In this paper we identify many striking elements in Leibniz (co)homology which arise from characteristic classes and K-theory. For a group G a field k of characteristic zero, it is shown that all primary characteristic classes, i.e. H^*(BG; k), naturally inject into certain Leibniz cohomology groups via an explicit chain map. Moreover, if f: A \to B is a homomorphism of algebras or rigns, the relative Leibniz homology groups HL_*(f) are defined, and if in addition f is surjective with nilpotent kernel, A and B algebras over the rationals, then there is a natural surjection
HL_{*+1}(gl(f)) \to HC_*(f), where HC_*(f) denotes relative cyclic homology, and gl(f): gl(A) \to gl(B) is the induced map on matrices. Here again, the surjection is realized via an explicit chain map, and offers a relation between Leibniz homology and K-theory.
We relate Leibniz homology to cyclic homology by studying a map from a long exact sequence in the Leibniz theory to the ISB periodicity sequence in the cyclic theory. This provides a setting by which the two theories can be compared via the 5-lemma. We then show that the Godbillon-Vey invariant, as detected by the Leibniz homology of formal vector fields, maps to the Godbillon-Vey invariant as detected by the cyclic homology of the universal enveloping algebra of these vector fields.
This article presents results being consistent with conjectures of J.-L. Loday about the existence and properties of a Leibniz homology for groups. Introducing L-sets we prove that (pointed) rack homology has properties this conjectural Leibniz homology should satisfy, namely the existence of a coZinbiel coalgebra structure on rack homology and the existence of a non trivial natural cocommutative coalgebra morphism from the rack homology of a group to its Eilenberg-MacLane homology. The end of the paper treats the particular cases of the linear group and of abelian groups. We prove the existen…
In this paper, we compute the Leibniz (co)homology of the affine indefinite orthogonal Lie algebra. This calculation generalizes a result \cite[corollary 4.5]{JL} obtained by Jerry Lodder. We construct several indefinite orthogonal invariants in terms of balanced tensors and provide the Leibniz homology in terms of these invariants.
Presented is a structure theorem for the Leibniz Homology, HL_*, of an Abelian extension of a simple real Lie algebra g. As applications, results are stated for affine extensions of the classical Lie algebras sl_n(R), so_n(R), and sp_n(R). Furthermore, HL_*(h) is calculated when h is the Lie algebra of the Poincare group as well as the Lie algebra of the affine Lorentz group. The general theorem identifies all of these in terms of g-invariants.
The universal central extensions and their extension kernels of the matrix Lie superalgebra sl(m, n, A), the Steinberg Lie superalgebra st(m, n, A) in category {\bf SLeib} of Leibniz superalgebras are determined under a weak assumption (compared with \cite{MP}) using the first Hochschild homology and the first cyclic homology group.
A Ronco algebra is a Leibniz algebra satisfying the identity: $$[[x,x],y]=0.$$ Based on properties of Leibniz homology, we give a proof an old and unpublished result of María Ronco, which describes free objects in these class of algebras. We also establish a relationship between Ronco algebras and so called Lie $μ$-algebras extending our previous result.
We construct a local characteristic map to a symplectic manifold M via certain cohomology groups of Hamiltonian vector fields. For each p in M, the Leibniz cohomology of the Hamiltonian vector fields on R^{2n} maps to the Leibniz cohomology of all Hamiltonian vector fields on M. For a particular extension g_n of the symplectic Lie algebra, the Leibniz cohomology of g_n is shown to be an exterior algebra on the canonical symplectic two-form. The Leibniz homology of g_n then maps to the Leibniz homology of Hamiltonian vector fields on R^{2n}.
In this paper, we compute the Leibniz homology of the Schrödinger algebra. We show that it is a graded vector space generated by tensors in dimensions $2n-2$ and $2n$. The Leibniz homology of the full Galilei algebra is also calculated.
In this paper we show that for a $n$-Filippov algebra $\g,$ the tensor power $\g^{\otimes n-1}$ is endowed with a structure of anti-symmetric co-representation over the Leibniz algebra $\g^{\wedge n-1}$. This co-representation is used to define two relative theories for Leibniz $n$-algebras with $n>2$ and obtain exact sequences relating them. As a result, we construct a spectral sequence for the Leibniz homology of Filippov algebras.
In this paper we apply homotopical localization to the framework of differential graded algebras over an operad. We get plus construction by performing nullification with respect to an universal acyclic algebra. This plus construction for Lie and Leibniz algebras up to homotopy yields two new homology theories for associative algebras. They are closely related to cyclic and Hochschild homologies. We also compute their first homology groups.
In this paper we study the variability and rigidity of secondary characteristic classes which arise from flat connections on a manifold. Considering the connection as a Lie-algebra valued one-form, we study the characteristic map from Lie algebra cohomology to de Rham cohomology of the manifold, and prove that if the Leibniz cohomology of the Lie algebra vanishes, then all secondary characteristic classes are rigid. In the case of codimension one foliations (with trivial normal bundle), we show that a characteristic map from the Leibniz cohomology of formal vector fields to the de Rham cohomol…
There is a notion of non-commutative Lie algebra called "Leibniz algebra", which is characterized by the condition: left bracketing is a derivation. The purpose of this article is to introduce and study a new notion of algebra, called "associative dialgebra", which is, to the Leibniz algebras, what associative algebras are to Lie algebras. An associative dialgebra is a vector space equipped with two associative operations satisfying three more conditions. For instance, any differential associative algebra gives rise to a dialgebra. In this article we construct and study a (co)homology theory f…
There is a series of cycles in the rational homology of the groups Out(F_n), first discovered by S. Morita, which have an elementary description in terms of finite graphs. The first two of these give nontrivial homology classes, and it is conjectured that they are all nontrivial. These cycles have natural lifts to the homology of Aut(F_n), which is stably trivial by a recent result of Galatius. We show that in fact a single application of the stabilization map from Aut(F_n) to Aut(F_(n+1)) kills the Morita classes, so that they disappear immediately after they appear.
Described the algebraic structure on the space of homotopy classes of cycles with marked topological flags of disks. This space is a non-commutative monoid, with an Abelian quotient corresponding to the group of singular homologies $H_k(M)$. For the marked flag contracted to a point the multiplication becomes commutative and the subgroup of spherical cycles corresponds to the usual homotopy group $π_k(M)$.
The goal of this paper is to present results which are consistent with conjectures about the Leibniz (co)homology for discrete groups stated by J. L. Loday. We show that rack cohomology has properties very close to the properties expected for the conjectural Leibniz cohomology. In particular, we prove the existence of a graded dendriform algebra structure on rack cohomology, and we construct a graded associative algebra morphism from group cohomology to rack cohomology which is injective in degree 1.
In this paper we construct "structural" pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the structural pre-braidings above, it gives a generalization and a unification of many known homology theories. All the constructions are categorified, resulting in particular in their super- and co-versions. L…
We prove without any assumption on the ground field that higher Hochschild homology groups do not vanish for two large classes of algebras whose global dimension in not finite.
In these notes, we survey the homology of the loop group Omega(K) of a compact group K, also known as the affine Grassmannian of a complex loop group. Using the Bott picture of H_*(Omega(K)), the homology algebra or Pontryagin ring, we obtain two new results: A. Factorization of affine Schubert homology classes. B. Definition of affine Schubert polynomials representing the affine Schubert homology classes in all types, in terms similar to ordinary Schubert polynomials.
We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.
We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral homology of a related associative algebra with involution. For the operads Comm, Assoc and Lie we compute this algebra explicitly, enabling us to apply known results on dihedral homology to the computation of hairy graph homology. In addition we determine the image in hairy graph …
Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{…
We prove that for "most" closed 3-dimensional manifolds $M$, the existence of a closed non singular one-form in a given cohomology class $u\in H^1 (M,\bf R)$ is equivalent to the fact that every twisted Alexander polynomial $Δ^H(M,u) \in {\bf Z}[G/\ker u]$ associated to a normal subgroup with finite index $H < π_1(M)$ has a unitary $u$-minimal term.
We show that for many semi-hyperbolic groups the decomposition into conjugacy classes of the Hochschild homology of the l^1-rapid decay group algebra is injective.
We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.
