Endomorphism kernel property for finite groups

summary:A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $\theta$ on $G$ is the kernel of an endomorphism on $G$. In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP

Endomorphisms and automorphisms of locally covariant quantum field theories

In the framework of locally covariant quantum field theory, a theory is described as a functor from a category of spacetimes to a category of *-algebras. It is proposed that the global gauge group of such a theory can be identified as the group of automorphisms of the defining functor. Consequently, multiplets of fields may be identified at the functorial level. It is shown that locally covariant theories that obey standard assumptions in Minkowski space, including energy compactness, have no proper endomorphisms (i.e., all endomorphisms are automorphisms) and have a compact automorphism group. Further, it is shown how the endomorphisms and automorphisms of a locally covariant theory may, in principle, be classified in any single spacetime. As an example, the endomorphisms and automorphisms of a system of finitely many free scalar fields are completely classified.Comment: v2 45pp, expanded to include additional results; presentation improved and an error corrected. To appear in Rev Math Phy

Classification of actions of discrete Kac algebras on injective factors

We will study two kinds of actions of a discrete amenable Kac algebra. The first one is an action whose modular part is normal. We will construct a new invariant which generalizes a characteristic invariant for a discrete group action, and we will present a complete classification. The second is a centrally free action. By constructing a Rohlin tower in an asymptotic centralizer, we will show that the Connes-Takesaki module is a complete invariant.Comment: 120 pages. Minor correction

Elliptic double affine Hecke algebras

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the $\tilde{C}_n$ version of the construction to construct a flat noncommutative deformation of the $n$th symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.Comment: 134 pages. v2: Added results on centers and generic Morita equivalence, plus a description of a conjectural construction of deformed Hilbert scheme

Curtis homomorphisms and the integral Bernstein center for GL_n

We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for GL_n(F) (that is, the center of the category of smooth W(k)[GL_n(F)]-modules, for F a p-adic field and k an algebraically closed field of characteristic l different from p) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for m at most n) implies the strong version of the conjecture. In a companion paper [HM] we show that the strong conjecture for n-1 implies the weak conjecture for n; thus the two papers together give an inductive proof of both conjectures. The upshot is a description of the integral Bernstein center for GL_n in purely Galois-theoretic terms; previous work of the author shows that such a description implies the conjectural "local Langlands correspondence in families" of Emerton and the author.Comment: Final accepted version. 36 pages. Note that the published version of Helm-Moss, "Converse theorems and the local Langlands correspondence in families" references an earlier version of this paper, and the section numbering has changed; in particular sections 9,10, and 11 of the referenced version correspond to sections 8,9, and 10 of the current version, respectivel

Morita Contexts, Idempotents, and Hochschild Cohomology - with Applications to Invariant Rings -

We investigate how to compare Hochschild cohomology of algebras related by a Morita context. Interpreting a Morita context as a ring with distinguished idempotent, the key ingredient for such a comparison is shown to be the grade of the Morita defect, the quotient of the ring modulo the ideal generated by the idempotent. Along the way, we show that the grade of the stable endomorphism ring as a module over the endomorphism ring controls vanishing of higher groups of selfextensions, and explain the relation to various forms of the Generalized Nakayama Conjecture for Noetherian algebras. As applications of our approach we explore to what extent Hochschild cohomology of an invariant ring coincides with the invariants of the Hochschild cohomology.Comment: 28 pages, uses conm-p-l.sty. To appear in Contemporary Mathematics series volume (Conference Proceedings for Summer 2001 Grenoble and Lyon conferences, edited by: L. Avramov, M. Chardin, M. Morales, and C. Polini

Tensor envelopes of regular categories

We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows: 1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N. 3. We determine all degree functions for which T(A,d) is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups S_n, the hyperoctahedral groups S_n\semidir Z_2^n, or the general linear groups GL(n,F_q) over a fixed finite field. This paper generalizes work of Deligne, who first constructed the interpolating category for the symmetric groups S_n. It also extends (and provides proofs for) a previous paper math.CT/0605126 on the special case of abelian categories.Comment: v1: 52 pages; v2: 52 pages, proof of Lemma 7.2 fixed, otherwise minor change