Category $\mathcal O$ for the Virasoro algebra: Cohomology and Koszulity

箭图模空间的Virasoro约束与表示

Virasoro constraints and representations for quiver moduli spaces

本文利用Joyce顶点代数研究带关系的箭图表示模空间的Virasoro约束，在光滑假设下构造了Virasoro代数的表示，并通过例外簇将其应用于del Pezzo曲面层模空间及Grassmannian。

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group $SL_q(2)$ as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges $c+\bar{c}=26$. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the…

Computation of cohomology of Lie conformal and Poisson vertex algebras

We develop methods for computation of Poisson vertex algebra cohomology. This cohomology is computed for the free bosonic and fermionic Poisson vertex (super)algebras, as well as for the universal affine and Virasoro Poisson vertex algebras. We establish finite dimensionality of this cohomology for conformal Poisson vertex (super)algebras that are finitely and freely generated by elements of positive conformal weight.

Adjoint cohomology of graded Lie algebras of maximal class

We compute explicitly the adjoint cohomology of two N-graded Lie algebras of maximal class (infinite dimensional filiform Lie algebras) m_0 and m_2. It is known that up to an isomorphism there are only three N-graded Lie algebras of the maximal class. The third algebra from this list is the "positive" part L_1 of the Witt (or Virasoro) algebra and its adjoint cohomology was computed earlier by Feigin and Fukhs. We show that the total space H*(m_j,m_j) is "almost" isomorphic to the completed tensor product of the algebra m_j by scalar cohomology space H^*(m_j), j=0,2.

层模空间上的 Virasoro 约束与顶点代数

Virasoro constraints on moduli of sheaves and vertex algebras

本文将层理论 Virasoro 约束重新表述为 Joyce 顶点代数中的初态，证明其在壁穿越下保持不变，并借此证明了任意曲线上无挠层模空间及特定曲面上的 Virasoro 约束猜想。

Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra

For each nonzero $h\in \mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy = h$. This gives a parametric family of subalgebras of the Weyl algebra $\mathsf{A}_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description the Hochschild cohomology $\mathsf{HH}^\bullet(\mathsf{A}_h)$ over a field of arbitrary characteristic. In case $\mathbb{F}$ has positive characteristic, the center of $\mathsf{A}_h$ is nontrivial and …

Root components for tensor product of affine Kac-Moody Lie algebra modules

Let g be an affine Kac-Moody Lie algebra and let $λ, μ$ be two dominant integral weights for g. We prove that under some mild restriction, for any positive root $β$, $V(λ)\otimes V(μ)$ contains $V(λ+μ-β)$ as a component, where $V(λ)$ denotes the integrable highest weight (irreducible) g-module with highest weight $λ$. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product $V(…

Filtered Lie conformal algebras whose associated graded algebras are isomorphic to that of general conformal algebra $gc_1$

Let $G$ be a filtered Lie conformal algebra whose associated graded conformal algebra is isomorphic to that of general conformal algebra $gc_1$. In this paper, we prove that $G\cong gc_1$ or ${\rm gr\,}gc_1$ (the associated graded conformal algebra of $gc_1$), by making use of some results on the second cohomology groups of the conformal algebra $\fg$ with coefficients in its module $M_{b,0}$ of rank 1, where $\fg=\Vir\ltimes M_{a,0}$ is the semi-direct sum of the Virasoro conformal algebra $\Vir$ with its module $M_{a,0}$. Furthermore, we prove that ${\rm gr\,}gc_1$ does not have a nontrivial…

关于亚纯开弦顶点代数左模的扩张，I

On the extensions of the left modules for a meromorphic open-string vertex algebra, I

本文研究亚纯开弦顶点代数左模的扩张，证明其与第一上同调类一一对应，并给出Ext^1维数的上界及一个Virasoro VOA的例子。