Dual coalgebras of twisted tensor products
Manuel L. Reyes
TL;DR
The paper develops a topological-duality framework to compute finite dual coalgebras of algebras formed by twisted tensor products. By treating the finite dual as a continuous dual on linearly topologized spaces with the cofinite topology, the authors prove that a continuous twisting map $\tau$ yields a cotwisting $\tau^\circ$ and $(A\otimes_\tau B)^\circ \cong A^\circ \otimes^{\tau^\circ} B^\circ$, with a practical criterion (centralize) ensuring continuity. This leads to concrete duality results for Ore extensions, smash products, and bitwisted tensor product bialgebras, and it is illuminated through detailed examples such as quantum planes, quantized Weyl algebras, and the Jordan plane, especially at roots of unity or in positive characteristic. The work provides a spectral perspective on noncommutative algebras built from subalgebras, enabling explicit descriptions of their finite duals in terms of cotwisted tensor products of distributions, with implications for understanding noncommutative geometries and quantum groups.
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.