A question on the construction of finite W-algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T04:49:05Z http://mathoverflow.net/feeds/question/39593 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39593/a-question-on-the-construction-of-finite-w-algebras A question on the construction of finite W-algebras Jan Weidner 2010-09-22T09:58:58Z 2010-12-08T14:24:44Z <p>In a well known construction of finite W-algebras, one first constructs a certain nilpotent subalgebra $\mathfrak{m}$ along with a character $\chi:\mathfrak{m}\rightarrow \mathbb{C}$. Then one defines</p> <p>$$U(\mathfrak{g},e)=(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}$$</p> <p>where $\mathfrak{m}_\chi$ is the set of all $m-\chi(m)$ and $\mathfrak{m}$ acts on $U(\mathfrak{g})$ by derivations, extending the adjoint action on $\mathfrak{g}$. Is this the same as </p> <p>$$U(\mathfrak{g})^{\mathfrak{m}}/(U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}?$$</p> <p>Of course one can reformulate this question and ask if the following cohomology group vanishes:<br> $$H^1(\mathfrak{m},U(\mathfrak{g})\mathfrak{m}_{\chi})=0?$$ Maybe this follows from some Lynch style vanishing, but I am not very familiar with these theorems.</p> http://mathoverflow.net/questions/39593/a-question-on-the-construction-of-finite-w-algebras/39651#39651 Answer by Ben Webster for A question on the construction of finite W-algebras Ben Webster 2010-09-22T20:27:31Z 2010-09-22T20:27:31Z <p>Look at Propositions 5.1 and 5.2 of Gan and Ginzburg's paper <a href="http://arxiv.org/abs/math/0105225" rel="nofollow">Quantization of Slodowy slices</a>. The "reason" behind the vanishing is its identification with algebraic deRham cohomology of an affine space.</p>