Roman Bezrukavnikov
We show that semi-infinite cohomology of a finite dimensional graded algebra (satisfying some additional requirements) are a particular case of a general categorical construction. The motivating example is provided by small quantum groups at a root of unity.
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We study the multiplication and comultiplication in equivariant cohomology of Sato Grassmannian
We describe a general setting for the definition of semi-infinite cohomology of finite dimensional algebras, and provide its categorical interpretation. We apply this interpretation to compute semi-infinite cohomology of some modules over the small group at a root of unity, generalizing an earlier result of S. Arkhipov (conjectured by B. Feigin).
We construct the quantum group $GL_q(2)$ as the semi-infinite cohomology of the tensor product of two braided vertex operator algebras based on the algebra $W_2$ with complementary central charges $c+\bar{c}=28$. The conformal field theory version of the Laplace operator on the quantum group is also obtained.
Let $B$ be a quantum algebra possessing a semiclassical limit $A$. We show that under certain hypotheses $B^e$ can be thought of as a deformation of the Poisson enveloping algebra of $A$, and we give a criterion for the Hochschild cohomology of $B$ to be a deformation of the Poisson cohomology of $A$ in the case that $B$ is Koszul. We verify that condition for the algebra of $2\times 2$ quantum matrices and calculate its Hochschild cohomology and the Poisson cohomology of its semiclassical limit.
A positive level Kazhdan-Lusztig functor is defined using Arkhipov-Gaitsgory duality for affine Lie algebras. The functor sends objects in the DG category of G(O)-equivariant positive level affine Lie algebra modules to objects in the DG category of modules over Lusztig's quantum group at a root of unity. We prove that the semi-infinite cohomology functor for positive level modules factors through the Kazhdan-Lusztig functor at positive level and the quantum group cohomology functor with respect to the positive part of Lusztig's quantum group.
We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne--Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne--Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes $X_h$. Boyarchenko's two conjectures are on the maximality of $X_h$ and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invarian…
This paper provides a homological algebraic foundation for generalizations of classical Hecke algebras introduced in math.QA/9805134. These new Hecke algebras are associated to triples of the form (A,B,e), where A is an associative algebra containing subalgebra B with character e. These algebras are connected with cohomology of associative algebras in the sense that for every left A-module V and right A-module W the Hecke algebra associated to triple (A,B,e) naturally acts in the B-cohomology and B-homology spaces of V and W, respectively. We also introduce the semi-infinite cohomology functor…
Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel of a smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finite dimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) is finitely generated and that H*(DG_r,M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras θ_r:H*(G_r,k) \otimes S(g) \to H*(DG_r,k) which offers an approach to support varieties for DG_r-modules. For many examples of interest, θ_r is injective and induces an isomorphism of associated reduced schemes. Additionally, for M an …
Let $S$ be the affine plane $\C^2$ together with an appropriate $\mathbb T = \C^*$ action. Let $\hil{m,m+1}$ be the incidence Hilbert scheme. Parallel to \cite{LQ}, we construct an infinite dimensional Lie algebra that acts on the direct sum $$\Wft = \bigoplus_{m=0}^{+\infty}H^{2(m+1)}_{\mathbb T}(S^{[m,m+1]})$$ of the middle-degree equivariant cohomology group of $\hil{m,m+1}$. The algebra is related to the loop algebra of an infinite dimensional Heisenberg algebra. In addition, we study the transformations among three different linear bases of $\Wft$. Our results are applied to the ring stru…
We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra.
On the semi-infinite cohomology of graded-unitary vertex algebras
本文研究了具有手性量子动量图的分级酉顶点代数的相对半无穷上同调，证明其具有类似紧致Kähler流形微分形式的结构且本身分级酉，为源自超对称量子场论的顶点算子代数确立了分级酉性，并建立了类似Lefschetz作用的USp(2)作用及形式性结果。
Let $\mathfrak{g}$ be a simple Lie algebra, and let $W_κ$ be the affine ${W}$-algebra associated to a principal nilpotent element of $\mathfrak{g}$ and level $κ$. We explain a duality between the categories of smooth ${W}$ modules at levels $κ+ κ_c$ and $-κ+ κ_c$, where $κ_c$ is the critical level. Their pairing amounts to a construction of semi-infinite cohomology for the ${W}$-algebra. As an application, we determine all homomorphisms between the Verma modules for ${W}$, verifying a conjecture from the conformal field theory literature of de Vos--van Driel. Along the way, we determine the li…
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…
Denote m_0 the infinite dimensional N-graded Lie algebra defined by basis e_i, i>= 1 and relations [e_1,e_i] = e_(i+1) for all i>=2. We compute in this article the bracket structure on H1(m_0,m_0), H2(m_0,m_0) and in relation to this, we establish that there are only finitely many true deformations of m_0 in each nonpositive weight, by constructing them explicitely. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.
The semi-infinite cohomology of Weyl modules with two singular points
本研究针对具有两个奇异点的Weyl模，计算其半无限上同调，发现其性质与Frenkel-Gaitsgory的Weyl模不同，进而引入新模以在sl(2)情形满足预期性质。
Denote $\fm_2$ the infinite dimensional $\N$-graded Lie algebra defined by the basis $e_i$ for $i\geq 1$ and by relations $[e_1,e_i]=e_{i+1}$ for all $i\geq 2$, $[e_2,e_j]=e_{j+2}$ for all $j\geq 3$. We compute in this article the bracket structure on $H^1(\fm_2,\fm_2)$, $H^2(\fm_2,\fm_2)$ and in relation to this, we establish that there are only finitely many true deformations of $\fm_2$ in each weight by constructing them explicitely. It turns out that in weight 0 one gets as non-trivial deformation only one formal non-converging deformation.
We compute the cohomology with trivial coefficients of two graded infinite-dimensional Lie algebras of maximal class, give explicit formulas for their representative cocycles. Also we discuss the relations with combinatorics and representation theory.
Hopf-Hecke algebras and Barbasch-Sahi algebras were defined by the first named author (2016) in order to provide a general framework for the study of Dirac cohomology. The aim of this paper is to explore new examples of these definitions and to contribute to their classification. Hopf-Hecke algebras are distinguished by an orthogonality condition and a PBW property. The PBW property for algebras such as the ones considered here has been of great interest in the literature and we extend this discussion by further results on the classification of such deformations and by a class of hitherto unex…
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