On the cohomology of the holomorph of a finite cyclic group
The mod 2 cohomology algebra of the holomorph of any finite cyclic group whose order is a power of 2 is determined.
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arXiv subjects
Johannes Huebschmann
Publications and source records attributed to Johannes Huebschmann.
Duality for Lie-Rinehart algebras and the modular class
We introduce a notion of duality for a Lie-Rinehart algebra giving certain bilinear pairings in its cohomology generalizing the usual notions of Poincaré duality in Lie algebra cohomology and de Rham cohomology. We show that the duality isomorphisms can be given by a cap product with a suitable fundamental class and hence may be taken natural in any reasonable sense. Thereafter, for a Lie-Rinehart algebra satisfying duality, we introduce a certain intrinsic module which provides a crucial ingredient for the construction of the bilinear pairings. This module determines a certain class, called modular class of the Lie-Rinehart algebra, which lies in a certain Picard group generalizing the abelian group of flat line bundles on a smooth manifold. Finally, we show that a Poisson algebra having suitable properties determines a certain module for the corrresponding Lie-Rinehart algebra and hence modular class whose square yields the module and characteristic class for its Lie-Rinehart algebra mentioned before. In particular, this gives rise to certain bilinear pairings in Poisson homology.
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Severi varieties and holomorphic nilpotent orbits
Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic--the chordal variety--in the corresponding complex projective space; the cubic and projective space are identified in terms of holomorphic nilpotent orbits. The projective space acquires an exotic Kähler structure with three strata, the cubic is an example of an exotic projective variety with two strata, and the corresponding Severi variety is the closed stratum in the exotic variety as well as in the exotic projective space. In the standard cases, these varieties arise also via Kähler reduction. An interpretation in terms of constrained mechanical systems is included.
math.DG↗
Lie-Rinehart algebras, descent, and quantization
A Lie-Rinehart algebra consists of a commutative algebra and a Lie algebra with additional structure which generalizes the mutual structure of interaction between the algebra of functions and the Lie algebra of smooth vector fields on a smooth manifold. Lie-Rinehart algebras provide the correct categorical language to solve the problem whether Kaehler quantization commutes with reduction which, in turn, may be seen as a descent problem.
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Kaehler spaces, nilpotent orbits, and singular reduction
For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, Jordan triple systems, symmetric domains, and pre-homogeneous spaces; in particular, in the world of singular Poisson geometry, the closures of principal holomorphic nilpotent orbits, positive definite hermitian JTS's, and certain pre-homogeneous spaces appear as different incarnations of the same structure. The space of representations of the fundamental group of a closed surface in a compact Lie group inherits a (positive) normal (stratified) Kaehler structure, as does the closure of a holomorphic nilpotent orbit in a semisimple Lie algebra of hermitian type. The closure of the principal holomorphic nilpotent orbit arises from a regular semisimple holomorphic orbit by contraction. Symplectic reduction carries a (positive) Kaehler manifold to a (positive) normal Kaehler space in such a way that the sheaf of germs of polarized functions thereupon coincides with the ordinary sheaf of germs of holomorphic functions. Symplectic reduction establishes a close relationship between singular reduced spaces and nilpotent orbits of the dual groups. Projectivization of holomorphic nilpotent orbits yields exotic stratified Kaehler structures on complex projective spaces and on certain complex projective varieties including complex projective quadrics. Physical examples are provided by certain reduced spaces arising from angular momentum zero.
math.DG↗
Formal solution of the master equation via HPT and deformation theory
We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant Lie algebra. To this end we endow the homology H(g) of any differential graded Lie algebra g with an sh-Lie structure such that g and H(g) are sh-equivalent. We discuss our solution of the master equation in the context of deformation theory. Given the extra structure appropriate to the extended moduli space of complex structures on a Calabi-Yau manifold, the known solutions result as a special case.
math.AG↗
Singularities and Poisson geometry of certain representation spaces
Playing off against each other the real and complex structures, we elucidate the local structure of certain representation spaces in the world of Poisson geometry. Particular cases of these spaces arise as moduli spaces of semistable holomorphic vector bundles on Riemann surfaces.
math.DG↗
Rinehart complexes and Batalin-Vilkovisky algebras
For a Lie-Rinehart algebra (A,L) such that, as an A-module, L is finitely generated and projective of finite constant rank, the relationship between generators of the Gerstenhaber bracket and connections on the highest A-exterior power of L given in an earlier paper arises from the canonical pairing between the exterior A-powers of L. Thus, given an exact generator for the corresponding Gerstenhaber algebra, the chain complex underlying the resulting Batalin-Vilkovisky algebra coincides with the Rinehart complex computing the corresponding Lie-Rinehart homology.
math.DG↗
Berikashvili's functor D and the deformation equation
Berikashvili's functor D defined in terms of twisting cochains is related to deformation theory, gauge theory, Chen's formal power series connections, and the master equation in physics. The idea is advertised that some unification and understanding of the links between these topics is provided by the notion of twisting cochain and the idea of classifying twisting cochains.
math.AT↗
Twilled Lie-Rinehart algebras and differential Batalin-Vilkovisky algebras
Twilled L(ie)-R(inehart) algebas generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an almost twilled pre-LR algebra, which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR-structures in terms of certain associated differential (bi)graded Lie and G(erstenhaber)-algebras; in particular, the G-algebra arising from an almost complex structure is a d(ifferential) G-algebra iff the almost complex structure is integrable. Such G-algebras, endowed with a generator turning them into a B(atalin)-V(ilkovisky) algebra, occur on the B-side of the mirror conjecture. We generalize a result of Koszul to those dG-algebras which arise from twilled LR-algebras. A special case thereof explains the relationship between holomorphic volume forms and exact generators for the corresponding dG-algebras and thus yields in particular a conceptual proof of the Tian-Todorov lemma. We give a differential homological algebra interpretation for twilled LR-algebras and by means of it we elucidate the notion of generator in terms of homological duality for differential graded LR-algebras. Finally we indicate how some of our results might be globalized by means of Lie groupoids.
math.DG↗
On the variation of the Poisson structures of certain moduli spaces
Given a Lie group G whose Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, we construct a Poisson algebra of continuous functions on a certain open subspace R of the space of representations in G of the fundamental group of a compact connected orientable topological surface with finitely many boundary circles; when G is compact and connected, R may be taken dense in the space of all representations. The space R contains spaces of representations where the values of those generators of the fundamental group which correspond to the boundary circles are constrained to lie in fixed conjugacy classes and, on these representation spaces, the Poisson algebra restricts to stratified symplectic Poisson algebras constructed elsewhere earlier. Hence the Poisson algebra on R gives a description of the variation of the stratified symplectic Poisson structures on the smaller representation spaces as the chosen conjugacy classes move.
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Extensions of Lie-Rinehart algebras and the Chern-Weil construction
A Chern-Weil construction for extensions of Lie-Rinehart algebras is introduced. This generalizes the classical Chern-Weil construction in differential geometry and yields characteristic classes for arbitrary extensions of Lie-Rinehart algebras. Some examples arising from spaces with singularities and from foliations are given that cannot be treated by means of the classical Chern-Weil construction.
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Lie-Rinehart algebras, Gerstenhaber algebras, and B-V algebras
For a Lie-Rinehart algebra (A,L), generators for the Gerstenhaber algebra Λ_A L correspond bijectively to right (A,L)-connections on A in such a way that B-V structures correspond to right (A,L)-module structures on A. When L is projective as an A-module, given an exact generator \partial, the homology of the B-V algebra (Λ_A L,\partial) coincides with that of L with coefficients in A with respect to the right (A,L)-module structure determined by \partial. When L is also of finite rank n, there are bijective correspondences between (A,L)-connections on Λ_A^nL and right (A,L)-connections on A and between left (A,L)- module structures on Λ_A^nL and right (A,L)-module structures on A. Hence there are bijective correspondences between (A,L)-connections on Λ_A^n L and generators for the Gerstenhaber bracket on Λ_A L and between (A,L)-module structures on Λ_A^n L and B-V algebra structures on Λ_A L. The homology of such a B-V algebra (Λ_A L,\partial) coincides with the cohomology of L with coefficients in Λ_A^n L, for the left (A,L)-module structure determined by \partial. Some applications are discussed.
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Extended moduli spaces and the Kan construction.II.Lattice gauge theory
Let $Y$ be a CW-complex with a single 0-cell, $K$ its Kan group, a model for the loop space of $Y$, and let $G$ be a compact, connected Lie group. We give an explicit finite dimensional construction of generators of the equivariant cohomology of the geometric realization of the cosimplicial manifold $\roman{Hom}(K,G)$ and hence of the space $\roman{Map}^o(Y,BG)$ of based maps from $Y$ to the classifying space $BG$. For a smooth manifold $Y$, this may be viewed as a rigorous approach to lattice gauge theory, and we show that it then yields, (i) when {$\roman{dim}(Y)=2$,} equivariant de Rham representatives of generators of the equivariant cohomology of twisted representation spaces of the fundamental group of a closed surface including generators for moduli spaces of semi stable holomorphic vector bundles on complex curves so that, in particular, the known structure of a stratified symplectic space results; (ii) when {$\roman{dim}(Y)=3$,} equivariant cohomology generators including the Chern-Simons function; (iii) when {$\roman{dim}(Y) = 4$,} the generators of the relevant equivariant cohomology from which for example Donaldson polynomials are obtained by evaluation against suitable fundamental classes corresponding to moduli spaces of ASD connections.
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Extended moduli spaces and the Kan construction
Let $Y$ be a CW-complex with a single 0-cell, let $K$ be its Kan group, a free simplicial group whose realization is a model for the space $ΩY$ of based loops on $Y$, and let $G$ be a Lie group, not necessarily connected. By means of simplicial techniques involving fundamental results of {\smc Kan's} and the standard $W$- and bar constructions, we obtain a weak $G$-equivariant homotopy equivalence from the geometric realization $|\roman{Hom}(K,G)|$ of the cosimplicial manifold $\roman{Hom}(K,G)$ of homomorphisms from $K$ to $G$ to the space $\roman{Map}^o(Y,BG)$ of based maps from $Y$ to the classifying space $BG$ of $G$ where $G$ acts on $BG$ by conjugation. Thus when $Y$ is a smooth manifold, the universal bundle on $BG$ being endowed with a universal connection, the space $|\roman{Hom}(K,G)|$ may be viewed as a model for the space of based gauge equivalence classes of connections on $Y$ for all topological types of $G$-bundles on $Y$ thereby yielding a rigorous approach to lattice gauge theory; this is illustrated in low dimensions.
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Symplectic and Poisson structures of certain moduli spaces. II. Projective representations of cocompact discrete planar groups
Let $G$ be a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction carried out in an earlier paper for the fundamental group of a closed surface may be extended to an arbitrary infinite orientation preserving cocompact planar discrete group of euclidean or non-euclidean motions $π$ and yields (i) a symplectic structure on a certain smooth manifold $\Cal M$ containing the space $\roman{Hom}(π,G)$ of homomorphisms and, furthermore, (ii) a hamiltonian $G$-action on $\Cal M$ preserving the symplectic structure together with a momentum mapping in such a way that the reduced space equals the space $\roman{Rep}(π,G)$ of representations. More generally, the construction also applies to certain spaces of projective representations. For $G$ compact, the resulting spaces of representations inherit structures of {\it stratified symplectic space\/} in such a way that the strata have finite symplectic volume . For example, {\smc Mehta-Seshadri} moduli spaces of semistable holomorphic parabolic bundles with rational weights or spaces closely related to them arise in this way by {\it symplectic reduction in finite dimensions\/}.
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Smooth structures on certain moduli spaces for bundles on a surface
Let $Σ$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, $ξ\colon P \to Σ$ a principal $G$-bundle, let $N(ξ)$ denote the moduli space of central Yang-Mills connections on $ξ$, for suitably chosen additional data, and let $\roman{Rep}_ξ(Γ,G)$ be the space of representations of the universal central extension $Γ$ of the fundamental group of $Σ$ in $G$ that corresponds to $ξ$. We construct smooth structures on $N(ξ)$ and $\roman{Rep}_ξ(Γ,G)$ and show that the assignment to a smooth connection $A$ of its holonomies with reference to suitable closed paths yields a diffeomorphism from $N(ξ)$ onto $\roman{Rep}_ξ(Γ,G)$; moreover we show that the derivative of the latter at the non-singular points of $N(ξ)$ amounts to a certain twisted integration mapping. Finally we examine the infinitesimal geometry of these moduli spaces with reference to the smooth structures and, for illustration, we show that, on the moduli space of flat $\roman{SU}(2)$-connections for a surface of genus two which, as a space, is just complex projective 3-space, our smooth structure looks rather different from the standard structure.
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Poisson structures on certain moduli spaces for bundles on a surface
Let $Σ$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $ξ\colon P \to Σ$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N(ξ)$ of central Yang- Mills connections, for appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a diffeomorphism from $N(ξ)$ onto a certain representation space $\roman{Rep}_ξ(Γ,G)$, with reference to suitable smooth structures $C^{\infty}(N(ξ))$ and $C^{\infty}(\roman{Rep}_ξ(Γ,G))$ where $Γ$ denotes the universal central extension of the fundamental group of $Σ$. Given an invariant symmetric bilinear form on $g^*$, we construct here Poisson structures on $C^{\infty}(N(ξ))$ and $C^{\infty}(\roman{Rep}_ξ(Γ,G))$ in such a way that the mentioned diffeomorphism identifies them. When the form on $g^*$ is non-degenerate the Poisson structures are compatible with the stratifications where $\roman{Rep}_ξ(Γ,G)$ is endowed with the corresponding stratification and, furthermore, yield structures of a {\it stratified symplectic space\/}, preserved by the induced action of the mapping class group of $Σ$.
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