The Cohomology Algebra of the Semi-Infinite Weil Complex

In 1993, Lian-Zuckerman constructed two cohomology operations on the BRST complex of a conformal vertex algebra with central charge 26. They gave explicit generators and relations for the cohomology algebra equipped with these operations in the case of the c = 1 model. In this paper, we describe another such example, namely, the semi-infinite Weil complex of the Virasoro algebra. The semi-infinite Weil complex of a tame -graded Lie algebra was defined in 1991 by Feigin-Frenkel, and they computed the linear structure of its cohomology in the case of the Virasoro algebra. We build on this result by giving an explicit generator for each non-zero cohomology class, and describing all algebraic relations in the sense of Lian-Zuckerman, among these generators.     

SAYD Modules over Lie-Hopf Algebras

In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.     

A van Est Isomorphism for Bicrossed Product Hopf Algebras

To any locally finite representation of a given double crossed sum (product) Lie algebra (group), we associate a stable anti Yetter-Drinfeld (SAYD) module over the bicrossed product Hopf algebra which arises from the semidualization procedure. We prove a van Est isomorphism between the relative Lie algebra cohomology of the total Lie algebra and the Hopf cyclic cohomology of the corresponding Hopf algebra with coefficients in the associated SAYD module.     

Twisted noncommutative equivariant cohomology: Weil and Cartan models

We propose Weil and Cartan models for the equivariant cohomology of noncommutative spaces which carry a covariant action of Drinfel’d twisted symmetries. The construction is suggested by the noncommutative Weil algebra of Alekseev and Meinrenken (2000) [5]; we show how to implement a Drinfel’d twist of their models in order to take into account the noncommutativity of the spaces we are acting on. We also provide basic examples and properties of the twisted noncommutative equivariant cohomology.     

Inverse backscattering in two dimensions

We interpretN=2 superconformal field theories (SCFTs) formulated by Kazama and Suzuki via Goddard-Kent-Olive (GKO) construction from a viewpoint of the Lie algebra cohomology theory for the affine Lie algebra. We determine the cohomology group completely in terms of a certain subset of the affine Weyl group. We find that this subset describing the cohomology group can be obtained from its classical counterpart by the action of the Dynkin diagram automorphisms. Some algebra automorphisms of theN=2 superconformal algebra are also formulated. Utilizing the algebra automorphisms, we study the field identification problem for the branching coefficient modules in the GKO-construction. Also the structure of the Poincaré polynomial defined for eachN=2 theory is revealed.Dedicated to Professor Noboru Tanaka on his sixtieth birthday     

Homogeneous generalized holomorphic structures

Homogeneous generalized holomorphic structures in the context of homogeneous principal fiber bundles are investigated. They are characterized in terms of Lie algebra data, and the generalized Dolbeault cohomology groups associated to a homogeneous generalized holomorphic vector bundle are identified with certain relative Lie algebra cohomology groups. We also provide some examples, using generalized flag manifolds.     

Differential Graded Cohomology and Lie Algebras¶of Holomorphic Vector Fields

This article continues work of B. L. Feigin [5] and N. Kawazumi [15] on the Gelfand-Fuks cohomology of the Lie algebra of holomorphic vector fields on a complex manifold. As this is not always an interesting Lie algebra (for example, it is 0 for a compact Riemann surface of genus greater than 1), one looks for other objects having locally the same cohomology. The answer is a cosimplicial Lie algebra and a differential graded Lie algebra (well known in Kodaira–Spencer deformation theory). We calculate the corresponding cohomologies and the result is very similar to the result of A. Haefliger [12], R. Bott and G. Segal [2] in the case of vector fields. Applications are in conformal field theory (for Riemann surfaces), deformation theory and foliation theory. Received: 25 February 1999 / Accepted: 20 July 1999     

A bigraded version of the Weil algebra and of the Weil homomorphism for Donaldson invariants. Elementary algebra and cohomology behind the Baulieu-Singer approach to Witten's topological Yang-Mills quantum field theory

We describe a bigraded generalization of the Weil algebra, of its basis and of the characteristic homomorphism which besides ordinary characteristic classes also maps on cohomology classes leading to Donaldson invariants in the appropriate context. Furthermore these cohomology classes exhaust the image of the generalized characteristic homomorphisms.     

Application of the cohomology of graded Lie algebras to formal deformations of Lie Algebras

The purpose of the Letter is to show how to use the cohomology of the Nijenhuis-Richardson graded Lie algebra of a vector space to construct formal deformations of each Lie algebra structure of that space. One then shows that the de Rham cohomology of a smooth manifold produces a family of cohomology classes of the graded Lie algebra of the space of smooth functions on the manifold. One uses these classes and the general construction above to provide one-differential formal deformations of the Poisson Lie algebra of the Poisson manifolds and to classify all these deformations in the symplectic case.     

Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra

We study the homology and cohomology groups of super Lie algebras of supersymmetries and of super Poincaré Lie algebras in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions ?11. For dimensions D=10,11 we describe also the cohomology of reduction of supersymmetry Lie algebra to lower dimensions. Our methods can be applied to extended supersymmetry Lie algebras.     

Algebraic approach to the geometry of the (super-) string model

We discuss the extension of constraint algebras to include subsidiary constraints within a larger algebra. The interplay between various mathematical aspects of this procedure is described. Tools from Lie algebra cohomology and differential geometry are used to gain new insights into BRS techniques for nonabelian constrained systems. We show that cohomology considerations restrict our formalism to non-semisimple constraint algebras, such as the (super-) string model; we illustrate the ideas by presenting concrete results for this case.     

On the 1-cohomology of Lie groups

Given a continous representation of a Lie group in a Banach space we study its 1-cohomology. We prove that the computation of the 1-cocycles can be reduced to that of the 1-cocycles of the differential of the representation. When the group is semi-simple and the representation is K-finite, we prove that the cohomology is equivalent to the cohomology of the Lie algebra representation on K-finite vectors. We prove, using Casimir operators, that there exist only a finite number of irreducible representation of a semi-simple Lie group with a non-trivial cohomology. Exemples of such representations are given.     

Lie Conformal Algebra Cohomology and the Variational Complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a \mathfrakg{\mathfrak{g}}-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.     

Differential geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained by considering that the (bosonic) generator x is invertible and by working with polynomials in ln x instead of polynomials in x. We construct the quantum Lie algebra associated with this extension and obtain its Hopf algebra structure and its dual Hopf algebra.     

A New Cohomology Theory Associated to Deformations of Lie Algebra Morphisms

We introduce a new cohomology theory related to deformations of Lie algebra morphisms. This notion involves simultaneous deformations of two Lie algebras and a homomorphism between them.This revised version was published online in March 2005 with corrections to the cover date.     

The Unitary Operator Corresponding to the General Two-Mode Coordinate-Momentum Mixed Transformation

We discuss the unitary operator corresponding to the general two-mode coordinate-momentum mixed transformation(q2,p2)→(Aq1 Bq2,Cq1 Dp2),where A,B,C and Dare arbitrary real numbers,Suitably selecting the parameters A,B,Cand D,we obtain a new two-mode bosonic realization of the SU(1,1) Lie algebra.We also study the squeezing effects of the squeezed vacuum associated with the new two-mode bosonic realization of the SU(1,1) Lie algebra.The results show that the new squeezed vacuum does not possess second-order squeezing,but exhibits higher-order squeezing.     

Some Remarks on the Cohomology of Krichever–Novikov Algebras

We calculate the continuous cohomology of the Lie algebra of meromorphic vector fields on a compact Riemann surface from the cohomology of the holomorphic vector fields on the open Riemann surface pointed in the poles. This cohomology has been given by Kawazumi. Our result shows the Feigin–Novikov conjecture.     

Some remarks on BRS transformations,anomalies and the cohomology of the Lie algebra of the group of gauge transformations

We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ? of gauge transformations. We examine the cohomology of the Lie algebra of ? and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.