×

Wei-Norman equations for classical groups via cominuscule induction. (English) Zbl 1323.22006

The Wei-Norman method was developed by its authors in order to reduce a system of linear differential equations with variable coefficients to a nonlinear one. The present paper is devoted to a unified approach based on the cominuscule induction, which works for all complex reductive Lie groups having no simple factors of type \(G_2\), \(F_4\) or \(E_8\). Moreover, for the case of simple Lie groups for which the cominuscule induction is not applicable, it is proved that employing the contact grading of the corresponding Lie algebra gives the Wei-Norman equations in the form of coupled first-order equations of at most fourth degree (compared to two in the cominuscule case). Since the contact grading exists for all simple Lie algebras, these results exhibit the structure of the Wei-Norman equations for all reductive complex Lie groups.

MSC:

22E30 Analysis on real and complex Lie groups
47D06 One-parameter semigroups and linear evolution equations
22E10 General properties and structure of complex Lie groups
34G10 Linear differential equations in abstract spaces
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Wei, J.; Norman, E., Lie algebraic solution of linear differential equations, J. Math. Phys., 4, 4, 575-581 (1963) · Zbl 0133.34202
[2] Charzyński, S.; Kuś, M., Wei-Norman equations for a unitary evolution, J. Phys. A, Math. Theor., 46, 26, 265208 (2013) · Zbl 1417.34036
[3] Charzyński, S.; Kuś, M., Wei-Norman equations for classical groups, J. Differ. Equ., 259, 1542-1559 (2015) · Zbl 1323.34045
[4] Kushner, A.; Lychagin, V.; Rubtsov, V., Contact Geometry and Non-Linear Differential Equations (2007), Cambridge University Press: Cambridge University Press Cambridge
[5] Richardson, R.; Röhrle, G.; Steinberg, R., Parabolic subgroups with abelian unipotent radical, Invent. Math., 110, 3, 649-671 (1992) · Zbl 0786.20029
[6] Čap, A.; Slovák, J., Background and general theory, (Parabolic Geometries. I. Parabolic Geometries. I, Mathematical Surveys and Monographs, vol. 154 (2009), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1183.53002
[7] Hwang, J.-M.; Mok, N., Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. Math., 131, 2, 393-418 (1998) · Zbl 0902.32014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.