Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.00470 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908493255016448 |
|---|---|
| author | Berenstein, Arkady Grigoriev, Dima |
| author_facet | Berenstein, Arkady Grigoriev, Dima |
| contents | 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-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 $A=U_q(w)$ results in mysterious "symplectomorphisms" of involved skew symmetric forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00470 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Valuations, bijections, and bases Berenstein, Arkady Grigoriev, Dima Rings and Algebras Representation Theory 16W60, 16Z10, 13F30 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-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 $A=U_q(w)$ results in mysterious "symplectomorphisms" of involved skew symmetric forms. |
| title | Valuations, bijections, and bases |
| topic | Rings and Algebras Representation Theory 16W60, 16Z10, 13F30 |
| url | https://arxiv.org/abs/2405.00470 |