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Derived subalgebras of centralisers and finite -algebras
| Content Provider | Scilit |
|---|---|
| Author | Premet, Alexander Topley, Lewis |
| Copyright Year | 2014 |
| Description | Journal: Compositio Mathematica Let $\mathfrak{g}=\mbox{Lie}(G)$ be the Lie algebra of a simple algebraic group $G$ over an algebraically closed field of characteristic $0$ . Let $e$ be a nilpotent element of $\mathfrak{g}$ and let $\mathfrak{g}_e=\mbox{Lie}(G_e)$ where $G_e$ stands for the stabiliser of $e$ in $G$ . For $\mathfrak{g}$ classical, we give an explicit combinatorial formula for the codimension of $[\mathfrak{g}_e,\mathfrak{g}_e]$ in $\mathfrak{g}_e$ and use it to determine those $e\in \mathfrak{g}$ for which the largest commutative quotient $U(\mathfrak{g},e)^{\mbox{ab}}$ of the finite $W$ -algebra $U(\mathfrak{g},e)$ is isomorphic to a polynomial algebra. It turns out that this happens if and only if $e$ lies in a unique sheet of $\mathfrak{g}$ . The nilpotent elements with this property are callednon-singularin the paper. Confirming a recent conjecture of Izosimov, we prove that a nilpotent element $e\in \mathfrak{g}$ is non-singular if and only if the maximal dimension of the geometric quotients $\mathcal{S}/G$ , where $\mathcal{S}$ is a sheet of $\mathfrak{g}$ containing $e$ , coincides with the codimension of $[\mathfrak{g}_e,\mathfrak{g}_e]$ in $\mathfrak{g}_e$ and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element $e$ in a classical Lie algebra $\mathfrak{g}$ the closed subset of Specm $U(\mathfrak{g},e)^{\mbox{ab}}$ consisting of all points fixed by the natural action of the component group of $G_e$ is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given. |
| Ending Page | 1548 |
| Starting Page | 1485 |
| ISSN | 00221295 |
| e-ISSN | 15705846 |
| DOI | 10.1112/s0010437x13007823 |
| Journal | Compositio Mathematica |
| Issue Number | 9 |
| Volume Number | 150 |
| Language | English |
| Publisher | Wiley-Blackwell |
| Publisher Date | 2014-07-09 |
| Access Restriction | Open |
| Subject Keyword | Journal: Compositio Mathematica Mathematical Physics |
| Content Type | Text |
| Resource Type | Article |
| Subject | Algebra and Number Theory |
