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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2503.12000 |
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| _version_ | 1866929761500004352 |
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| author | Pan, Zhennan Han, Gang |
| author_facet | Pan, Zhennan Han, Gang |
| contents | Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. Let $P$ be a non-commutative Poisson algebra over some algebraically closed field of characteristic zero. For any $z\in P$, there exist four subalgebras of $P$ associated with the inner derivation $ad_z$ on $P$. Based on the relationships between these four subalgebras, elements of $P$ can be divided into eight types. We will mainly focus on two types of non-commutative Poisson algebras: the usual Poisson algebras and the associative algebras with the commutator as the Poisson bracket. The following problems are studied for such non-commutative Poisson algebras: how the type of an element changes under homomorphisms between non-commutative Poisson algebras, how the type of an element changes after localization, and what the type of the elements of the form $z_1 \otimes z_2$ and $z_1 \otimes 1 + 1 \otimes z_2$ is in the tensor product of non-commutative Poisson algebras $P_1\otimes P_2$. As an application of above results, one knows that Dixmier Conjecture for $A_1$ holds under certain conditions. Some properties of the Weyl algebras are also obtained, such as the commutativity of certain subalgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12000 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Types of elements in non-commutative Poisson algebras and Dixmier Conjecture Pan, Zhennan Han, Gang Rings and Algebras Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. Let $P$ be a non-commutative Poisson algebra over some algebraically closed field of characteristic zero. For any $z\in P$, there exist four subalgebras of $P$ associated with the inner derivation $ad_z$ on $P$. Based on the relationships between these four subalgebras, elements of $P$ can be divided into eight types. We will mainly focus on two types of non-commutative Poisson algebras: the usual Poisson algebras and the associative algebras with the commutator as the Poisson bracket. The following problems are studied for such non-commutative Poisson algebras: how the type of an element changes under homomorphisms between non-commutative Poisson algebras, how the type of an element changes after localization, and what the type of the elements of the form $z_1 \otimes z_2$ and $z_1 \otimes 1 + 1 \otimes z_2$ is in the tensor product of non-commutative Poisson algebras $P_1\otimes P_2$. As an application of above results, one knows that Dixmier Conjecture for $A_1$ holds under certain conditions. Some properties of the Weyl algebras are also obtained, such as the commutativity of certain subalgebras. |
| title | Types of elements in non-commutative Poisson algebras and Dixmier Conjecture |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2503.12000 |