MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

4 added 4 characters in body

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

$$U(\mathfrak{g},e)=(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}$$

Where

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 $\mathfrak{g}$ \mathfrak{g}$. Is this the same as $$U(\mathfrak{g})^{\mathfrak{m}}/(U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}$$$U(\mathfrak{g})^{\mathfrak{m}}/(U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}?$$Of course one can reformulate this question and ask if the following cohomology group vanishes:$$H^1(\mathfrak{m},U(\mathfrak{g})\mathfrak{m}_{\chi})=0$$H^1(\mathfrak{m},U(\mathfrak{g})\mathfrak{m}_{\chi})=0?$$ Maybe this follows from some Lynch style vanishing, but I am not very familiar with these theorems.

3 edited tags
2 edited body

In a well known construction of finite W-algebras, one first constructs a certain nilpotent subalgebra $\mathfrak{m}$ along with a character $\chi:\mathcal{m}\rightarrow \chi:\mathfrak{m}\rightarrow \mathbb{C}$. Then one defines

$$U(\mathfrak{g},e)=(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}$$

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

$$U(\mathfrak{g})^{\mathfrak{m}}/(U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}$$

Of course one can reformulate this question and ask if the following cohomology group vanishes:
$$H^1(\mathfrak{m},U(\mathfrak{g})\mathfrak{m}_{\chi}))=0$$ $H^1(\mathfrak{m},U(\mathfrak{g})\mathfrak{m}_{\chi})=0$\$ Maybe this follows from some Lynch style vanishing, but I am not very familiar with these theorems.

1