Hochschild cohomology of a strongly homotopy commutative algebra

The Hochschild cohomology of a DG algebra A with coefficients in itself is, up to a suspension of degrees, a graded Lie algebra. The purpose of this paper is to prove that a certain DG Lie algebra of derivations appears as a finite codimensional graded sub Lie algebra of this Lie algebra when A is a strongly homotopy commutative algebra whose homology is concentrated in finitely many degrees. This result has interesting implications for the free the loop space homology which we explore here as well.     

Separated Lie models and the homotopy Lie algebra

A simply connected topological space X has homotopy Lie algebra π_∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.     

Graded lie algebras of depth one

To each simply connected topological space is associated a graded Lie algebra; the rational homotopy Lie algebra. The Avramov-Felix conjecture says that for a space of finite Ljusternik-Schnirelmann category this Lie algebra contains a free Lie subalgebra on two generators. We prove the conjecture in the case when the Lie algebra has depth one.     

Infinite-dimensional super Lie groups

A super Lie group is a group whose operations are G^∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G^∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators.In this context, we prove that if h is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group G, then h is the super Lie algebra of a sub-super Lie group of G. Additionally, we show that if g is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group G such that the super Lie algebra g is in fact the super Lie algebra of G. We also show that if H is a closed sub-super Lie group of a super Lie group G, then G→G/H is a principal fiber bundle.We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G^∞ category.     

Symmetry reduction of sh-Lie structures and of local functionals

Reduced sh-Lie structures have been studied for the case when a Lie group acts on the fibers of a vector bundle while preserving the base space of the bundle. In this paper we investigate how one obtains a reduced sh-Lie structure using the ideas of symmetry reduction where the action of the Lie group is transversal to the fibers of the bundle. We also show how local functionals are reduced using these ideas.     

Derivations, the Lawrence–Sullivan interval and the Fiorenza–Manetti mapping cone

We describe the rational homotopy type of any component of the based mapping space map^*(X,Y) as an explicit L _∞ algebra defined on the (desuspended and positive) derivations between Quillen models of X and Y. When considering the Lawrence–Sullivan model of the interval, we obtain an L _∞ model of the contractible path space of Y. We then relate this, in a geometrical and natural manner, to the L _∞ structure on the Fiorenza–Manetti mapping cone of any differential graded Lie algebra morphism, two in principal different algebraic objects in which Bernoulli numbers appear.     

Combinatorial Lie bialgebras of curves on surfaces

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6) (1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.     

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.     

Some examples of homotopy Π-algebras

The homotopy Π-algebra of a pointed topological space, X, consists of the homotopy groups of X together with the additional structure of the primary homotopy operations. We extend two well-known results for homotopy groups to homotopy Π-algebras and look at some examples illustrating the depth of structure on homotopy groups; from graded group to graded Lie ring, to Π-algebra and beyond. We also describe an abstract Π-algebra and give three abstract Π-algebra structures on the homotopy groups of the loop space of X which can be realized as the homotopy Π-algebras of three different spaces.     

Lie ideals of graded associative algebras

We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We also show that the Lie algebra Der _gr (A) of graded derivations of a graded semiprime associative algebra is strongly non-degenerate (modulo a certain ideal containing the center of Der _gr (A)).     

Formality in generalized Kähler geometry

We prove that no nilpotent Lie algebra admits an invariant generalized Kähler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kähler case, while it is never formal for a generalized complex structure on a nilpotent Lie algebra.     

Hom-Lie algebra structures on semi-simple Lie algebras

Hom-Lie algebras can be considered as a deformation of Lie algebras. In this note, we prove that the hom-Lie algebra structures on finite-dimensional simple Lie algebras are trivial. We find when a finite-dimensional semi-simple Lie algebra admits non-trivial hom-Lie algebra structures and the isomorphic classes of non-trivial hom-Lie algebras are determined.     

On the homotopy of simplicial algebras over an operad

According to a result of H. Cartan, the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this article, we show how to extend this result to other kinds of algebras. For instance, we prove that the homotopy of a simplicial Lie algebra is equipped with the structure of a restricted Lie algebra.

     

The homotopy Lie algebra of classifying spaces

Let X be a 1-connected CW-complex of finite type and L_x its rational homotopy Lie algebra. In this work, we show that there is a spectral sequence whose E² term is the Lie algebra Ext_ULx(Q, L_x), and which converges to the homotopy Lie algebra of the classifying space B autX. Moreover, some terms of this spectral sequence are related to derivations of L_x and to the Gottlieb group of X.     

Two observations about free graded hopf algebras

This paper records two results about graded Hopf algebras that do not appear to be stated explicitly in the literature. Let B be a graded set, graded by the positive integers. Let V be the graded vector space with basis B over a field K of characteristic zero and V^'=K⊕V, where K is in grading zero. Let L ne the free graded Lie algebra on B over K and let T be the free graded tensor algebra on B. The first result is the "graded Witt formula" giving the dimension of the subspace of L in each grading. The second result is the observation that any graded coassociative, co-commutative comultiplication Δ:V^'→V^'⊗V^', with co-unit the projection V¹→K. extends to a graded Hopf algebra structure on T that is in fact isomorphic to the natural graded Hopf algebra structure on T. In the ungraded case the statement analogous to the second result is false.     

A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker-Campbell-Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.     

Lie Derivations of Reflexive Algebras

A Lie derivation is called standard if it is a sum of a derivation and a linear map with image in the center vanishing on commutators. In this paper we show that Lie derivations of a reflexive algebra on a Banach space are standard if is a nest, or has the non-trivial smallest element, or has the non-trivial greatest element. This work was supported by NNSFC (No. 10771154) and PNSFJ (No. BK2007049).     

Exponentiation of infinite dimensional Z-graded lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace £ of the dual space V ^*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.     

A cohomology for vector valued differential forms

A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Frölicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of traceless vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed.