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@article{VVGUM_2017_20_3_a4, author = {B. Zegarli\'nski}, title = {Crystallographic {Groups} for {H{\"o}rmander} {Fields}}, journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie}, pages = {43--64}, publisher = {mathdoc}, volume = {20}, number = {3}, year = {2017}, language = {en}, url = {http://geodesic.mathdoc.fr/item/VVGUM_2017_20_3_a4/} }
B. Zegarliński. Crystallographic Groups for H{\"o}rmander Fields. Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 43-64. http://geodesic.mathdoc.fr/item/VVGUM_2017_20_3_a4/
[1] A. Agrachev, U. Boscain, J.\;-P. Gauthier, F. Rossi, “The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups”, J. Funct. Analysis, 256 (2009), 2621–2655 | DOI | MR | Zbl
[2] C. Altafini, “A Matrix Lie Group of Carnot Type for Filiform Sub-Riemannian Structures and its Application to Control Systems in Chained Form”, Proceedings of the Summer School on Differential Geometry, Universidade de Coimbra, Coimbra, 1999, 59–66 | MR | Zbl
[3] J.\;M. Ancochea, R. Campoamor, “Characteristically Nilpotent Lie Algebras: A Survey”, Extracta Mathematicae, 16:2 (2001), 153–210 | MR | Zbl
[4] H. Bahouri, I. Gallagher, The heat kernel and frequency localized functions on the Heisenberg group, 2008, arXiv: 0804.0340 [math.AP] | MR
[5] D. Bakry, F. Baudoin, M. Bonnefont, D. Chafai, “On gradient bounds for the heat kernel on the Heisenberg group”, J. Func. Analysis, 255 (2008), 1905–1938 | DOI | MR | Zbl
[6] D. Bakry, M. Emery, “Diffusions hypercontractives”, Sem. de Probab. XIX, Lecture Notes in Math., 1123, Springer–Verlag, Berlin, 1985, 177–206 | DOI | MR
[7] D. Basu, “Formal Equivalence of the Hydrogen Atom and Harmonic Oscillator and Factorization of the Bethe-Salpeter Equations”, J. Math. Phys., 12 (1971), 1474–1483 | DOI
[8] J.\;J. Benedetto, R.\;L. Benedetto, “The Construction of Wavelet Sets”, Wavelets and Multi-Scale Analysis: from theory to application, Birkhäuser, Boston, 2011, 17–56 | MR
[9] I. Benjamini, I. Chavel, A. Feldman, “Heat Kernel Lower Bounds on Riemannian Manifolds Using the Old Idea of Nash”, Proc. of the London Math. Soc., s3–72 (1) (1996), 215–240 | DOI | MR | Zbl
[10] A. Berenstein, Y. Burman, “Dunkl Operators and Canonical Invariants of Reflection Groups”, Symmetry, Integrability and Geometry: Methods and Applications, SIGMA 5 (2009), 057 | MR | Zbl
[11] J.\;M. A. Bermúdez, O.\;R. C. Stursberg, “Classification of $(n - 5)$-filiform Lie algebras”, Linear Algebra and its Applications, 336 (2001), 167–180 | DOI | MR | Zbl
[12] R. Black, W. Hudelson, L. Lackney, J. Rohal, J. Adler, J. Palagallo, “Self-similar Tilings of Nilpotent Lie Groups”, Univ Acron., 2006 http://jamesrohal.com/blog/wp-content/uploads/2008/10/reu-2006-final-write-up.pdf
[13] J.\;D. Blanchard, K.\;R. Steffen, “Crystallographic Haar-type Composite Dilation Wavelets”, Wavelets and Multi-Scale Analysis: from theory to application, Birkhäuser, Boston, 2011, 83–108 | MR
[14] M. Boiteux, “The three-dimensional hydrogen atom as a restricted four-dimensional harmonic oscillator”, Physica, 65 (1973), 381–395 | DOI | MR
[15] A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians, Springer–Verlag, Berlin; Heidelberg, 2007, 802 pp. | MR | Zbl
[16] U. Boscain, J.\;-P. Gauthier, F. Rossi, Hypoelliptic heat kernel on 3-step nilpotent Lie groups, 2010, arXiv: 1002.0688v1 [math.AP]
[17] L. Boza, E.\;M. Fedriani, J. Nuñez, “A new method for classifying complex Filiform Lie algebras”, Applied Mathematics and Computation, 121 (2001), 169–175 | DOI | MR | Zbl
[18] M. Bridson, A. Hafliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 2007, 643 pp. http://www.math.psu.edu/petrunin/papers/akp-papers/bridson.haefliger.pdf | MR
[19] J.\;M. Cabezas, E. Pastor, L.\;M. Camacho, J.\;R. Gomez, A. Jimenez–Merchan, J. Reyes, I. Rodriguez, “Chapter 3. Some Problems About Nilpotent Lie Algebras”, Ring Theory And Algebraic Geometry, CRC Press, Boca Raton, 2001, 79–106 | MR
[20] J.\;L. Cardoso, R. Alvarez-Nodarse, “Recurrence relations for radial wavefunctions for the Nth-dimensional oscillators and hydrogenlike atoms”, J. Phys. A: Math. Gen., 36 (2003), 2055–2068 | DOI | MR | Zbl
[21] H.\;A. Cerdeira, “Hydrogen atom in the four-dimensional oscillator representation: matrix elements of $z^{n}$”, J. Phys. A: Math. Gen., 18 (1985), 2719–2727 http://iopscience.iop.org/0305-4470/18/14/022 | DOI | MR
[22] P.\;A. Clarkson, “Painlevé Equations — Nonlinear Special Functions”, Orthogonal Polynomials and Special Functions, Springer, Berlin; Heidelberg, 2006, 331–411 | DOI | MR | Zbl
[23] L.\;J. Corwin, F.\;P. Greenleaf, Representations of nilpotent Lie groups and their applications, Cambridge Univ. Press, Cambridge, 1990, 280 pp. | MR | Zbl
[24] J. Cygan, “Heat kernels for class 2 nilpotent groups”, Studia Math., 64:3 (1979), 227–238 | DOI | MR | Zbl
[25] E.\;B. Davies, Heat kernels and spectral theory, Cambridge University Press, Cambridge, 1989, 197 pp. | MR | Zbl
[26] J. Dixmier, “Sur les representations unitaires des groupes de Lie nilpotents. I”, Amer. J. Math., 81 (1959), 160–170 | DOI | MR | Zbl
[27] J. Dixmier, “Sur les representations unitaires des groupes de Lie nilpotents. II”, Bull. Soc. Math. France, 85 (1957), 325–388 | DOI | MR | Zbl
[28] J. Dixmier, “Sur les representations unitaires des groupes de Lie nilpotents. III”, Canad. J. Math., 10 (1958), 321–348 | DOI | MR | Zbl
[29] J. Dixmier, “Sur les representations unitaires des groupes de Lie nilpotents. IV”, Canad. J. Math., 11 (1959), 321–344 | DOI | MR | Zbl
[30] J. Dixmier, “Sur les representations unitaires des groupes de Lie nilpotents. V”, Bull. Soc. Math. France, 87 (1959), 65–79 | DOI | MR | Zbl
[31] J. Dixmier, “Sur les representations unitaires des groupes de Lie nilpotents. VI”, Canad. J. Math., 12 (1960), 324–352 | DOI | MR | Zbl
[32] J. Dixmier, “Sur les representations unitaires des groupes de Lie algebriques”, Ann. Inst. Fourier (Grenoble), 7 (1957), 315–328 | DOI | MR | Zbl
[33] F. Dragoni, V. Kontis, B. Zegarliński, “Ergodicity of Markov Semigroups with Hormander type generators in Infinite Dimensions”, Journal of Potential Analysis, 37 (2012), 199–227, arXiv: 1012.0257v1 [math.AP] | DOI | MR | Zbl
[34] B.\;K. Driver, T. Melcher, “Hypoelliptic heat kernel inequalities on Lie groups”, Stoch. Process. Appl., 118 (2008), 368–388 | DOI | MR
[35] B.\;K. Driver, T. Melcher, “Hypoelliptic heat kernel inequalities on the Heisenberg group”, J. Func. Analysis, 221 (2005), 340–365 | DOI | MR | Zbl
[36] C.\;F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Cambridge University Press, Cambridge, 2001, 408 pp. | MR | Zbl
[37] P. Etingof, Lectures on Calegero — Moser Systems, 2006, arXiv: 0606233 [math.AP]
[38] P. Etingof, X. Ma, Lecture Notes on Cherednik Algebras, 2010, arXiv: 1001.0432v4 [math.AP]
[39] G. Felder, A.\;P. Veselov, “Action of Coxeter Groups on Harmonic Polynomials and KZ Equations”, Moscow Math Journal, 3:4 (2003), 1269–1291 | MR | Zbl
[40] A.\;S. Fokas, A.\;R. Its, A.\;A. Kapaev, V.\;Yu. Novokshenov, Painleve Transcendents: The Riemann-Hilbert Approach, AMS, Providence, RI, 2006, 553 pp. | MR | Zbl
[41] G.\;B. Folland, Harmonic analysis in phase space, Princeton Univ. Press, Princeton, 1989, 288 pp. | MR | Zbl
[42] H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Springer-Verlag, Berlin; Heidelberg, 2005, 199 pp. | MR | Zbl
[43] L. Gallardo, M. Yor, “Some Remarkable Properties of the Dunkl Martingales”, Lecture Notes in Mathematics, 1874 (2006), 337–356 | DOI | MR | Zbl
[44] B. Gaveau, “Principle de moindre action, propogation de la chaleur et estimees sous-elliptiques sur certains groups nilpotents”, Acta Math., 139 (1997), 95–153 | DOI | MR
[45] J.\;R. Gómez, A. Jimenéz-Merchán, Y. Khakimdjanov, “Low-dimensional filiform Lie algebras”, J. Pure and Appl. Algebra, 130 (1998), 133–158 | DOI | MR
[46] J.\;R. Gómez, A. Jimenéz-Merchán, Y. Khakimdjanov, “Symplectic structures on the filiform Lie algebras”, J. Pure and Appl. Algebra, 156 (2001), 15–31 | DOI | MR
[47] M.\;-P. Gong, “Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and $R$)”, University of Waterloo Thesis, Waterloo, Canada, 1998 http://etd.uwaterloo.ca/etd/mpgong1998.pdf | MR
[48] M. Grayson, R. Grossman, “Nilpotent Lie algebras and vector fields”, Symbolic Computation: Applications to Scientific Computing, SIAM, Philadelphia, 1989, 77–96 | DOI | MR
[49] A. Grigor'yan, “Gaussian Upper Bounds for the Heat Kernel on Arbitrary Manifolds”, J. Differential Geometry, 45 (1997), 33–52 | DOI | MR | Zbl
[50] A. Hulanicki, “The distribution of energy of the Brownian motion in Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group”, Studia Math., 56 (1976), 165–173 | DOI | MR | Zbl
[51] A.\;A. Kirillov, “Unitary representations of nilpotent Lie groups”, Russian Math. Surveys, 17 (1962), 53–104 | DOI | MR | Zbl
[52] A.\;A. Kirillov, Lectures on the Orbit Method, AMS, Providence, RI, 2004, 408 pp. | MR | Zbl
[53] M. Kuczma, Functional equations in a single variable, PWN, Warszawa, 1968, 383 pp. | MR | Zbl
[54] H.\;-Q. Li, “Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg”, J. Funct. Analysis, 236 (2006), 369–394 | DOI | MR | Zbl
[55] Y.\;F. Liu, Y.\;A. Lei, J.\;Y. Zeng, “Reply to the comment by Stahlhofen on “Factorization of the radial Schrödinger equation and four kinds of raising and lowering operators of hydrogen atoms and isotropic harmonic oscillators””, Physics Letters A, 241 (1998), 300–302 | DOI | MR | Zbl
[56] H. Liu, Y. Liu, H. Wang, “Multiresolution Analysis, Self-Similar Tilings and Haar Wavelets on the Heisenberg”, Acta Mathematica Scientia, 29:5 (2009), 1251–1266 | DOI | MR | Zbl
[57] H. Liu, L. Peng, “Admissible Wavelets Associated with the Heisenberg Group”, Pacific Journal of Mathematics, 180 (1997), 101–123 | DOI | MR | Zbl
[58] L. Magnin, “Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras”, Algebr. Represent. Theor., 13 (2010), 723–753 | DOI | MR | Zbl
[59] L. Magnin, “Research Article Adjoint and Trivial Cohomologies of Nilpotent Complex Lie Algebras of Dimension $\leq$ 7”, International J. of Math. and Math. Sci., 2008 (2008), 805305 | DOI | MR | Zbl
[60] M. Noumi, Y. Yamada, “Affine Weyl group symmetries in Painleve type equations”, Toward the exact WKB analysis of differential equations, linear or non-linear, Kyoto University Press, Kyoto, 2000, 245–259 | MR | Zbl
[61] K. Okamoto, “Studies on the Painleve equations. I”, Ann. Mat. Pura Appl., 146 (1987), 337–381 | DOI | MR | Zbl
[62] K. Okamoto, “Studies on the Painleve equations. II”, Jap. J. Math., 13 (1987), 47–76 | DOI | MR | Zbl
[63] K. Okamoto, “Studies on the Painleve equations. III”, Math. Ann., 275 (1986), 221–256 | DOI | MR
[64] K. Okamoto, “Studies on the Painleve equations. IV”, Funkcial. Ekvac., 30 (1987), 305–332 | MR | Zbl
[65] M.\;A. Olshanetsky, A.\;M. Perelomov, “Classical integrable finite-dimensional systems related to Lie algebras”, Phys. Reports, 71 (1981), 313–400 | DOI | MR
[66] E.\;M. Opdam, “Lecture Notes on Dunkl Operators for Real and Complex Reflection Groups”, Mathematical Society of Japan, 2013 http://projecteuclid.org/euclid.msjm/1389985758
[67] Painlevè transcendents, http://en.wikipedia.org/wiki/Painleve_transcendents
[68] M. Rösler, “Dunkl Operators: Theory and Applications”, Lecture Notes in Math, 1817 (2003), 93–136, Springer-Verlag, Berlin; Heidelberg | DOI | MR
[69] M. Rösler, “Generalized Hermite polynomials and the heat equation for Dunkl operators”, Commun. Math. Phys., 192 (1998), 519–542 | DOI | MR
[70] A. Sakka, “Bäcklund transformations for Painleve I and II equations to Painleve-type equations of second order and higher degree”, Phys. Lett. A., 300 (2002), 228–232 | DOI | MR | Zbl
[71] C. Seeley, “7-dimensional nilpotent Lie algebras”, Trans. Amer. Math. Soc., 335 (1993), 479–496 | MR | Zbl
[72] T. Skjelbred, T. Sund, “On the classification of nilpotent Lie algebras”, Comptes Rendus Acad. Sc. Paris serie A, 286 (1978), 241–242 | MR | Zbl
[73] M. Suzuki, N. Tahara, K. Takano, “Hierarchy Backlund transformation groups of the Painleve systems”, J. Math. Soc. Japan Vol., 56:4 (2004), 1221–1224 | DOI | MR
[74] K.\;A. Umlauf, Uber Die Zusammensetzung Der Endlichen Continuierlichen Transformationsgruppen, Insbesondre Der Gruppen Vom Range Null (1891), Kessinger Publ., Whitefish, Montana, 2010, 82 pp.
[75] M. Vergne, “Cohomologie des algtbres de Lie nilpotentes. Application a l'etude de la variete des algebres de Lie nilpotentes”, Bull. Sot. Math. France, 98 (1970), 81–116 | DOI | MR | Zbl
[76] B. Warhurst, “Contact and Quasiconformal Mappings on real model of filiform groups”, Bull. Austral. Math. Soc., 68 (2003), 329–343 | DOI | MR | Zbl
[77] Q. Yang, Multiresolution Analysis on Non-Abelian Locally Compact Groups., 1999 www.nlc-bnc.ca/ojour/s4/f2/dsk1/tape9/PQDD_0018/NQ43523.pdf | MR
[78] B. Zegarliński, “Analysis on Extended Heisenberg Group”, Annales de la Faculte des Sciences de Toulouse, XX:2 (2011), 379–405 | DOI | Zbl
[79] G.\;-J. Zeng, S.\;-L. Zhou, S.\;-M. Ao, F.\;-S. Jiang, “Transformation between a hydrogen atom and a harmonic oscillator of arbitrary dimensions”, J. Phys. A: Math. Gen., 30 (1997), 1775–1783 | DOI | MR | Zbl
[80] F. Zhu, “The heat kernel and the Riesz transform on the quaternionic Heisenberg groups”, Pacific J. Math., 209 (2003), 175–199 | DOI | MR | Zbl