Holstein–Primakoff representations.PACS Nos.: 02.20.Sv, 02.30.Ik, 02.40.Tt

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.

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