Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.     

Strong integrality of quantum invariants of 3-manifolds

We prove that the quantum -invariant of an arbitrary 3-manifold is always an algebraic integer if the order of the quantum parameter is co-prime with the order of the torsion part of . An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-spheres. Here we also generalize Habiro's result to all rational homology 3-spheres.

     

Knot homology via derived categories of coherent sheaves II, $\mathfrak{sl}_{m}$ case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov–Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu’s by homological mirror symmetry.     

A unified quantum <Emphasis Type="Italic">SO</Emphasis>(3) invariant for rational homology 3-spheres

Given a rational homology 3-sphere M with |H ₁(M,ℤ)|=b and a link L inside M, colored by odd numbers, we construct a unified invariant I _M,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten–Reshetikhin–Turaev invariants of the pair (M,L). If b=1 and L=∅, I _M coincides with Habiro’s invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined by I _M . Important applications are the new Ohtsuki series (perturbative expansions of I _M ) dominating quantum SO(3) invariants at roots of unity whose order is not a power of a prime. These series are not known to be determined by the LMO invariant.     

Calculating the Number of Functions with a Given Endomorphism

An iterative procedure is proposed for calculating the number of k-valued functions of n variables such that each one has an endomorphism different from any constant and permutation. Based on this procedure, formulas are found for the number of three-valued functions of n variables such that each one has nontrivial endomorphisms. For any arbitrary semigroup of endomorphisms, the power is found of the set of all three-valued functions of n variables such that each one has endomorphisms from a specified semigroup.     

The tail of a positivity-preserving semigroup

With certain assumptions a representation theorem is proved for the elements of σS, where Σ is an abelian semigroup of, endomorphisms of a real vector space, andS is a convex antisymmetric cone. Application is made to chacterization of nonnegative harmonic functions on bounded Lipschitz domains, of Hausdorff-Stieltjes moment sequences, and of “bilateral Laplace transforms” on locally compact abelian groups, Euclidean motion groups, and noncompact semi-simple Lie groups. Uniqueness of the representation is proved in both the Euclidean motion and the semi-simple cases. Alfred P. Sloan Fellow. Research supported by NSF GP 18961.     

Spectral synthesis in spaces invariant under composition semigroups

If φt is a continuously differentiable composition semigroup of analytic endomorphisms of the disk , then all closed subspaces of Hol( ) invariant with respect to this semigroup (acting by compositions) admit spectral synthesis. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 55–77. Translated by V. I. Vasyunin.     

Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds

We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and non-compact spaces, as well as with periodic and non-periodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the non-planar parts of the Green functions is understood simply in terms of off-diagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the non-planar part of the one-loop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing. Communicated by Vincent Rivasseau submitted 22/12/04, accepted 22/03/05     

(0,1)-Matrices and semigroups of ring endomorphisms

In this paper we investigate semigroups of ring endomorphisms of several classes of rings. As one result we find that the Green relations in the endomorphism semigroup, End R, for a ring R in a given class, are restrictions of those in the transformation semigroup .     

On generating countable sets of endomorphisms

Sierpiski proved that every countable set of mappings on an infinite set X is contained in a 2-generated subsemigroup of the semigroup of all mappings on X. In this paper we prove that every countable set of endomorphisms of an algebra which has an infinite basis (independent generating set) is contained in a 2-generated subsemigroup of the semigroup of all endomorphisms of .     

Three-manifold invariants associated with restricted quantum groups

We show a simple relation between Witten–Reshetikhin–Turaev SU(2) invariant and the Hennings invariant associated with the restricted quantum ${{\mathfrak{sl}_{2}}}$ . These invariants are defined in very different methods: the former uses the representation theory of quantum ${{\mathfrak{sl}_{2}}}$ while the latter uses the integral of the Hopf algebra. But they turn out to be the same for most rational homology 3-spheres up to a sign. This relation can be used to prove the integrality of the former invariant.     

On finite groups acting on ℤ<Subscript>2</Subscript>-homology 3-spheres

We consider finite groups G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. It is known that every finite group G admits actions on rational homology 3-spheres (and even free actions). On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted (and close to the class of finite subgroups of the orthogonal group SO(4), acting on the 3-sphere). In the present paper, we consider the intermediate case of ₂-homology 3-spheres (i.e., with the ₂-homology of the 3-sphere where ₂ denote the integers mod two; we note that these occur much more frequently in 3-dimensional topology than the integer ones). Our main result is a list of finite nonsolvable groups G which are the candidates for orientation-preserving actions on ₂-homology 3-spheres. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group or the binary dodecahedral group most of these groups are subgroups of the orthogonal group SO(4) and hence admit actions on S³. Roughly, in the case of ₂-homology 3-spheres the groups PSL(2,5) and SL(2,5) get replaced by the groups PSL(2,q) and SL(2,q), for an arbitrary odd prime power q. We have many examples of actions of the groups PSL(2,q) and SL(2,q) on ₂-homology 3-spheres, for various small values of q (constructed as regular coverings of suitable hyperbolic 3-orbifolds and 3-manifolds, using computer-supported methods to calculate the homology of the coverings). We think that all of them occur but have no method to prove this at present (in particular, the exact classification of the finite nonsolvable groups admitting actions on ₂-homology 3-spheres remains still open).     

The Universal Covering of an Inverse Semigroup

We examine an inverse semigroup T in terms of the universal locally constant covering of its classifying topos . In particular, we prove that the fundamental group of coincides with the maximum group image of T. We explain the connection between E-unitary inverse semigroups and locally decidable toposes, characterize E-unitary inverse semigroups in terms of a kind of geometric morphism called a spread, characterize F-inverse semigroups, and interpret McAlister’s “P-theorem” in terms of the universal covering.     

The crystal commutor and Drinfeld’s unitarized <Emphasis Type="Italic">R</Emphasis>-matrix

Drinfeld defined a unitarized R-matrix for any quantum group . This gives a commutor for the category of representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer’s construction agrees with Drinfeld’s commutor. We then describe the action of Drinfeld’s commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor. P. Tingley was supported by the RTG grant DMS-0354321.     

Zamolodchikov-Faddeev algebras for Yangian doubles at level 1

The representation theory of centrally extended Yangian doubles is investigated. The intertwining operators are constructed for infinite dimensional representations of , which are deformed analogs of the highest weight representations of the affine algebra at level 1. We give bosonized expressions for the intertwining operators and verify that they generate an algebra isomorphic to the Zamolodchikov-Faddeev algebra for the SU(2)-invariant Thirring model. From them, we compose L-operators by Miki’s method and verify that they coincide with L-operators constructed from the universal R-matrix. The matrix elements of the product of these operators are calculated explicitly and are shown to satisfy the quantum (deformed) Knizhnik-Zamolodchikov equation associated with the universal R-matrix for . This paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 25–45. January, 1997.     

Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup

We prove hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup on the entire algebra of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain. Then we deduce a logarithmic Sobolev inequality for the semigroup and gain an elementary estimate of the best constant.     

Mutative hyperbolic homology 3-spheres with the same Floer homology

We construct a large class of finitely many hyperbolic homology 3-spheres making the following invariants equal, simultaneously, the integral homology, the quantum SU(2) invariants, the hyperbolic volume, the hyperbolic isometry group, the -invariant, the Chern-Simons invariant, and the Floer homology.     

Automorphisms of linear semigroups over division rings

Given an-dimensional right vector spaceV over a division ring we denote byS the semigroup of the endomorphisms ofV and designate this semigroup as alinear semigroup. First we prove that every automorphism ofS can be written asT→fTf ⁻¹, wheref∶V→V is a semilinear homeomorphism. Furthermore, we show that every isomorphism between maximal compact subsemigroups ofS is also of this type.     

Homomorphisms of extensions of a semigroup within a variety

LetV be a semigroup variety containing all commutative semigroups such that the law of exponents (xy) ⁿ =xⁿyⁿ fails inV for everyn > 1. For every semigroupS V such that the reflection of the semigroup obtained fromS by an adding unity has only one idempotent there exists a semigroupT V extendingS without non-trivial endomorphisms. In more general, the full subcategory ofV formed by all extensions ofS withinV is universal.Presented by B. M. Schein.     

A quasianalyticity property for monogenic solutions of small divisor problems

We discuss the quasianalytic properties of various spaces of functions suit-able for one-dimensional small divisor problems. These spaces are formed of functions ¹-holomorphic on certain compact sets K _j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K _j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K _j ’s.