Polytopal linear algebra

TL;DR

This paper explores the algebraic structure of polytopal algebras, comparing them to vector spaces, and investigates their automorphisms, homomorphisms, and related K-theoretic properties, extending previous research in the field.

Contribution

It introduces the polytopal Picard group, examines automorphisms and homomorphisms of polytopal algebras, and proposes a conjecture on their structure, advancing the understanding of polyhedral algebraic objects.

Findings

The polytopal Picard group is trivial for fields.

Support for the conjecture on homomorphisms of polytopal algebras.

Further confirmation via homomorphisms on Veronese singularities.

Abstract

We investigate similarities between the category of vector spaces and that of polytopal algebras, containing the former as a full subcategory. In Section 2 we introduce the notion of a polytopal Picard group and show that it is trivial for fields. The coincidence of this group with the ordinary Picard group for general rings remains an open question. In Section 3 we survey some of the previous results on the automorphism groups and retractions. These results support a general conjecture proposed in Section 4 about the nature of arbitrary homomorphisms of polytopal algebras. Thereafter a further confirmation of this conjecture is presented by homomorphisms defined on Veronese singularities. This is a continuation of the project started in our papers "Polytopal linear groups" (J. Algebra 218 (1999), 715--737), "Polytopal linear retractions" preprint, math.AG/0001049) and "Polyhedral algebras, arrangements of toric varieties, and their groups" (preprint, http://www.mathematik.uni-osnabrueck.de/K-theory/0232/index.html). The higher $K$-theoretic aspects of polytopal linear objects will be treated in "Polyhedral $K$-theory" (in preparation).