Braids and crossed modules
We describe Artin's braid group on a (fixed) finite number of strings as a crossed module over itself. In particular, we interpret the braid relations as crossed module structure relations.
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arXiv subjects
Johannes Huebschmann
Publications and source records attributed to Johannes Huebschmann.
Homological perturbations, equivariant cohomology, and Koszul duality
Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various invariants of a certain geometric situation, is cut to a small model, and the higher homotopies can then be dealt with concisely in the language of sh-structures (strong homotopy structures). This amounts to precise ways of handling the requisite additional structure encapsulating the various coherence conditions. Given e. g. two augmented differential graded algebras A and B, an sh-map from A to B is a twisting cochain from the reduced bar construction of A to B and, in this manner, the class of morphisms of augmented differential graded algebras is extended to that of sh-morphisms. In the present paper, we explore small models for equivariant (co)homology via differential homological algebra techniques including homological perturbation theory which, in turn, is a standard tool to handle sh-structures. Koszul duality, for a finite type exterior algebra on odd positive degree generators, then comes down to a duality between the category of sh-modules over that algebra and that of sh-comodules over its reduced bar construction. This kind of duality relies on the extended functoriality of the differential graded Tor-, Ext-, Cotor-, and Coext functors, extended to the appropriate sh-categories. We construct the small models as certain twisted tensor products and twisted Hom-objects. These are chain and cochain models for the chains and cochains on geometric bundles and are compatible with suitable additional structure.
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Equivariant cohomology over Lie groupoids and Lie-Rinehart algebras
Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general Lie-Rinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the associated standard constructions. This extends a characterization of equivariant de Rham cohomology in terms of derived functors developed earlier for the special case where the Lie groupoid is an ordinary Lie group, viewed as a Lie groupoid with a single object; in that theory over a Lie group, the ordinary Bott-Dupont-Shulman-Stasheff complex arises as an a posteriori object. We prove that, given a locally trivial Lie groupoid G and a smooth G-manifold f over the space B of objects of G, the resulting G-equivariant de Rham theory of f boils down to the ordinary equivariant de Rham theory of a vertex manifold relative to the corresponding vertex group, for any vertex in the space B of objects of G; this implies that the equivariant de Rham cohomology introduced here coincides with the stack de Rham cohomology of the associated transformation groupoid whence this stack de Rham cohomology can be characterized as a relative derived functor. We introduce a notion of cone on a Lie-Rinehart algebra and in particular that of cone on a Lie algebroid. This cone is an indispensable tool for the description of the requisite monads.
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Line bundles on moduli and related spaces
Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the principal G-circle bundles with connection on P having the given relative 2-form as curvature. Given a compact Lie group K, a biinvariant Riemannian metric on K, and a closed Riemann surface S of genus s, when we apply the construction to the particular case where f is the familiar relator map from a product of 2s copies of K to K we obtain the principal K-circle bundles on the associated extended moduli spaces which, via reduction, then yield the corresponding line bundles on possibly twisted moduli spaces of representations of the fundamental group of S in K, in particular, on moduli spaces of semistable holomorphic vector bundles or, more precisely, on a smooth open stratum when the moduli space is not smooth. The construction also yields an alternative geometric object, distinct from the familiar gerbe, representing the fundamental class in the third integral cohomology group of K or, equivalently, the first Pontrjagin class of the classifying space of K.
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Relative homological algebra, equivariant de Rham theory, and Koszul duality
Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham cohomology in terms of a suitable relative differential graded Ext, defined on the appropriate category of (G,Cg)-modules. The meaning of "relative" is made precise via the dual standard construction associated with the monad involving the aforementioned functor and the associated forgetful functor. The corresponding infinitesimal equivariant cohomology is the relative differential Ext over Cg relative to g. The functor under discussion decomposes into two functors, the functor which determines differentiable cohomology in the sense of Hochschild-Mostow and the functor which determines the infinitesimal equivariant theory, suitably interpreted. This functor decomposition, in turn, entails an extension of a Decomposition Lemma due to Bott. Appropriate models for the differential graded Ext involving a comparison between a suitably defined simplicial Weil coalgebra and the Weil coalgebra dual to the familiar ordinary Weil algebra yield small models for equivariant de Rham cohomology including the standard Weil and Cartan models for the special case where the group G is compact and connected. Koszul duality in de Rham theory results from these considerations in a straightforward manner.
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Kirillov's character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing
Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric. The complexification G of K inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and left and right translation turn the Hilbert space of square-integrable holomorphic functions on G relative to a suitable measure into a unitary (KxK)-representation. We establish the statement of the Peter-Weyl theorem for this Hilbert space to the effect that this Hilbert space contains the vector space of representative functions on G as a dense subspace and that the assignment to a holomorphic function of its Fourier coefficients yields an isomorphism of Hilbert algebras from the convolution algebra on G onto the algebra which arises from the endomorphism algebras of the irreducible representations of K by the appropriate operation of Hilbert space sum. Consequences are (i) a holomorphic Plancherel theorem and the existence of a uniquely determined unitary isomorphism between the space of square-integrable functions on K and the Hilbert space of holomorphic functions on G, and (ii) a proof that this isomorphism coincides with the corresponding Blattner-Kostant-Sternberg pairing map, multiplied by a suitable constant. We then show that the spectral decomposition of the energy operator on the Hilbert space of holomorphic functions on G associated with the metric on K refines to the Peter-Weyl decomposition of this Hilbert space in the usual manner and thus yields the decomposition of this Hilbert space into irreducible isotypical (KxK)-representations. Among our crucial tools is Kirillov's character formula. Our methods are geometric and independent of heat kernels, which are used by B. C. Hall to obtain many of these results [Journal of Functional Analysis 122 (1994), 103--151], [Comm. in Math. Physics 226 (2002), 233--268].
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On the construction of A-infinity structures
We relate a construction of Kadeishvili's establishing an A-infinity-structure on the homology of a differential graded algebra or more generally of an A-infinity algebra with certain constructions of Chen and Gugenheim. Thereafter we establish the links of these constructions with subsequent developments.
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The universal Hopf algebra associated with a Hopf-Lie-Rinehart algebra
We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure.
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The Lie algebra perturbation lemma
Let g be a differential graded Lie algebra and suppose given a contraction of chain complexes of g onto a general chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S" to the given Lie algebra g, and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on g onto S" which is natural in the data. This extends a result established in a joint paper of the author with J. Stashef [Forum math. 14 (2002), 847-868, math.AG/9906036] where only the particular where M is the homology of g has been explored.
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Origins and breadth of the theory of higher homotopies
Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. We recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940's. Prompted by the failure of the Alexander-Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor's classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of An-spaces and Ainfty-spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950's, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic observation is that higher homotopy structures behave much better relative to homotopy than strict structures, and HPT enables one to exploit this observation in various concrete situations which, in particular, leads to the effective calculation of various invariants which are otherwise intractable. Higher homotopies abound but they are rarely recognized explicitly and their significance is hardly understood; at times, their appearance might at first glance even come as a surprise, for example in the Kodaira-Spencer approach to deformations of complex manifolds or in the theory of foliations.
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Stratified Kaehler structures on adjoint quotients
Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and Kaehler reduction with reference to the adjoint action yields a stratified Kaehler structure on the resulting adjoint quotient. Exploiting classical invariant theory, in particular bisymmetric functions and variants thereof, we explore the singular Poisson-Kaehler geometry of this quotient. Among other things we prove that, for various compact groups, the real coordinate ring of the adjoint quotient is generated, as a Poisson algebra, by the real and imaginary parts of the fundamental characters. We also show that singular Kaehler quantization of the geodesic flow on the reduced level yields the irreducible algebraic characters of the complexified group.
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Classical phase space singularities and quantization
Simple classical mechanical systems and solution spaces of classical field theories involve singularities. In certain situations these singularities can be understood in terms of stratified Kaehler spaces. We give an overview of a research program whose aim is to develop a holomorphic quantization procedure on stratified Kaehler spaces.
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Singular Poisson-Kaehler geometry of certain adjoint quotients
The Kaehler quotient of a complex reductive Lie group relative to the conjugation action carries a complex algebraic stratified Kaehler structure which reflects the geometry of the group. For the group SL(n,C), we interpret the resulting singular Poisson-Kaehler geometry of the quotient in terms of complex discriminant varieties and variants thereof.
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Slices for lifted tangent and cotangent actions
Given a Lie group G, a G-manifold M, and a point b of M with compact stabilizer, we construct slices for the lifted tangent and cotangent actions at a pre-image of b in terms of a slice for the G-action on M at the point b. We interpret the slice for the lifted cotangent action in terms of a symplectic slice and in terms of a Witt-Artin-decomposition.
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Minimal free multi models for chain algebras
Let R be a local ring and A a connected differential graded algebra over R which is free as a graded R-module. Using homological perturbation theory techniques, we construct a minimal free multi model for A having properties similar to that of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute multi refers to the category of multicomplexes.
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Singular Poisson-Kaehler geometry of Scorza varieties and their secant varieties
Each Scorza variety and its secant varieties in the ambient projective space are identified, in the realm of singular Poisson-Kaehler geometry, in terms of projectivizations of holomorphic nilpotent orbits in suitable Lie algebras of hermitian type, the holomorphic nilpotent orbits, in turn, being affine varieties. The ambient projective space acquires an exotic Kaehler structure, the closed stratum being the Scorza variety and the closures of the higher strata its secant varieties. In this fashion, the secant varieties become exotic projective varieties. In the rank 3 case, the four regular Scorza varieties coincide with the four critical Severi varieties. In the standard cases, the Scorza varieties and their secant varieties arise also via Kaehler reduction. An interpretation in terms of constrained mechanical systems is included.
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Kaehler quantization and reduction
Exploiting a notion of Kaehler structure on a stratified space introduced elsewhere we show that, in the Kaehler case, reduction after quantization coincides with quantization after reduction: Key tools developed for that purpose are stratified polarizations and stratified prequantum modules, the latter generalizing prequantum bundles. These notions encapsulate, in particular, the behaviour of a polarization and that of a prequantum bundle across the strata. Our main result says that, for a positive Kaehler manifold with a hamiltonian action of a compact Lie group, when suitable additional conditions are imposed, reduction after quantization coincides with quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond but the (invariant) unreduced and reduced quantum observables as well. Over a stratified space, the appropriate quantum phase space is a costratified Hilbert space in such a way that the costratified structure reflects the stratification. Examples of stratified Kaehler spaces arise from the closures of holomorphic nilpotent orbits including angular momentum zero reduced spaces, and from representations of compact Lie groups. For illustration, we carry out Kaehler quantization on various spaces of that kind including singular Fock spaces.
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Higher homotopies and Maurer-Cartan algebras: Quasi-Lie-Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras
Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber-, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra-the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra-and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector fields on the "space of leaves" and for the algebra of multivector fields are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including as well the Hodge-de Rham spectral sequence.
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