Generalization of Tangential Complexes of Weight Three and Their Connections with Grassmannian Complex

Abstract

Following earlier work by Gangl, Cathelineaue, and others, Siddiqui defined the Siegel's cross-ratio identity and Goncharov's triple ratios over the truncated polynomial ring F[epsilon](gamma). They used these constructions to introduce both dialogarithmic and trilogarithmic tangential complexes of first order. They proposed various maps to relate first-order tangent complex to the Grassmannian complex. Later, we extended all the notions related to dialogarithmic complexes to a general order n. Now, this study is aimed to generalize all of the constructions associated to trilogarithmic tangential complexes to higher orders. We also propose motphisms between the tangent to Goncharov's complex and Grassmannian subcomplex for general order. Moreover, we connect both of these complexes by demonstrating that the resulting diagrams are commutative. In this generalization process, the classical Newton's identities are used. The results reveal that the tangent group TB3n (F) of a higher order and defining relations are feasible for all orders.