Related papers: Higher localized analytic indices and strict defor…

The first main result of this paper is to prove that the convergence of Lott's delocalized eta invariant holds for all differential operators with a sufficiently large spectral gap at zero. Furthermore, to each delocalized cyclic cocycle,…

In this paper we shall show the equivalence of algebraic and analytic localisation for algebras of smooth deformation quantization for several situations. The proofs are based on old work by Whitney, Malgrange and Tougeron on the…

We develop here a concept of deformed algebras and their related groups through two examples. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of…

In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the…

In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper etale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it…

We provide the geometrical meaning of the ${\cal N}=4$ superconformal index. With this interpretation, the ${\cal N}=4$ superconformal index can be realized as the partition function on a Scherk-Schwarz deformed background. We apply the…

Let H be a graded Hecke algebra with complex deformation parameters and Weyl group W. We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to…

In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then…

We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation.…

Let G be a semisimple algebraic group over a field k. We introduce the higher Tits indices of G as the set of all Tits indices of G over all field extensions K/k. In the context of quadratic forms this notion coincides with the notion of…

Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $\eta\colon Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the associated…

In this note, we use give some algebraic applications of a previous result by the author which compares the deformations parameterized by the Maurer-Cartan elements of a differential graded Lie algebra, and a differential graded Lie…

A strict quantization of a compact symplectic manifold $S$ on a subset $I\subseteq\R$, containing 0 as an accumulation point, is defined as a continuous field of $C^*$-algebras $\{A_{\hbar}\}_{\hbar\in I}$, with $A_0=C_0(S)$, and a set of…

Boundary groupoids were introduced by the second author, which can be used to model many analysis problems on singular spaces. In order to investigate index theory on boundary groupoids, we introduce the notion of {\em a deformation from…

We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss (2009) who relied on the…

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product…

In [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral $\int_{X}\tau^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ for a flag manifold $X=\Fl_{r_1, ..., r_I}({\Bbb C}^n)$ via an extended mirror principle…

We prove that if $\Gamma$ is a group of polynomial growth then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocyle we thus define a higher analogue of Lott's…

This is a survey of the relationship between C*-algebraic deformation quantization and the tangent groupoid in noncommutative geometry, emphasizing the role of index theory. We first explain how C*-algebraic versions of deformation…

We develop a deformation framework for $C^*$-algebras equipped with a coaction of a locally compact quantum group, formulated intrinsically at the level of spectral subspaces determined by the coaction. The construction is defined…