arXiv subjects
Publications and source records attributed to Sergey Arkhipov.
We construct a geometric version of BRST cohomology complex of a chiral module over a Lie-* algebra using the language of differential graded Lie algebroids in the category of D-modules on a compact curve $X$.
In this note we give a definition of semiinfinite cohomology for Tate Lie algebras using the language of Differential Graded Lie algebroids with Curvature (CDG Lie algebroids).
In this paper we construct some quantum analogues of the global Cousin complex for the flag variety in positive characteristic. Just like in the positive characteristic case, we obtain some remarkable resolutions of the contragradient modules over the small quantum group. We use these resolutions to calculate semiinfinite cohomology of the small quantum group with coefficients in contragradient Weyl modules. It turns out that the described graded space of semiinfinite cohomology is naturally isomorphic to the space of distributions on the cotangent bundle of the flag variety with support in the union of conormal bundles to the Schubert cells with coefficients in the pull-back of a line bundle on the flag variety.
The paper is devoted to the proof of the following conjecture due to B. Feigin. Let $\frak u_\ell$ be the small quantum group a the primitive $\ell$-th root of unity. Then it is known that the usual $Ext$ algebra of the trivial $\frak u_\ell$-module is isomorphic to the algebra of regular functions on the nilpotent variety $\cal N$ in the corresponding simple Lie algebra $\frak g$ (see [GK]). Consider semiinfinite cohomology of the trivial $\frak u_\ell$-module introduced in [Ar1], [Ar2]. It was shown in [Ar2] that the $Ext$ algebra of the trivial $\frak u_\ell$-module acts naturally on semiinfinite cohomology. Moreover semiinfinite cohomology space of the trivial $\frak_\ell$-module is equipped with a natural $\frak g$-module structure. B. Feigin conjectured that the described $\frak g$-module and $F(\cal N)$-module structures coincide with the ones on the space of local cohomology of the structure sheaf on $\cal N$ with support in the standard positive nilpotent subalgebra $\frak n^+\subset\cal N\subset \frak g$. We give a detailed proof of the conjecture. Moreover we generalize the statement and describe semiinfinite cohomology of contragradient Weyl modules in terms of local cohomology of certain coherent sheaves on the nilpotent cone and on its desingularization $T^*(G/B)$. To do this we construct a certain specialization of the quantum BGG resolution of the simple module $L(λ)$ defined for generic values of the quantizing parameter into the root of unity called the quasi-BGG complex. The complex conists of direct sums of quasi-Verma modules and provides conjecturally a resolution of the Weyl module $W(λ)$.
It is known that the semi-infinite cohomology spaces of the infinitely twisted nilpotent subalgebra in an affine Lie algebra $g$ with coefficients in an integrable simple module over the affine Lie algebra have a base enumerated by elements of the corresponding affine Weyl group graded by the semiinfinite length function. Let $U$ be the affine quantum group corresponding to $g$. It is possible to define a subalgebra in $U$ being the quantum analogue of the universal enveloping algebra of the infinitely twisted nilpotent subalgebra in $g$. In this paper we prove that for general values of the parameter $v$ the semiinfinite cohomology of this associative algebra with coefficients in an integrable simple module over $U$ coincides with the one of the corresponding Lie subalgebra in $g$ with coefficients in the corresponding $g$-module.
We introduce the techniques of semiregular bimodules over a Lie algebra with respect to a Lie subalgebra. Using this techniques in the case of affine Lie algebras we introduce twisting functors on the categories of modules. These functors are enumerated by elements of the affine Weyl group corresponding to the chosen affine Lie algebra and map Verma modules to so called twisted Verma modules. Applying twisting functors to the BGG resolution of an integrable simple module we obtain a sequence of complexes called the twisted BGG resolutions. We show that the twisted BGG resolutions form an inductive system of complexes. It turns out that the limit complex of the inductive system is exactly the semi-infinite BGG resolution introduced by B. Feigin and E. Frenkel and consisting of direct sums of Wakimoto modules.
We describe semiinfinite cohomology of associative algebras in terms of Koszul (or bar) duality. Consider an associative algebra $A$ and two its subalgebras $B$ and $N$ such that $A=B\otimes N$ as a vector space. We prove that the endomorphism algebra of the semiregular $A$-module appears naturally in semiinfinite cohomology theory as a ``two times Koszul dual'' to the algebra $A$. We compare semiinfinite cohomology of universal enveloping algebras with the well-known Lie algebra semiinfinite cohomology. A new description of the critical 2-cocycle is provided. As a consequence we obtain another proof of the fact that additive categories generated by Verma modules over affine Lie algebras on dual levels are antiequivalent.
In this paper we develop a new homology theory of associative algebras called semiinfinite cohomology in a derived category setting. We show that in the case of small quantum groups the zeroth semiinfinite cohomology of the trivial module is closely related to the conformal blocks' spaces. We provide a description of the semiinfinite cohomology spaces of the trivial module over a small quantum group in terms of distributions on the nilpotent cone of the corresponding semisimple Lie algebra.