scholarly journals Residues and homology for pseudodifferential operators on foliations

2004 ◽  
Vol 94 (1) ◽  
pp. 75 ◽  
Author(s):  
M.-T. Benameur ◽  
V. Nistor
Keyword(s):  
Diophantine Approximation ◽  
Pseudodifferential Operators ◽  
Hochschild Homology ◽  
Foliated Manifold ◽  
Homology Groups ◽  
General Properties ◽  
Poisson Homology ◽  
Foliated Manifolds

We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras of pseudodifferential operators along the leaves of a foliation.

scholarly journals Homology and residues of adiabatic pseudodifferential operators

2004 ◽  
Vol 175 ◽  
pp. 171-221 ◽  
Author(s):  
Sergiu Moroianu
Keyword(s):  
Existence And Uniqueness ◽  
Pseudodifferential Operators ◽  
Hochschild Homology ◽  
Total Space ◽  
Homology Groups ◽  
Noncommutative Residue

AbstractWe compute the Hochschild homology groups of the adiabatic algebra Ψa(X), a deformation of the algebra of pseudodifferential operators Ψ(X) when X is the total space of a fibration of closed manifolds. We deduce the existence and uniqueness of traces on Ψa(X) and some of its ideals and quotients, in the spirit of the noncommutative residue of Wodzicki and Guillemin. We introduce certain higher homological versions of the residue trace. When the base of the fibration is S1, these functionals are related to the η function of Atiyah-Patodi-Singer.

scholarly journals TRIPLY-GRADED LINK HOMOLOGY AND HOCHSCHILD HOMOLOGY OF SOERGEL BIMODULES

2007 ◽  
Vol 18 (08) ◽  
pp. 869-885 ◽  
Author(s):  
MIKHAIL KHOVANOV
Keyword(s):  
Group Action ◽  
Braid Group ◽  
Polynomial Algebra ◽  
Homology Theory ◽  
Hochschild Homology ◽  
Polynomial Algebras ◽  
Homology Groups ◽  
Link Homology ◽  
Soergel Bimodules ◽  
Braid Group Action

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

scholarly journals Hochschild Homology Groups of System Quiver Algebras of Minimal Wild Type

2012 ◽  
Vol 25 ◽  
pp. 2081-2088
Author(s):  
Zhibing Liu ◽  
Huixia Wang ◽  
Guiju Liu ◽  
Fang He
Keyword(s):  
Hochschild Homology ◽  
Wild Type ◽  
Homology Groups

TRANSVERSALLY BIHARMONIC MAPS BETWEEN FOLIATED RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 19 (08) ◽  
pp. 981-996 ◽  
Author(s):  
YUAN-JEN CHIANG ◽  
ROBERT A. WOLAK
Keyword(s):  
Riemannian Manifolds ◽  
Positive Constant ◽  
Conservation Law ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Biharmonic Maps ◽  
Foliated Manifold ◽  
Transverse Curvature ◽  
Biharmonic Map ◽  
Foliated Manifolds

We generalize the notions of transversally harmonic maps between foliated Riemannian manifolds into transversally biharmonic maps. We show that a transversally biharmonic map into a foliated manifold of non-positive transverse curvature is transversally harmonic. Then we construct examples of transversally biharmonic non-harmonic maps into foliated manifolds of positive transverse curvature. We also prove that if f is a stable transversally biharmonic map into a foliated manifold of positive constant transverse sectional curvature and f satisfies the transverse conservation law, then f is a transversally harmonic map.

scholarly journals An application of the Morse theory to foliated manifolds

1974 ◽  
Vol 54 ◽  
pp. 165-178 ◽  
Author(s):  
Kazuhiko Fukui
Keyword(s):  
Critical Points ◽  
Morse Theory ◽  
Topological Properties ◽  
Foliated Manifold ◽  
Variation Equation ◽  
Codimension One ◽  
Foliated Manifolds ◽  
The Given

In [5], R. Thorn has started the study of the foliated structures by using the Morse theory. Recently K. Yamato [7] has studied the topological properties of leaves of a codimension one foliated manifold by investigating the “critical points” of variation equation of the given one-form.

scholarly journals On (co)homology of Frobenius Poisson algebras

2014 ◽  
Vol 14 (2) ◽  
pp. 371-386 ◽  
Author(s):  
Can Zhu ◽  
Fred Van Oystaeyen ◽  
Yinhuo Zhang
Keyword(s):  
Bilinear Form ◽  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Poisson Algebra ◽  
Hochschild Homology ◽  
Poisson Algebras ◽  
Poisson Homology ◽  
Poisson Cohomology ◽  
Frobenius Algebras

AbstractIn this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisky algebra.

An equality on Hochschild homology groups

1997 ◽  
Vol 42 (8) ◽  
pp. 624-627 ◽  
Author(s):  
Pu Zhang ◽  
Shaoxue Liu
Keyword(s):  
Hochschild Homology ◽  
Homology Groups

scholarly journals On the group of diffeomorphisms of foliated manifolds

2020 ◽  
Vol 30 (1) ◽  
pp. 49-58
Author(s):  
A.Ya. Narmanov ◽  
A.N. Zoyidov
Keyword(s):  
Differential Geometry ◽  
Lie Group ◽  
Closed Subgroup ◽  
Compact Open Topology ◽  
Topological Groups ◽  
Foliated Manifold ◽  
Group Diff ◽  
Group Of Diffeomorphisms ◽  
Foliated Manifolds

Now the foliations theory is intensively developing branch of modern differential geometry, there are numerous researches on the foliation theory. The purpose of our paper is study the structure of the group DiffF(M) of diffeomorphisms and the group IsoF(M) of isometries of foliated manifold (M,F). It is shown the group DiffF(M) is closed subgroup of the group Diff(M) of diffeomorphisms of the manifold M in compact-open topology and also it is proven the group IsoF(M) is Lie group. It is introduced new topology on DiffF(M) which depends on foliation F and called F- compact open topology. It's proven that some subgroups of the group DiffF(M) are topological groups with F-compact open topology.

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