Hyperholomorphism

Hyperholomorphism is a possible generalization of the concept of holomorphism from complex numbers to more general Clifford algebras , based on the concept of the Cauchy-Riemann differential equations .

definition

Let it denote a Clifford algebra with the algebraic base . The system is a basis in ${\displaystyle Cl_{p,q}}$ ${\displaystyle 1;\,e_{1},\dots ,e_{n};\,e_{1}e_{2},\dots e_{n-1}e_{n};\dots ;\,e_{1}e_{2}\dots e_{n}}$ ${\displaystyle \{e_{1},\dots ,e_{n}\}}$

${\displaystyle \mathbb {R} ^{p,q}}$with . ${\displaystyle e_{i}^{2}=-1,\,i=1,\dots ,q;\,e_{i}^{2}=1,\;i=p+1,\dots ,p+q=n}$

The operator

${\displaystyle D=\sum _{i=1}^{n}e_{i}\partial _{i}}$,

that acts in space , where is a region, is called a Dirac operator . The operator is called the Cauchy-Riemann type operator. The symbols indicate the partial derivatives. A function is now called hyperholomorphic if it satisfies one of the following equations: ${\displaystyle C^{1}(G,Cl_{p,q})}$ ${\displaystyle G\subset \mathbb {R} ^{n}}$ ${\displaystyle \partial =\partial _{0}+D}$ ${\displaystyle \partial _{i};i=0,1,...,n}$ ${\displaystyle f}$

${\displaystyle Df=0,(fD)=0,\partial f=0,(f\partial )=0}$.

Since Clifford algebras are generally non-commutative algebras , the application from the left and from the right provides different classes of functions. Often the more precise terms - (left) -holomorphic or - (right) -holomorphic are used. If it is clear from which side the use of the operators is to be understood, the additions to the left and right are also omitted. Instead of the term holomorphic , monogenic or Cl-analytical is sometimes used. Important Clifford algebras are the algebra of complex numbers (commutative), the algebra of real quaternions, and Pauli algebra . ${\displaystyle Cl_{p,q}}$ ${\displaystyle Cl_{p,q}}$ ${\displaystyle Cl_{0,1}}$ ${\displaystyle Cl_{0,2}}$ ${\displaystyle Cl_{3,0}}$

literature

• Brackx, F .; Delanghe; R., Sommen, F. (1982) Clifford analysis , Pitman Res. Notes Math. Ser. 76
• Gürlebeck K. and Sprößig W. (1997) Quaternionic and Clifford Calculus for Physicists and Engineers , Series: Mathematical Methods in Practice, Wiley \ & Sons, Chichester
• Kravchenko VV; Shapiro, MV (1996) Integral representations for spatial models of mathematical physics , Pitman Res. Notes Math Ser. 351
• K. Gürlebeck, K. Habetha , W. Sprössig (2006) Function theory in the plane and in space , Birkhäuser