An infinite differential ideal
The author presents an infinite differential ideal which can be applied to structural problems in relativity and twistor theory. Suppose \(\mathbb A\) to be a Grassmann \(\mathbb Z\)-algebra with twist map \(\tau_{m, n}=(-1)^{mn}\), \((m,n)\in\mathbb Z^2,\) and polynomial over the set \(S\) in the grade zero component of \(\mathbb A\) (as a Grassmann algebra) with compatible gradings. Then \(\mathbb A\) with the induced \(\mathbb Z^2\) grading is called a \(\mathbb GS\)-algebra. Let \(\mathbb A^n\) be the grade \(n\)-component of \(\mathbb A\) as a Grassmann algebra, \(\mathbb A_m\) be the grade \(m\)-component of \(\mathbb A\) as an algebra polynomial over \(S\) and \(\mathbb A_m^n=\mathbb A^n\cap\mathbb A_m.\) In the appropriate way, the spaces of derivations \(\mathbb D_\mathbb Z(\mathbb A)\) and \(\mathbb D_{\mathbb Z^2}(\mathbb A)\) are introduced. For a \(\mathbb GS\)-pair \((\mathbb A',\mathbb A),\) we denote by \(\mathbb D(\mathbb A'|\mathbb A)\) the subalgebra of \(\mathbb D(\mathbb A')\) consisting of all derivations in \(\mathbb D(\mathbb A')\) that annihilate \(\mathbb A.\) For any \((m,n)\in \mathbb Z^2,\) we denote by \(\mathbb D_m(\mathbb A'|\mathbb A)\), \(\mathbb D^n(\mathbb A'|\mathbb A)\) and \(\mathbb D_m^n(\mathbb A'|\mathbb A)\) the respective intersections of \(\mathbb D_m(\mathbb A')\), \(\mathbb D^n(\mathbb A')\) and \(\mathbb D_m^n(\mathbb A')\) with \(\mathbb D(\mathbb A'|\mathbb A).\) The main result of the paper is the deduction of the formulas which consistently give recursive definitions for a sequence \((x),\) a pair of sequences \((x,r),\) and a triple of sequences \((x,r,\lambda),\) where \(x\equiv\{x_n\in\mathbb A^0_n\}\), \(r\equiv\{r_n\in \mathbb D^0_n(\mathbb A'|\mathbb A)\}\) and \(\lambda\equiv\{\lambda_n\in\mathbb A^{-1}_n\}\). The bulk of the text involves giving the requisite definitions and setting the stage for these formulas.
scientific article; zbMATH DE number 1434522