Abstract
The tangential k-Cauchy–Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables, respectively. In this paper, we introduce a Lie group that the Heisenberg group can be imbedded into and call it generalized complex Heisenberg. We investigate quaternionic analysis on the generalized complex Heisenberg. We also give the Penrose integral formula for k-CF functions and construct the tangential k-Cauchy-Fueter complex.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R J Baston, M G Eastwood. The Penrose transform. Its interaction with representation theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989.
R J Baston. Quaternionic complexes, J Geom Phys, 1992, 8(1–4): 29–52.
J Bureš, A Damiano, I Sabadini. Explicit resolutions for the complex of several Fueter operators, J Geom Phys, 2007, 57(3): 765–775.
J Bureš, V Souček. Complexes of invariant operators in several quaternionic variables, Complex Var Elliptic Equ, 2006, 51(5–6): 463–485.
F Colombo, V Souček, D C Struppa. Invariant resolutions for several Fueter operators, J Geom Phys, 2006, 56(7): 1175–1191.
M Dunajski. Solitons, instantons, and twistors, Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2010.
M G Eastwood, R Penrose, R O Wells Jr. Cohomology and massless fields, Comm Math Phys, 1981, 78: 305–351.
L J Mason, N M J Woodhouse. Integrability, Self-Duality, and Twistor Theory, London Mathematical Society Monographs New Series, The Clarendon Press, Oxford University Press, New York, 1996.
G Z Ren, Y Shi, W Wang. The tangential k-Cauchy–Fueter operator and k–CF functions over the Heisenberg group, Adv Appl Clifford Algebr, 2020, 30: 1–20.
G Z Ren, W Wang. Anti-self-dual connections over the 5D Heisenberg group and the twistor method, J Geom Phys, 2023, 183(53): 104699.
Y Shi, W Wang. The tangential k-Cauchy–Fueter complexes and Hartogs’ phenomenon over the right quaternionic Heisenberg group, Ann Mat Pura Appl, 2020, 199(4): 651–680.
H Y Wang, N X Sun, X L Bian. A version of Schwarz lemma associated to the k-Cauchy–Fueter operator, Adv Appl Clifford Algebr, 2021, 31(4): 64.
W Wang. The k-Cauchy–Fueter complexes, Penrose transformation and Hartogs’ phenomenon for quaternionic k-regular functions, J Geom Phys, 2010, 60: 513–530.
R S Ward, R O Wells Jr. Twistor geometry and field theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1990.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest The authors declare no conflict of interest.
Additional information
Supported by National Nature Science Foundation in China(12101564, 11971425, 11801508), Nature Science Foundation of Zhejiang province(LY22A010013), and Domestic Visiting Scholar Teacher Professional Development Project(FX2021042).
Rights and permissions
About this article
Cite this article
Ren, Gz., Shi, Y. & Kang, Qq. The tangential k-Cauchy–Fueter type operator and Penrose type integral formula on the generalized complex Heisenberg group. Appl. Math. J. Chin. Univ. 39, 181–190 (2024). https://doi.org/10.1007/s11766-024-4942-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-024-4942-6
Keywords
- the generalized complex Heisenberg group
- the tangential k-Cauchy–Fueter type operator
- Penrose-type integral formula