$\D$-locally nilpotent algebras, their ideal structure and simplicity criteria
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910482754961408 |
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| author | Bavula, V. V. |
| author_facet | Bavula, V. V. |
| contents | The class of $\D$-locally nilpotent algebras (introduced in the paper) is a wide generalization of the algebras of differential operators on commutative algebras. Examples includes all the rings $\CD (A)$ of differential operators on commutative algebras (in arbitrary characteristic), all subalgebras of $\CD (A)$ that contain the algebra $A$, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions $A[x_1, \ldots , x_n ; \d_1 , \ldots , \d_n]$, certain generalized Weyl algebras, and others.
In \cite{SimCrit-difop}, simplicity criteria are given for the algebras differential operators on commutative algebras (it was a long standing problem). The aim of the paper is to describe the ideal structure of $\D$-locally nilpotent algebras and as a corollary to give simplicity criteria for them (it is a generalization of the results of \cite{SimCrit-difop}). Examples are considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07569 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\D$-locally nilpotent algebras, their ideal structure and simplicity criteria Bavula, V. V. Rings and Algebras 13N10, 16S32, 16D30, 13N15, 14J17, 14B05, 16D25 The class of $\D$-locally nilpotent algebras (introduced in the paper) is a wide generalization of the algebras of differential operators on commutative algebras. Examples includes all the rings $\CD (A)$ of differential operators on commutative algebras (in arbitrary characteristic), all subalgebras of $\CD (A)$ that contain the algebra $A$, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions $A[x_1, \ldots , x_n ; \d_1 , \ldots , \d_n]$, certain generalized Weyl algebras, and others. In \cite{SimCrit-difop}, simplicity criteria are given for the algebras differential operators on commutative algebras (it was a long standing problem). The aim of the paper is to describe the ideal structure of $\D$-locally nilpotent algebras and as a corollary to give simplicity criteria for them (it is a generalization of the results of \cite{SimCrit-difop}). Examples are considered. |
| title | $\D$-locally nilpotent algebras, their ideal structure and simplicity criteria |
| topic | Rings and Algebras 13N10, 16S32, 16D30, 13N15, 14J17, 14B05, 16D25 |
| url | https://arxiv.org/abs/2406.07569 |