Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum me...Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator D： C^∞（R^4n W4n） →4 C^∞（R^4n, W4n）, where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B, the bicomplex number B, and the Clifford algebra R0,4n. The operator D is a square root of the Laplacian in R^4n, introduced by the formula D = ∑j=0^3=0 Kjδzj with Kj being the basis of B, and δzj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B×R0,4n whose definition involves a delicate construction of the bicomplex Witt basis. The introduction of the operator D can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.展开更多
The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional o...The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional octonion analysis is initiated in this article,which extends the theory of several complex variables,such as the Bochner–Martinelli formula,the theory of non-homogeneous Cauchy–Riemann equations,and the Hartogs principle,to the non-commutative and non-associative realm.展开更多
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11371337） and the Research Fund for the Doctoral Program of Higher Education （China） （Grant No. 20123402110068）.
文摘Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator D： C^∞（R^4n W4n） →4 C^∞（R^4n, W4n）, where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B, the bicomplex number B, and the Clifford algebra R0,4n. The operator D is a square root of the Laplacian in R^4n, introduced by the formula D = ∑j=0^3=0 Kjδzj with Kj being the basis of B, and δzj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B×R0,4n whose definition involves a delicate construction of the bicomplex Witt basis. The introduction of the operator D can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.
基金This work was supported by the NNSF of China(11071230),RFDP(20123402110068).
文摘The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional octonion analysis is initiated in this article,which extends the theory of several complex variables,such as the Bochner–Martinelli formula,the theory of non-homogeneous Cauchy–Riemann equations,and the Hartogs principle,to the non-commutative and non-associative realm.
