Skip to main content
MathOverflow

Return to Question

Notice removed Canonical answer required by CommunityBot
Bounty Ended with no winning answer by CommunityBot
edited tags
Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286
Notice added Canonical answer required by Pulcinella
Bounty Started worth 50 reputation by Pulcinella
added 1 character in body
Source Link
Pulcinella
Pulcinella
  • 5.7k
  • 1
  • 15
  • 55

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $\mathfrak{g}((t))$ using the invariant bilinear form $\kappa$ on $\mathfrak{g}$. Write $V_\kappa(\mathfrak{g})=\text{Ind}^{\widehat{\mathfrak{g}}_\kappa}_{\mathfrak{g}[[t]]\oplus \mathbf{C}c}\mathbf{C}$ for the Verma module of $\widehat{\mathfrak{g}}_\kappa$, where on $\mathbf{C}$, $\mathfrak{g}[[t]]$ acts by $0$ on $\mathbf{C}$ and the central element $c$ acts by $1$.

Is there a space (topological space, stack, etc.) $X$ whose cohomology is $H^*(X)=V_\kappa(\mathfrak{g})$?

For instance, if $Y$ is a smooth algebraic surface then $X=\coprod_{n\ge 0}\text{Hilb}^nY$ has cohomology of the form $V_\kappa(\mathfrak{h})$, where the vector space $\mathfrak{h}=H^*(Y)$ is viewed as a commutative Lie algebra with form $\kappa(\alpha,\beta)=\int_Y\alpha\wedge\beta$. I am curious if there is a construction for non-commutative Lie algebras.

If the natural thing is not to consider cohomology but some other functor from spaces to vector spaces, that is fine too (though ideally it would generalise the above example).

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $\mathfrak{g}((t))$ using the invariant bilinear form $\kappa$ on $\mathfrak{g}$. Write $V_\kappa(\mathfrak{g})=\text{Ind}^{\widehat{\mathfrak{g}}_\kappa}_{\mathfrak{g}[[t]]\oplus \mathbf{C}c}\mathbf{C}$ for the Verma module of $\widehat{\mathfrak{g}}_\kappa$, where $\mathfrak{g}[[t]]$ acts by $0$ on $\mathbf{C}$ and the central element $c$ acts by $1$.

Is there a space (topological space, stack, etc.) $X$ whose cohomology is $H^*(X)=V_\kappa(\mathfrak{g})$?

For instance, if $Y$ is a smooth algebraic surface then $X=\coprod_{n\ge 0}\text{Hilb}^nY$ has cohomology of the form $V_\kappa(\mathfrak{h})$, where the vector space $\mathfrak{h}=H^*(Y)$ is viewed as a commutative Lie algebra with form $\kappa(\alpha,\beta)=\int_Y\alpha\wedge\beta$. I am curious if there is a construction for non-commutative Lie algebras.

If the natural thing is not to consider cohomology but some other functor from spaces to vector spaces, that is fine too (though ideally it would generalise the above example).

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $\mathfrak{g}((t))$ using the invariant bilinear form $\kappa$ on $\mathfrak{g}$. Write $V_\kappa(\mathfrak{g})=\text{Ind}^{\widehat{\mathfrak{g}}_\kappa}_{\mathfrak{g}[[t]]\oplus \mathbf{C}c}\mathbf{C}$ for the Verma module of $\widehat{\mathfrak{g}}_\kappa$, where on $\mathbf{C}$, $\mathfrak{g}[[t]]$ acts by $0$ and the central element $c$ acts by $1$.

Is there a space (topological space, stack, etc.) $X$ whose cohomology is $H^*(X)=V_\kappa(\mathfrak{g})$?

For instance, if $Y$ is a smooth algebraic surface then $X=\coprod_{n\ge 0}\text{Hilb}^nY$ has cohomology of the form $V_\kappa(\mathfrak{h})$, where the vector space $\mathfrak{h}=H^*(Y)$ is viewed as a commutative Lie algebra with form $\kappa(\alpha,\beta)=\int_Y\alpha\wedge\beta$. I am curious if there is a construction for non-commutative Lie algebras.

If the natural thing is not to consider cohomology but some other functor from spaces to vector spaces, that is fine too (though ideally it would generalise the above example).

Source Link
Pulcinella
Pulcinella
  • 5.7k
  • 1
  • 15
  • 55

$X$ with $H^*(X)=$affine Verma module?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $\mathfrak{g}((t))$ using the invariant bilinear form $\kappa$ on $\mathfrak{g}$. Write $V_\kappa(\mathfrak{g})=\text{Ind}^{\widehat{\mathfrak{g}}_\kappa}_{\mathfrak{g}[[t]]\oplus \mathbf{C}c}\mathbf{C}$ for the Verma module of $\widehat{\mathfrak{g}}_\kappa$, where $\mathfrak{g}[[t]]$ acts by $0$ on $\mathbf{C}$ and the central element $c$ acts by $1$.

Is there a space (topological space, stack, etc.) $X$ whose cohomology is $H^*(X)=V_\kappa(\mathfrak{g})$?

For instance, if $Y$ is a smooth algebraic surface then $X=\coprod_{n\ge 0}\text{Hilb}^nY$ has cohomology of the form $V_\kappa(\mathfrak{h})$, where the vector space $\mathfrak{h}=H^*(Y)$ is viewed as a commutative Lie algebra with form $\kappa(\alpha,\beta)=\int_Y\alpha\wedge\beta$. I am curious if there is a construction for non-commutative Lie algebras.

If the natural thing is not to consider cohomology but some other functor from spaces to vector spaces, that is fine too (though ideally it would generalise the above example).