Technique of tensors with values in the Atiyah-Kähler algebra in the new representation of Dirac-Yang-Mills equations
In this paper the technique of so-called \(\Lambda\)-tensor calculus is presented, i.e., a tool which is used in the formulation of relativistic equations of field theory. This is a continuation of former work of the author. Using coordinate notation in a Spin 4-manifold \(X\), the author defines locally \(\Lambda\)-tensors and some natural operations related to them, that lead to the notion of \(\Lambda T\)-algebra. Indeed, the set \(\Lambda\) of 1-forms on \(X\) is endowed, besides the exterior multiplication, with a Clifford multiplication that naturally extends to tensors with values in \(\Lambda\). This is refered as the Atiyah-Kähler algebra \(\Lambda T\). On \(\Lambda T\) one can naturally define the differential, the codifferential and the Hodge star operator. For \(U\in \Lambda\), the first theorem gives the invariant subspaces of the operator \(F_U:\Lambda \rightarrow \Lambda\) defined by \[ F_U(V)=U^* V U. \] The second theorem gives for a pseudo-Riemannian space equipped with a compatible affine connection, the local expression of the affine curvature and some other tensors together with their transformation under a local Pin-transformation. Finally, using the \(\Lambda T\) language and the previous results, the author introduces briefly a Dirac-Yang-Mills equation in the context of 4-dimensional spinor manifold.
scientific article; zbMATH DE number 5136630