Cohomology Of Complex Research Articles

Cohomology of Hom-Lie superalgebras and q-deformed Witt superalgebra

Hom-Lie algebra (superalgebra) structure appeared naturally in q-deformations, based on σ-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of α k -derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted osp(1, 2) superalgebra and q-deformed Witt superalgebra.

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.

On the Dirac cohomology of complex lie group representations

This note studies the problem of classifying all the irreducible unitary representations with nonzero Dirac cohomology for a complex Lie group G. We reduce it to the classification of spherical ones with nonzero Dirac cohomology on the Levi level. Then in the spherical unitary dual, by computing spin norm and utilizing Vogan pencil, we show how to further reduce the classification to fairly few candidate representations.

Characteristic cycles and decomposition numbers

We observe that the characteristic cycle of a D-module gives bounds for decomposition numbers of intersection cohomology complexes.

Lefschetz operator and local Langlands mod $\ell$: The regular case

Let p and ` be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod ` cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vigneras’ local Langlands correspondence modulo ` for unipotent representations.

On the Hall algebra of an elliptic curve, II

This paper is a sequel to math.AG/0505148, where the Hall algebra U^+_E of the category of coherent sheaves on an elliptic curve E defined over a finite field was explicitly described, and shown to be a two-parameter deformation of the ring of diagonal invariants R^+=C[x_1^{\pm 1}, ..., y_1,...]^{S_{\infty}} (in infinitely many variables). In the present work, we study a geometric version of this Hall algebra, by considering a convolution algebra of perverse sheaves on the moduli spaces of coherent sheaves on E. This allows us to define a "canonical basis" B of U^+_E in terms of intersection cohomology complexes. We also give a characterization of this basis in terms of an involution, a lattice and a certain PBW-type basis. Finally we provide some (loose) interpretation of the above construction in terms of Schubert varieties in (nonexistent!) "double affine grassmanian", and in terms of character sheaves of loop groups (both of type A).

New Calabi–Yau orbifolds with mirror Hodge diamonds

We prove a representation-theoretic version of Borisov–Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When X is the Fermat quintic in P4, and X˜∗ is a Sym5-equivariant, toric resolution of its mirror X∗, we deduce that for any subgroup Γ of the alternating group A5, the Γ-Hilbert schemes Γ-Hilb(X) and Γ-Hilb(X˜∗) are smooth Calabi–Yau threefolds with (explicitly computed) mirror Hodge diamonds.

Graph homology and graph configuration spaces

If R is a commutative ring, M a compact R-oriented manifold and G a finite graph without loops or multiple edges, we consider the graph configuration space MG and a Bendersky–Gitler type spectral sequence converging to the homology H*(MG, R). We show that its E1 term is given by the graph cohomology complex CA(G) of the graded commutative algebra A = H*(M, R) and its higher differentials are obtained from the Massey products of A, as conjectured by Bendersky and Gitler for the case of a complete graph G. Similar results apply to the spectral sequence constructed from an arbitrary finite graph G and a graded commutative DG algebra \({\mathcal{A}}\).

Théorie de Lubin–Tate non abélienne ℓ-entière

For two distinct primes p and l, we investigate the Z_l-cohomology of the Lubin-Tate towers of a p-adic field. We prove that it realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible supercuspidal representations parametrized by a Z_l-algebra R, in a way compatible with extension of scalars. When R is a field of characteristic l, this gives a cohomological realization of the Langlands-Vigneras correspondence for supercuspidals, and a new proof of its existence. When R runs over complete local algebras, this provides bijections between deformations of matching mod-l representations. Roughly speaking, we can decompose "the supercuspidal part" of the l-integral cohomology as a direct sum, indexed by irreducible supercuspidals \pi mod l, of tensor products of universal deformations of \pi and of its two mates. Besides, we also get a virtual realization of both the semi-simple Langlands-Vigneras correspondence and the l-modular Langlands-Jacquet transfer for all representations, by using the cohomology complex and working in a suitable Grothendieck group.

Opérateur de Lefschetz sur les tours de Drinfeld et Lubin–Tate

AbstractWe define and study a Lefschetz operator on the equivariant cohomology complex of the Drinfeld and Lubin–Tate towers. For ℓ-adic coefficients we show how this operator induces a geometric realization of the Langlands correspondence composed with the Zelevinski involution for elliptic representations. Combined with our previous study of the monodromy operator, this suggests a possible extension of Arthur’s philosophy for unitary representations occurring in the intersection cohomology of Shimura varieties to the possibly non-unitary representations occurring in the cohomology of Rapoport–Zink spaces. However, our motivation for studying the Lefschetz operator comes from the hope that its geometric nature will enable us to realize the mod-ℓ Langlands correspondence due to Vignéras. We discuss this problem and propose a conjecture.

A note on exact forms on almost complex manifolds

Reformulations of Donaldson’s “tamed to compatible” question are obtained in terms of spaces of exact forms on a compact almost complex manifold (M, J). In dimension 4, we show that J admits a compatible symplectic form if and only if J admits tamed symplectic forms with arbitrarily given J-anti-invariant parts. Some observations about the cohomology of J-modified de Rham complexes are also made.

Modular intersection cohomology complexes on flag varieties

We present a combinatorial procedure (based on the W-graph of the Coxeter group) which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given by Kazhdan–Lusztig basis elements. Our procedure exploits the existence and uniqueness of parity sheaves. In particular we are able to show that the characters of all intersection cohomology complexes with coefficients in a field on the flag variety of type A n for n < 7 are given by Kazhdan–Lusztig basis elements. By results of Soergel, this implies a part of Lusztig’s conjecture for SL(n) with n ≤ 7. We also give examples where our techniques fail. In the appendix by Tom Braden examples are given of intersection cohomology complexes on the flag varities for SL(8) and SO(8) which have torsion in their stalks or costalks.

Solvable group representations and free divisors whose complements are [formula omitted]ʼs

We apply previous results on the representations of solvable linear algebraic groups to construct a new class of free divisors whose complements are K ( π , 1 ) ʼs. These free divisors arise as the exceptional orbit varieties for a special class of “block representations” and have the structure of determinantal arrangements. Among these are the free divisors defined by conditions for the (modified) Cholesky-type factorizations of matrices, which contain the determinantal varieties of singular matrices of various types as components. These complements are proven to be homotopy tori, as are the Milnor fibers of these free divisors. The generators for the complex cohomology of each are given in terms of forms defined using the basic relative invariants of the group representation.

Symmetries of massless vertex operators in [formula omitted

The worldsheet sigma model of the superstring in AdS 5 × S 5 has a one-parameter family of flat connections parametrized by the spectral parameter. The corresponding Wilson line is not BRST invariant for an open contour, because the BRST transformation leads to boundary terms. These boundary terms define a cohomological complex associated with the endpoint of the contour. We study the cohomology of this complex for Wilson lines in some infinite-dimensional representations. We find that for these representations the cohomology is nontrivial at the ghost number 2. This implies that it is possible to define a BRST invariant open Wilson line. The central point in the construction is the existence of massless vertex operators transforming exactly covariantly under the action of the global symmetry group. In flat space, massless vertices transform covariantly only up to adding BRST-exact terms. But in AdS we show that it is possible to define vertices so that they transform exactly covariantly.

SPHERICAL HALL ALGEBRAS OF CURVES AND HARDER-NARASIMHAN STRATAS

We show that the characteristic function <TEX>$1S_{\underline{\alpha}}$</TEX> of any Harder-Narasimhan strata <TEX>$S{\underline{\alpha}}\;{\subset}\;Coh_X^{\alpha}$</TEX> belongs to the spherical Hall algebra <TEX>$H_X^{sph}$</TEX> of a smooth projective curve X (defined over a finite field <TEX>$\mathbb{F}_q$</TEX>). We prove a similar result in the geometric setting: the intersection cohomology complex IC(<TEX>${\underline{S}_{\underline{\alpha}}$</TEX>) of any Harder-Narasimhan strata <TEX>${\underline{S}}{\underline{\alpha}}\;{\subset}\;{\underline{Coh}}_X^{\underline{\alpha}}$</TEX> belongs to the category <TEX>$Q_X$</TEX> of spherical Eisenstein sheaves of X. We show by a simple example how a complete description of all spherical Eisenstein sheaves would necessarily involve the Brill-Noether stratas of <TEX>${\underline{Coh}}_X^{\underline{\alpha}}$</TEX>.

LOCAL COMMUTATORS AND DEFORMATIONS IN CONFORMAL CHIRAL QUANTUM FIELD THEORIES

We study the general form of Möbius covariant local commutation relations in conformal chiral quantum field theories and show that they are intrinsically determined up to structure constants, which are subject to an infinite system of constraints. The deformation theory of these commutators is controlled by a cohomology complex, whose cochain spaces consist of linear maps that are subject to a complicated symmetry property, a generalization of the anti-symmetry of the Lie algebra case.

Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory

We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to "light modes", i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the $A_{\infty}$-subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of $A_{\infty}$-algebras. The associated generalized Maurer-Cartan equation gives the analogue of the Yang-Mills equation for a wide class of VOAs. Applying this construction to an example of VOA generated by $\beta$-$\gamma$ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory.

Representations on Hessenberg Varieties and Young's Rule

We combinatorially construct the complex cohomology (equivariant and ordinary) of a family of algebraic varieties called regular semisimple Hessenberg varieties. This construction is purely in terms of the Bruhat order on the symmetric group. From this a representation of the symmetric group on the cohomology is defined. This representation generalizes work of Procesi, Stembridge and Tymoczko. Here a partial answer to an open question of Tymoczko is provided in our two main result. The first states, when the variety has multiple connected components, this representation is made up by inducing through a parabolic subgroup of the symmetric group. Using this, our second result obtains, for a special family of varieties, an explicit formula for this representation via Young's rule, giving the multiplicity of the irreducible representations in terms of the classical Kostka numbers. Nous construisons la cohomologie complexe (équivariante et ordinaire) d'une famille de variétés algébriques appelées variétés régulières semisimples de Hessenberg. Cette construction utilise exclusivement l'ordre de Bruhat sur le groupe symétrique, et on en déduit une représentation du groupe symétrique sur la cohomologie. Cette représentation généralise des résultats de Procesi, Stembridge et Tymoczko. Nous offrons ici une réponse partielle à une question de Tymoczko grâce à nos deux résultats principaux. Le premier déclare que lorsque la variété a plusieurs composantes connexes, cette représentation s'obtient par induction à travers un sous-groupe parabolique du groupe symétrique. Nous en déduisons notre deuxième résultat qui fournit, pour une famille spéciale de variétés, une formule explicite pour cette représentation par la règle de Young, et donne ainsi la multiplicité des représentations irréductibles en termes des nombres classiques de Kostka.

Quantization of quasi-Lie bialgebras

We construct quantization functors of quasi-Lie bialgebras. We establish a bijection between this set of quantization functors, modulo equivalence and twist equivalence, and the set of quantization functors of Lie bialgebras, modulo equivalence. This is based on the acyclicity of the kernels of natural morphisms between the universal versions of Lie algebra cohomology complexes for quasi-Lie and Lie bialgebras. The proof of this acyclicity consists of several steps, ending up in the acyclicity of a complex related to free Lie algebras, namely, the universal version of the Lie algebra cohomology complex of a Lie algebra in its enveloping algebra, viewed as the left regular module. Using the same arguments, we also prove the compatibility of quantization functors of quasi-Lie bialgebras with twists, which allows us to recover our earlier results on compatibility of quantization functors with twists in the case of Lie bialgebras.

Topics on n-ary Algebraic Structures

We review the basic definitions and properties of two types of n-ary structures, the Generalized Lie Algebras (GLA) and the Filippov (≡ n-Lie) algebras (FA), as well as those of their Poisson counterparts, the Generalized Poisson (GPS) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology complexes relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends Whitehead’s lemma to all n ≥ 2, n = 2 being the original Lie algebra case. Some comments onn-Leibniz algebras are also made.