Quasi-Poisson Modules and Harish-Chandra AD-Modules
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910270047125504 |
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| author | Yousofzadeh, Malihe |
| author_facet | Yousofzadeh, Malihe |
| contents | We introduce the notion of quasi-Poisson modules over Lie-Rinehart pairs and prove that for the Lie-Rinehart pair $(\dot A,\dot\fk)$ in which $\dot A=\bbbc[t_1^{\pm1},\ldots,t_m^{\pm1}]\ot\Lam_n$ and $\dot\fk={\rm Der}(\dot A)$, there is a one-to-one correspondence between simple cuspidal quasi-Poisson modules over $(\dot A,\dot\fk)$ and simple cuspidal Harish-Chndra $A\fk$-modules for $A:=\bbbc[t_0^{\pm1}]\ot \dot A$ and $\fk:={\rm Der}(A).$ We also classify simple cuspidal quasi-Poisson modules over the Lie-Rinehart pair $(\dot A,\dot\fk)$ and show that each such module is a tensor module $\dot A\ot Ω$ for an admissible $\frak{gl}(m+1,n)$-module $Ω$ via a prescribed action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16950 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quasi-Poisson Modules and Harish-Chandra AD-Modules Yousofzadeh, Malihe Representation Theory We introduce the notion of quasi-Poisson modules over Lie-Rinehart pairs and prove that for the Lie-Rinehart pair $(\dot A,\dot\fk)$ in which $\dot A=\bbbc[t_1^{\pm1},\ldots,t_m^{\pm1}]\ot\Lam_n$ and $\dot\fk={\rm Der}(\dot A)$, there is a one-to-one correspondence between simple cuspidal quasi-Poisson modules over $(\dot A,\dot\fk)$ and simple cuspidal Harish-Chndra $A\fk$-modules for $A:=\bbbc[t_0^{\pm1}]\ot \dot A$ and $\fk:={\rm Der}(A).$ We also classify simple cuspidal quasi-Poisson modules over the Lie-Rinehart pair $(\dot A,\dot\fk)$ and show that each such module is a tensor module $\dot A\ot Ω$ for an admissible $\frak{gl}(m+1,n)$-module $Ω$ via a prescribed action. |
| title | Quasi-Poisson Modules and Harish-Chandra AD-Modules |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2605.16950 |