arXiv math/05011172005-01math.QA

The Cohomology Algebra of the Semi-infinite Weil Complex

Andrew R. Linshaw

备注：: Minor editorial changes, references added
发表信息：: Commun. Math. Phys. 267, 13-23 (2006).
DOI：: 10.1007/s00220-006-0062-9

英文摘要：

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 Z-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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

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; we show that one can implement a Drinfel'd twist of their models in order to take into account the noncommutativity of the spaces we are acting on.

2009-01-13hep-thmath-phmath.MP

Cyclic cohomology of Hopf algebras, and a non-commutative Chern-Weil theory

REVISED VERSION: We have re-organized the paper, and included some new results. Most important, we prove that the (truncated) Weil complexes compute the cyclic cohomology of the Hopf algebra (see the new Theorem 7.3). We also include a short discussion on the uni-modulare case, and the computation for $H= U_q(sl_2)$. THE OLD ABSTRACT: We give a construction of Connes-Moscovici's cyclic cohomology for any Hopf algebra equipped with a twisted antipode. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the …

1999-08-05math-phmath.KTmath.MP

Chiral Equivariant Cohomology I

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Qu…

2006-07-10math.DGmath.QA

Superalgebraic interpretation of quantization maps of Weil algebras

In 1998, A.Alekseev and E.Meinrenken construct an explicit $G$-differential space homomorphism $\mathcal{Q}$, called the quantization map, between the Weil algebra $\Weil{\g}= \sym{\co{\g}} \otimes \ext{\co{\g}}$ and $\NWeil{\g}=\U{\g} \otimes \Cl{\g}$ (which they called the noncommutative Weil algebra) for any quadratic Lie algebra $\g$. They showed that $\mathcal{Q}$ induces an algebra isomorphism between the basic cohomology rings $H^{\ast}_{bas}(\Weil{g})$ and $H^{\ast}_{bas}(\NWeil{\g})$. In this paper, I interpret the quantization map $\mathcal{Q}$ as the super Duflo map between the symm…

2005-06-07math-phmath.MPmath.QA

Cyclic cohomology of Lie algebras

In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module over it. This complex is quasi-isomorphic to the Hopf cyclic complex of the enveloping algebra of the Lie algebra with coefficients in the corresponding module. It is shown that the (truncated) Weil algebra, the Weil algebra with generalized coefficients defined by Alekseev-Meinre…

2011-08-13math.DGmath.KTmath.QA

Equivariant Chern classes in Hopf cyclic cohomology

We present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometric realization of the Hopf cyclic characteristic classes of foliations.

2015-02-09math.DGmath.QA

更多相关论文收起更多论文

Transgression and Clifford algebras

Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, >..., p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/<p_1, ..., p_r>)$ is isomorphic to a Clifford algebra $\text{Cl}(P,B)$, where the (possibly degenerate) bilinear form $B$ depends on $W$. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of $W$ given by the quantized Weil algebra $\qWg = \Ug \otimes \Clg$ for $\Lieg$ a reductive Lie algebra…

2007-12-06math.QAmath.RT

Equivariant cohomology and tensor categories

We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of results in equivariant cohomology, including the Chern-Weil theorem for an arbitrary rigid Lie algebra object. For a quadratic Lie algebra object we obtain a proof of the Duflo isomorphism along the lines of Alekseev-Meinrenken, thereby generalizing their result to Lie superalgebras.

2008-02-07math.DGmath.QA

Some cohomology operators in 2-D field theory

It is typical for a semi-infinite cohomology complex associated with a graded Lie algebra to occur as a vertex operator (or chiral) superalgebra where all the standard operators of cohomology theory, in particular the differential, are modes of vertex operators (fields). Although vertex operator superalgebras -with the inherent Virasoro action- are regarded as part of Conformal Field Theory (CFT), a VOSA may exhibit a square-zero operator (often, but not always, the semi-infinite cohomology differential) for which the Virasoro algebra acts trivially in the cohomology. Capable of shedding its C…

1993-07-26hep-thmath.QA

Pairings in Hopf-cyclic cohomology of algebras and coalgebras with coefficients

This paper is concerned with the theory of cup-products in Hopf-type cyclic cohomology of algebras and coalgebras. Here we give detailed proofs of the statements, announced in our previous paper. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving noncommutative Weil algebra. Then we use a generalization of Quillen and Crainic's construction to define the cup-product. We discuss the relation of the introduced cup-product and $S$-operations on cyclic cohomology. After this we describe the relation of this type of product and bivariant cyclic cohomolog…

2006-10-20math.KTmath.QA

Quantum Cohomology of a Product

The operation of tensor product of Cohomological Field Theories (or algebras over genus zero moduli operad) introduced in an earlier paper by the authors is described in full detail, and the proof of a theorem on additive relations between strata classes is given. This operation is a version of the Kuenneth formula for quantum cohomology. In addition, rank one CohFT's are studied, and a generalization of Zograf's formula for Weil-Petersson volumes is suggested.

1995-02-13alg-geommath.AGmath.QA

Weil positivity and Trace formula, the archimedean place

We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula of the paper "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function". (Selecta Math. 5 (1999), no. 1, 29--106). We explore in great details the simplest case of the single archimedean place. The root of the positivity is the trace of the scaling action compressed onto the orthogonal complement of the range of the cutoff projections associated to the cutoff in phase space, for cutoff parameter equal to 1. We express th…

2020-06-24math.NTmath.OAmath.QA

Geometric construction of Hopf cyclic characteristic classes

In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have found a Hopf algebra H(n), playing the role of a `quantum structure group' for the `space of leaves' of a codimension n foliation, whose Hopf cyclic cohomology is canonically isomorphic to the Gelfand-Fuks cohomology of the Lie algebra of formal vector fields. However, with a few low-dimensional exceptions, no explicit construction was known for its Hopf cycli…

2015-02-09math.DGmath.QA