Valuations, bijections, and bases

TL;DR

This paper develops a theory of valuations for various algebras, including noncommutative ones, establishing bijections between valuation semigroups that generalize classical correspondences and reveal deep structural symmetries.

Contribution

It introduces a new framework for valuations in noncommutative algebras and constructs canonical bijections between valuation semigroups, extending classical Jordan-Hölder theory.

Findings

Constructed valuations for algebras with zero divisors.

Established bijections between valuation semigroups as analogs of Jordan-Hölder correspondences.

Demonstrated applications to quantum Schubert cells and related symplectomorphisms.

Abstract

The aim of this paper is to build a theory of commutative and noncommutative {\it injective} valuations of various algebras (including algebras with zero divisors). The targets of our valuations are (well-)ordered commutative and noncommutative (partial and entire) semigroups including any sub-semigroups of the free monoid $F_n$ on $n$ generators and various quotients. When the range of a valuation of an algebra $A$ is a finitely generated (partial) semigroup, we construct a generalization of the standard monomial bases in $A$, which seems to be new in noncommutative case. Quite remarkably, for any pair of well-ordered valuations one has a canonical bijection between the valuation semigroups, which serves as an analog of the celebrated Jordan-H\"older correspondences and these bijections are ``almost" homomorphisms of the involved semigroups. A spectacular demonstration of this remarkable property of JH-bijections for quantum Schubert cells $A=U_q(w)$ results in mysterious "symplectomorphisms" of involved skew symmetric forms.