The symplectic Fueter-Sce theorem. (English) Zbl 1455.53090
In general terms, the Fueter(-Sce-Qian) theorem is a result that allows to produce a hypercomplex regular function from a complex holomorphic one. The application of such a theorem to polynomials plays an important role in representation theory.
In this rather technical paper, the authors prove a symplectic version of the Fueter theorem for spinor-valued functions (see Section 2 for the formal definitions). Instead of the classic Dirac operator, they consider an operator \(D_s\), already defined in [H. De Bie et al., J. Geom. Phys. 75, 120–128 (2014; Zbl 1279.30051)]. This is a first-order differential operator, acting on functions \(f(x, y)\) on \(\mathbb{R}^{2n}\) taking values in the spinor representation for the symplectic Lie algebra \(\mathfrak{sp}(2n)\), which commutes with the regular action of this Lie algebra. As highlighted in the paper, a main differences to the classic case (centered around the Dirac operator and the spin group) will be the fact that one cannot start from holomorphic functions \(f(z)\); instead one has to consider other special functions as a starting point.
In this rather technical paper, the authors prove a symplectic version of the Fueter theorem for spinor-valued functions (see Section 2 for the formal definitions). Instead of the classic Dirac operator, they consider an operator \(D_s\), already defined in [H. De Bie et al., J. Geom. Phys. 75, 120–128 (2014; Zbl 1279.30051)]. This is a first-order differential operator, acting on functions \(f(x, y)\) on \(\mathbb{R}^{2n}\) taking values in the spinor representation for the symplectic Lie algebra \(\mathfrak{sp}(2n)\), which commutes with the regular action of this Lie algebra. As highlighted in the paper, a main differences to the classic case (centered around the Dirac operator and the spin group) will be the fact that one cannot start from holomorphic functions \(f(z)\); instead one has to consider other special functions as a starting point.
Reviewer: Amedeo Altavilla (Ancona)
MSC:
| 53D05 | Symplectic manifolds (general theory) |
| 53C27 | Spin and Spin\({}^c\) geometry |
| 15A67 | Applications of Clifford algebras to physics, etc. |
| 30G35 | Functions of hypercomplex variables and generalized variables |
Citations:
Zbl 1279.30051References:
| [1] | Brackx, F.; Delanghe, R.; Sommen, F., Clifford Analysis, Research Notes in Mathematics (1982), London: Pitman, London · Zbl 0529.30001 |
| [2] | Cahen, M.; Gutt, S.; Rawnsley, J., Symplectic Dirac operators and \(Mp^c\)-structures, Gen. Relativ. Gravit., 43, 3593-3617 (2011) · Zbl 1270.81117 · doi:10.1007/s10714-011-1239-x |
| [3] | Cahen, M., Gutt, S.: \(Spin^c, Mp^c\) and symplectic Dirac operators. In: Kielanowski, P., et al. (eds.) Geometric Methods in Physics, XXXI Workshop 2012, Trends in Mathematics. Birkhäuser, pp. 13-28 (2013) · Zbl 1403.53064 |
| [4] | Cahen, M.; Gutt, S.; La Fuente Gravy, L.; Rawnsley, J., On \(Mp^c\)-structures and symplectic Dirac operators, J. Geom. Phys., 86, 434-466 (2014) · Zbl 1325.53066 · doi:10.1016/j.geomphys.2014.09.006 |
| [5] | Colombo, F.; Sabadini, I.; Sommen, F., The inverse Fueter mapping theorem, Commun. Pure Appl. Anal., 10, 4, 1165-1181 (2011) · Zbl 1258.30022 · doi:10.3934/cpaa.2011.10.1165 |
| [6] | De Bie, H.; Holíková, M.; Somberg, P., Basic aspects of symplectic Clifford analysis for the symplectic Dirac operator, Adv. Appl. Clifford Algebras, 27, 2, 1103-1132 (2017) · Zbl 1367.53042 · doi:10.1007/s00006-016-0696-4 |
| [7] | De Bie, H.; Somberg, P.; Souček, V., The metaplectic Howe duality and polynomial solutions for the symplectic Dirac operator, J. Geom. Phys., 75, 120-128 (2014) · Zbl 1279.30051 · doi:10.1016/j.geomphys.2013.09.005 |
| [8] | Delanghe, R.; Sommen, F.; Souček, V., Clifford Analysis and Spinor Valued Functions (1992), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 0747.53001 |
| [9] | Eelbode, D.; Souček, V.; Van Lancker, P., Gegenbauer polynomials and the Fueter theorem, Complex Var. Ellipt. Equ., 59, 6, 826-840 (2014) · Zbl 1307.30091 · doi:10.1080/17476933.2013.787531 |
| [10] | Eelbode, D.; Souček, V.; Van Lancker, P., The Fueter theorem by representation theory, AIP Conf. Proc., 1479, 1, 340-343 (2012) · doi:10.1063/1.4756132 |
| [11] | Frink, O.; Krall, HL, A new class of orthogonal polynomials: the Bessel polynomials, Trans. Am. Math. Soc., 65, 1, 100-115 (1948) · Zbl 0031.29701 |
| [12] | Fueter, R., Die Funktionentheorie der Differentialgleichungen \(\Delta u = 0\) und \(\Delta \Delta u = 0\) mit vier reellen Variablen, Comment. Math. Helv., 7, 307-330 (1935) · Zbl 0012.01704 · doi:10.1007/BF01292723 |
| [13] | Gilbert, J.; Murray, MAM, Clifford Algebras and Dirac Operators in Harmonic Analysis (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0733.43001 |
| [14] | Haydys, A.: Generalized Seiberg-Witten equations and hyperKähler geometry, Ph.D. thesis, Georg-August University of Göttingen (2006) |
| [15] | Holíková, M.; Křižka, L.; Somberg, P., \( \widetilde{\rm SL}(3,\mathbb{R})\) and the symplectic Dirac operator, Arch. Math., 52, 2041-2052 (2016) |
| [16] | Hohloch, S.; Noetzel, G.; Salamon, DA, Hypercontact structures and Floer homology, Geom. Topol., 13, 5, 2543-2617 (2009) · Zbl 1220.53099 · doi:10.2140/gt.2009.13.2543 |
| [17] | Kou, KI; Qian, T.; Sommen, F., Generalizations of Fueter’s Theorem, Methods Appl. Anal., 9, 2, 273-290 (2002) · Zbl 1079.30066 |
| [18] | Nita, A.: Essential Self-Adjointness of the Symplectic Dirac Operators. Mathematics Graduate Theses and Dissertations, vol. 45 (2016) |
| [19] | Peña Peña, D.; Sommen, F., Fueter’s theorem: the saga continues, J. Math. Anal. Appl., 365, 1, 29-35 (2012) · Zbl 1183.30056 · doi:10.1016/j.jmaa.2009.10.006 |
| [20] | Qian, T., Generalization of Fueter’s result to \(\mathbb{R}^{m+1} \), Rend. Mat. Acc. Lincei, 9, 111-117 (1997) · Zbl 0909.30036 |
| [21] | Sce, M., Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica, 23, 220-225 (1957) · Zbl 0084.28302 |
| [22] | Sommen, F., On a generalization of Fueter’s theorem, Z. Anal. Anw., 19, 899-902 (2000) · Zbl 1030.30039 · doi:10.4171/ZAA/988 |
| [23] | Walpuski, T., A compactness theorem for Fueter sections, Comment. Math. Helv., 92, 4, 751-776 (2017) · Zbl 1383.58009 · doi:10.4171/CMH/423 |
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.
