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Relative Dixmier property for Poisson algebras

Hongdi Huang, Zahra Nazemian, Xin Tang, Xingting Wang, Yanhua Wang, James J. Zhang

TL;DR

The paper introduces the relative Dixmier property for Poisson algebras and develops a robust framework—via $u$-invariants, Poisson valuations, and complementary-space theory—to study when Poisson algebras satisfy the Dixmier property or its relative versions. It proves that uniparameter simple Poisson tori have the Dixmier property (and are $R$-Dixmier for connected graded Poisson domains $R$), and extends these ideas to tensor products of Poisson algebras built from i.s. potentials, establishing Dixmier behavior for distinct Adams degrees. The authors also connect these structural insights to relative cancellation and the non-existence of Hopf coactions, and they develop integral-closure–based tools (closures and complementary spaces) to analyze Poisson automorphism groups, culminating in methods to construct Poisson algebras with trivial automorphism groups. Overall, the work broadens the class of Poisson algebras known to satisfy the Dixmier property and provides a versatile toolkit for understanding Poisson automorphisms and endomorphisms through relative and graded perspectives.

Abstract

Dixmier property concerns the bijectivity of endomorphisms for algebras. We introduce a relative Dixmier property, which is a generalization of the Dixmier property. This new concept has applications in proving that several classes of Poisson algebras possess the Dixmier property, as well as in other topics such as the cancellation problem and the non-existence of Hopf coactions.

Relative Dixmier property for Poisson algebras

TL;DR

The paper introduces the relative Dixmier property for Poisson algebras and develops a robust framework—via -invariants, Poisson valuations, and complementary-space theory—to study when Poisson algebras satisfy the Dixmier property or its relative versions. It proves that uniparameter simple Poisson tori have the Dixmier property (and are -Dixmier for connected graded Poisson domains ), and extends these ideas to tensor products of Poisson algebras built from i.s. potentials, establishing Dixmier behavior for distinct Adams degrees. The authors also connect these structural insights to relative cancellation and the non-existence of Hopf coactions, and they develop integral-closure–based tools (closures and complementary spaces) to analyze Poisson automorphism groups, culminating in methods to construct Poisson algebras with trivial automorphism groups. Overall, the work broadens the class of Poisson algebras known to satisfy the Dixmier property and provides a versatile toolkit for understanding Poisson automorphisms and endomorphisms through relative and graded perspectives.

Abstract

Dixmier property concerns the bijectivity of endomorphisms for algebras. We introduce a relative Dixmier property, which is a generalization of the Dixmier property. This new concept has applications in proving that several classes of Poisson algebras possess the Dixmier property, as well as in other topics such as the cancellation problem and the non-existence of Hopf coactions.
Paper Structure (14 sections, 30 theorems, 104 equations)

This paper contains 14 sections, 30 theorems, 104 equations.

Table of Contents

  1. Some examples of Poisson algebras
  2. Preliminaries
  3. $u$-invariants
  4. Relative Dixmier property for uniparameter simple Poisson torus
  5. Poisson $u$-invariants
  6. Poisson valuations and relative Dixmier property
  7. Other applications: relative cancellation and non-existence of Hopf coactions
  8. Tensor products of Poisson algebras
  9. Integral closure and complementary space
  10. Closure of $A$
  11. Complementary spaces and complement modules
  12. Dixmier property vs Complementary spaces
  13. Field extension
  14. Poisson algebras with trivial automorphism groups

Key Result

Theorem 2

Let $\Omega_1,\cdots,\Omega_d$ be a set of i.s. potentials of distinct Adams degrees with $\deg \Omega_i\geq 5$ for all $i$. Let $\xi_1,\cdots,\xi_d$ be a set of scalars in $\Bbbk$. Let $P$ be the Poisson algebra $P_{\Omega_1-\xi_1}\otimes \cdots \otimes P_{\Omega_d-\xi_d}$ and let $Q$ be the fracti

Theorems & Definitions (78)

  • Example 1
  • Theorem 2
  • Definition 4
  • Example 5
  • Theorem 6
  • Corollary 7
  • Theorem 9
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • ...and 68 more