The orbit method for the Virasoro algebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912337803345920 |
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| author | Pham, Tuan Anh |
| author_facet | Pham, Tuan Anh |
| contents | Let $W = \mathbb{C}[t, t^{-1}]\partial_t$ be the Witt algebra of algebraic vector fields on $\mathbb{C}^\times$ and let $V\!ir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In 2023, Petukhov and Sierra showed that Poisson primitive ideals of $\mathrm{S}(W)$ and $\mathrm{S}(V\!ir)$ can be constructed from elements of $W^*$ and $V\!ir^*$ of a particular form, called local functions. In this paper, we show how to use a local function on $W$ or $V\!ir$ to construct a representation of the Lie algebra. We further show that the annihilators of these representations are new completely prime primitive ideals of $\mathrm{U}(W)$ and $\mathrm{U}(V\!ir)$. We use this to define a Dixmier map from the Poisson primitive spectrum of $\mathrm{S}(V\!ir)$, respectively $\mathrm{S}(W)$, to the primitive spectrum of $\mathrm{U}(V\!ir)$, respectively $\mathrm{U}(W)$, successfully extending the orbit method from finite-dimensional solvable Lie algebras to our countable-dimensional setting.
Our method involves new ring homomorphisms from $\mathrm{U}(W)$ to the tensor product of a localized Weyl algebra and the enveloping algebra of a finite-dimensional solvable subquotient of $W$. We further show that the kernels of these homomorphisms are intersections of the primitive ideals constructed from natural subsets of $W^*$. As a corollary, we disprove the conjecture that any primitive ideal of $\mathrm{U}(W)$ is the kernel of some map from $\mathrm{U}(W)$ to the first Weyl algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The orbit method for the Virasoro algebra Pham, Tuan Anh Rings and Algebras Representation Theory 17B68, 16D60 (Primary), 16S30, 17B10 (Secondary) Let $W = \mathbb{C}[t, t^{-1}]\partial_t$ be the Witt algebra of algebraic vector fields on $\mathbb{C}^\times$ and let $V\!ir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In 2023, Petukhov and Sierra showed that Poisson primitive ideals of $\mathrm{S}(W)$ and $\mathrm{S}(V\!ir)$ can be constructed from elements of $W^*$ and $V\!ir^*$ of a particular form, called local functions. In this paper, we show how to use a local function on $W$ or $V\!ir$ to construct a representation of the Lie algebra. We further show that the annihilators of these representations are new completely prime primitive ideals of $\mathrm{U}(W)$ and $\mathrm{U}(V\!ir)$. We use this to define a Dixmier map from the Poisson primitive spectrum of $\mathrm{S}(V\!ir)$, respectively $\mathrm{S}(W)$, to the primitive spectrum of $\mathrm{U}(V\!ir)$, respectively $\mathrm{U}(W)$, successfully extending the orbit method from finite-dimensional solvable Lie algebras to our countable-dimensional setting. Our method involves new ring homomorphisms from $\mathrm{U}(W)$ to the tensor product of a localized Weyl algebra and the enveloping algebra of a finite-dimensional solvable subquotient of $W$. We further show that the kernels of these homomorphisms are intersections of the primitive ideals constructed from natural subsets of $W^*$. As a corollary, we disprove the conjecture that any primitive ideal of $\mathrm{U}(W)$ is the kernel of some map from $\mathrm{U}(W)$ to the first Weyl algebra. |
| title | The orbit method for the Virasoro algebra |
| topic | Rings and Algebras Representation Theory 17B68, 16D60 (Primary), 16S30, 17B10 (Secondary) |
| url | https://arxiv.org/abs/2504.14670 |