Dual coalgebras of twisted tensor products

TL;DR

This paper explores when the finite dual coalgebra of a twisted tensor product of algebras can be expressed as a cotwisted tensor product of their finite dual coalgebras, using topological duality and monoidal functors.

Contribution

It provides a new interpretation of finite duals as topological duals and establishes conditions for their cotwisted tensor product structure, applicable to various algebraic constructions.

Findings

Finite dual coalgebra of twisted tensor product can be cotwisted under certain conditions.

Continuous dual is a strong monoidal functor on linearly topologized vector spaces.

Applicable to Ore extensions, smash products, and crossed product bialgebras.

Abstract

We investigate cases where the finite dual coalgebra of a twisted tensor product of two algebras is a cotwisted tensor product of their respective finite dual coalgebras. This is achieved by interpreting the finite dual as a topological dual; in order to prove this, we show that the continuous dual is a strong monoidal functor on linearly topologized vector spaces whose open subspaces have finite codimension. We describe a sufficient condition for the result on finite dual coalgebras to be applied, and we this condition to particular constructions including Ore extensions, smash product algebras, and crossed product bialgebras.