K-Theory and Homology
In this paper we prove that Loday--Quillen--Tsygan Theorem generalizes to the case of coalgebras. Specifically, we show that the Chevalley--Eilenberg--Lie homology of the Lie coalgebra of infinite matrices over a coassociative coalgebra $C$…
We present a generalized Lyapunov Schmidt reduction scheme for diffeomorphisms living on a finite dimensional real vector space V which transform under real one dimensional characters of an arbitrary compact group with linear action V.…
We survey some of the known results on the relation between the homology of the {\em full} Hecke algebra of a reductive $p$-adic group $G$, and the representation theory of $G$. Let us denote by $\CIc(G)$ the full Hecke algebra of $G$ and…
Let F be a nonarchimedean local field and let GL(N) = GL(N,F). We prove the existence of parahoric types for GL(N). We construct representative cycles in all the homology classes of the chamber homology of GL(3).
We show that the graded commutative ring structure of the Hochschild cohomology HH*(A) is trivial in case A is a triangular quadratic string algebra. Moreover, in case A isgentle, the Lie algebra structure on HH*(A) is also trivial.
We define and study an equivariant $E$-theory with respect to coactions of Hopf $C^*$-algebras; we prove the Baaj-Skandalis duality in this setting. We show that the corresponding equivariant $KK$-theory of Baaj and Skandalis enjoys an…
In the paper we describe the C*-algebras of noncommutative spherical tight frames over some C*-algebras and then apply to study the noncommutative version of the universal classifying space.
This is the second part of the article [math.KT/0408094]. In the first paper, we used the underlying coalgebra structure to develop a cyclic theory. In this paper we define a dual theory by using the algebra structure. We define a cyclic…
The Z_2-graded Cech cohomology theory is considered in the framework of noncommutative geometry over complex number field and in particular the homotopy invariance and Morita invariance are proven.
Certain cocycles constructed by Connes are characters of $p$-summable Fredholm modules. In this article, we establish some consequences of the universal properties which these characters enjoy. Our main technical result is that the entire…
These notes aim to give an introduction to a few aspects of noncommutative geometry.
Let k be a characteristic zero field, C a k-algebra and M a square zero two sided ideal of C. We obtain a new mixed complex, simpler that the canonical one, giving the Hochschild and cyclic homologies of C relative to M. This complex…
The cyclic (co)homology of Hopf algebras is defined by Connes and Moscovici [math.DG/9806109] and later extended by Khalkhali et.al [math.KT/0306288] to admit stable anti-Yetter-Drinfeld coefficient module/comodules. In this paper we will…
We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result…
We construct a cofibrantly generated Quillen model structure on the category of small differential graded categories. ----- Nous construisons une structure de categorie de modeles de Quillen a engendrement cofibrant sur la categorie des…
Dieter Happel asked the following question: If the $n$-th Hochschild cohomology group of a finite dimensional algebra $\Gamma$ over a field vanishes for all sufficiently large $n$, is the global dimension of $\Gamma$ finite? We give a…
Twisted complex $K$-theory can be defined for a space $X$ equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C$^*$-algebras. Up to equivalence, the twisting corresponds to an element of $H^3(X;\Z)$. We…
In this paper we develop the obstruction theory for lifting complexes, up to quasi-isomorphism, to derived categories of flat nilpotent deformations of abelian categories. As a particular case we also obtain the corresponding obstruction…
We show that the cyclotomic trace map for smooth varieties over number rings can be interpreted as a regulator map and hence are related to special values of $\zeta$-functions.
This is an overview of results from our experiment of merging two seemingly unrelated disciplines - higher algebraic K-theory of rings and the theory of lattice polytopes. The usual K-theory is the ``theory of a unit simplex''. A conjecture…