Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

Cyclic cohomology and algebra extensions

We construct a spectral sequence to study cyclic cohomology for an extension A = R/I of algebras. When R is a free algebra we describe the cyclic cohomology of A in terms of traces defined on R or powers of I. Explicit cyclic cocycles representing the cyclic cohomology class belonging to the trace are constructed as analogues of Chern character and Chern-Simons forms.Dedicated to Alexander Grothendieck     

A Dixmier-Douady theorem for Fell algebras

We generalise the Dixmier-Douady classification of continuous-trace C^?-algebras to Fell algebras. To do so, we show that C^?-diagonals in Fell algebras are precisely abelian subalgebras with the extension property, and use this to prove that every Fell algebra is Morita equivalent to one containing a diagonal subalgebra. We then use the machinery of twisted groupoid C^?-algebras and equivariant sheaf cohomology to define an analogue of the Dixmier-Douady invariant for Fell algebras, and to prove our classification theorem.     

Nilpotent n-Lie Algebras

We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I ² nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

Chiral differential operators: Formal loop group actions and associated modules

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of 2-dimensional σ-models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a “formal loop group action” on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said “formal loop group actions” and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of “formal loop spaces”. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus.     

Finite vs affine W-algebras

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra Zhu_ΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu_H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu_H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established. “I am an old man, and I know that a definition cannot be so complicated.” I.M. Gelfand (after a talk on vertex algebras in his Rutgers seminar)     

Betti Number Behavior for Nilpotent Lie Algebras

We discuss three general problems concerning the cohomology of a (real or complex) nilpotent Lie algebra: first of all, determining the Betti numbers exactly; second, determining the distribution these Betti numbers follow; and finally, estimating the size of the individual cohomology spaces or the total cohomology space. We show how spectral sequence arguments can contribute to a solution in a concrete setting. For one-dimensional extensions of a Heisenberg algebra, we determine the Betti numbers exactly. We then show that some families in this class have a M-shaped Betti number distribution, and construct the first examples with an even more exotic Betti number distribution. Finally, we discuss the construction of (co)homology classes for split metabelian Lie algebras, thus proving the Toral Rank Conjecture for this class of algebras.     

阶化Cartan型李代数的上同调

本文利用广义限制李代数的概念和应用Frobenius代数的一些性质来研究广义限制李代数的广义限制完备上同调,并利用广义限制上同调与通常上同调的关系尝试着给出一种计算系数为不可约模的阶化Cartan型李代数上同调的方法.     

Cohomology of Oriented Tree Diagram Lie Algebras

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this article, we use Hodge Laplacian to study the cohomology of these Lie algebras. The “total rank conjecture” and “b ₂-conjecture” for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler–Poincaré principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.     

Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves

We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on \mathbbP²{\mathbb{P}^2} . The top non-vanishing equivariant Chern classes of the cohomology of these complexes yield actions of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology of the moduli spaces. In this way we obtain a geometric realization of the boson-fermion correspondence and related vertex operators.     

Deformations of algebras and cohomology of fixed point sets

In [13] it is shown that under certain conditions the cohomology algebra of the fixed point set of a space with group action is in an algebraic sense a deformation of the cohomology algebra of the space itself. Here we attempt to prove a converse of the above statement, i.e. we try to realize geometrically a given algebraic deformation of a (commutative) graded algebras as the cohomology algebra of the fixed point set of a suitable space with group action. The first part of this note in a sense reduces this realization problem in equivariant topology to a non-equivariant problem while the second part uses Sullivan's theory of minimal models to actually obtain a converse for S¹-actions, where cohomology is taken with rational coefficients.     

Quasi-Piecewise-Koszul Modules

In this article we first study an equivariant cyclic cohomology for weak H-module agebras over a weak Hopf algebra H with a bijective antipode. Then we define an equivariant K-theory for weak quantum Yetter–Drinfeld algebras over H and establish a generalized Connes' pairing between the equivariant cyclic cohomology and the equivariant K-theory. As an application we consider our theory for groupoids.     

Hochschild Cohomology of <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We compute the Hochschild cohomology of any block of q-Schur algebras. We focus on the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of q-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all q-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derive equivalence.     

Generalized Poisson brackets and lie algebras of type H in characteristic 0

The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials . We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras. Received December 20, 1996; in final form September 15, 1997     

Baxter algebras and Hopf algebras

By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.

     

André–Quillen cohomology of algebras over an operad

We study the André–Quillen cohomology with coefficients of an algebra over an operad. Using resolutions of algebras coming from the Koszul duality theory, we make this cohomology theory explicit and we give a Lie theoretic interpretation. For which operads is the associated André–Quillen cohomology equal to an Ext-functor? We give several criteria, based on the cotangent complex, to characterize this property. We apply it to homotopy algebras, which gives a new homotopy stable property for algebras over cofibrant operads.     

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W（2, 2） Algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.