Geodesic
Canonical bases for $\mathfrak{sl}(2,{\mathbb{C}})$-modules of spherical monogenics in dimension 3
Archivum mathematicum, Tome 46 (2010) no. 5, pp. 339-349 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as ${\mathfrak{sl}}(2,{\mathbb{C}})$-modules. As finite-dimensional irreducible ${\mathfrak{sl}}(2,{\mathbb{C}})$-modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.
Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as ${\mathfrak{sl}}(2,{\mathbb{C}})$-modules. As finite-dimensional irreducible ${\mathfrak{sl}}(2,{\mathbb{C}})$-modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.
Classification : 17B10, 30G35, 33C50
Keywords: spherical monogenics; orthogonal basis; Legendre polynomials; $\mathfrak{sl}(2, {\mathbb{C}})$-module 
@article{ARM_2010_46_5_a4,
     author = {L\'avi\v{c}ka, Roman},
     title = {Canonical bases for $\mathfrak{sl}(2,{\mathbb{C}})$-modules of spherical monogenics in dimension 3},
     journal = {Archivum mathematicum},
     pages = {339--349},
     year = {2010},
     volume = {46},
     number = {5},
     mrnumber = {2753988},
     zbl = {1249.30136},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2010_46_5_a4/}
}
TY  - JOUR
AU  - Lávička, Roman
TI  - Canonical bases for $\mathfrak{sl}(2,{\mathbb{C}})$-modules of spherical monogenics in dimension 3
JO  - Archivum mathematicum
PY  - 2010
SP  - 339
EP  - 349
VL  - 46
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/ARM_2010_46_5_a4/
LA  - en
ID  - ARM_2010_46_5_a4
ER  - 
%0 Journal Article
%A Lávička, Roman
%T Canonical bases for $\mathfrak{sl}(2,{\mathbb{C}})$-modules of spherical monogenics in dimension 3
%J Archivum mathematicum
%D 2010
%P 339-349
%V 46
%N 5
%U http://geodesic.mathdoc.fr/item/ARM_2010_46_5_a4/
%G en
%F ARM_2010_46_5_a4
Lávička, Roman. Canonical bases for $\mathfrak{sl}(2,{\mathbb{C}})$-modules of spherical monogenics in dimension 3. Archivum mathematicum, Tome 46 (2010) no. 5, pp. 339-349. http://geodesic.mathdoc.fr/item/ARM_2010_46_5_a4/

[1] Bock, S.: Über funktionentheoretische Methoden in der räumlichen Elastizitätstheorie. Ph.D. thesis, University Weimar, 2010, German, http://e-pub.uni-weimar.de/frontdoor.php?source_opus=1503

[2] Bock, S., Gürlebeck, K.: On an orthonormal basis of solid spherical monogenics recursively generated by anti–holomorphic $\bar{z}$–powers. Proc. of ICNAAM 2009, AIP Conference Proceedings (Simos, T. E., Psihoyios, G., Tsitouras, Ch., eds.), vol. 1168, 2009, pp. 765–768. | Zbl

[3] Bock, S., Gürlebeck, K.: On a generalized Appell system and monogenic power series. Math. Methods Appl. Sci. 33 (4) (2010), 394–411. | MR | Zbl

[4] Bock, S., Gürlebeck, K., Lávička, R., Souček, V.: Gelfand–Tsetlin bases for spherical monogenics in dimension 3. preprint.

[5] Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman, London, 1982. | MR | Zbl

[6] Brackx, F., Schepper, H. De, Lávička, R., Souček, V.: Gelfand–Tsetlin Bases of Orthogonal Polynomials in Hermitean Clifford Analysis. preprint.

[7] Brackx, F., Schepper, H. De, Lávička, R., Souček, V.: The Cauchy–Kovalevskaya Extension Theorem in Hermitean Clifford Analysis. preprint.

[8] Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups. Springer, New York, 1985. | MR

[9] Cação, I.: Constructive approximation by monogenic polynomials. Ph.D. thesis, Univ. Aveiro, 2004.

[10] Cação, I., Malonek, H. R.: Remarks on some properties of monogenic polynomials. Proc. of ICNAAM 2006 (Simos, T. E., Psihoyios, G., Tsitouras, Ch., eds.), Wiley–VCH, Weinheim, 2006, pp. 596–599.

[11] Cação, I., Malonek, H. R.: On a complete set of hypercomplex Appell polynomials. Proc. of ICNAAM 2008 (Timos, T. E., Psihoyios, G., Tsitouras, Ch., eds.), vol. 1048, 2008, AIP Conference Proceedings, pp. 647–650.

[12] Delanghe, R.: On homogeneous polynomial solutions of the Riesz system and their harmonic potentials. Complex Var. Elliptic Equ. 52 (10–11) (2007), 1047–1061. | MR | Zbl

[13] Delanghe, R., Lávička, R., Souček, V.: The Gelfand–Tsetlin bases for Hodge–de Rham systems in Euclidean spaces. preprint.

[14] Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor–Valued Functions. Kluwer Academic Publishers, Dordrecht, 1992. | MR

[15] Falcão, M. I., Cruz, J. F., Malonek, H. R.: Remarks on the generation of monogenic functions. Proc. of IKM 2006 (Gürlebeck, K., Könke, C., eds.), Bauhaus–University Weimar, 2006, ISSN 1611–4086.

[16] Falcão, M. I., Malonek, H. R.: Generalized exponentials through Appell sets in $\mathbb{R}^{n+1}$ and Bessel functions. Proc. of ICNAAM 2007 (Simos, T. E., Psihoyios, G., Tsitouras, Ch., eds.), vol. 936, 2007, AIP Conference Proceedings, pp. 750–753.

[17] Gilbert, J. E., Murray, M. A. M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, 1991. | MR | Zbl

[18] Gürlebeck, K., Morais, J.: On the development of Bohr’s phenomenon in the context of quaternionic analysis and related problems. arXiv:1004.1188v1 [math.CV], 2010.

[19] Gürlebeck, K., Morais, J.: Real–Part Estimates for Solutions of the Riesz System in $\mathbb{R}^3$. arXiv:1004.1191v1 [math.CV], 2010.

[20] Gürlebeck, K., Morais, J.: On monogenic primitives of Fueter polynomials. Proc. of ICNAAM 2006 (Simos, T. E., Psihoyios, G., Tsitouras, Ch., eds.), Wiley–VCH, Weinheim, 2006, pp. 600–605.

[21] Gürlebeck, K., Morais, J.: On the calculation of monogenic primitives. Adv. Appl. Clifford Algebras 17 (3) (2007), 481–496. | DOI | MR | Zbl

[22] Gürlebeck, K., Morais, J.: Bohr type theorem for monogenic power series. Comput. Methods Funct. Theory 9 (2) (2009), 633–651. | DOI | MR

[23] Sommen, F.: Spingroups and spherical means III. Rend. Circ. Mat. Palermo (2) Suppl. 1 (1989), 295–323. | MR | Zbl

[24] Sudbery, A.: Quaternionic analysis. Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–225. | DOI | MR | Zbl

[25] Van Lancker, P., : Spherical monogenics: An algebraic approach. Adv. Appl. Clifford Algebra 19 (2009), 467–496. | DOI | MR | Zbl

[26] Zeitlinger, P.: Beiträge zur Clifford Analysis und deren Modifikation. Ph.D. thesis, University Erlangen, 2005.