2024/05/01 by Arkady Berenstein, Berenstein, Arkady, Dima Grigoriev +1
Mathematics · #13F30 #16W60 #16Z10 #FOS: Mathematics #Mathematics and Applications #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2405.00470
openalex publication_date 2024/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Fn 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-Hölder 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=Uq(w) results in mysterious "symplectomorphisms" of involved skew symmetric forms.