Duality for Multimodules

TL;DR

This paper introduces duals for multimodules and explores their properties, including semi-adjunctions, involutions, tensor products, and differential operators, proposing a new algebraic framework involving colored properads.

Contribution

It presents the first systematic development of duals for multimodules and their algebraic structures, including semi-adjunctions and properads, expanding the theoretical foundation.

Findings

Defined duals for multimodules and established semi-adjunctions.

Developed notions of generalized morphisms, involutions, and tensor products.

Constructed an involutive colored properad of multimodules.

Abstract

Suitable duals of multimodules are introduced and used to provide transposition contravariant right semi-adjunctions (and dualitites under reflexivity). Several additional notions on multimodules are discussed: generalized morphisms and involutions; tensor products and contractions; inner products; first-order differential operators. A construction of an (involutive) colored properad of multimodules is suggested.