standard forms of local nets in quantum field theory nets of von neumann algebras with local algebras associated with all compact convex causally complete subsets of spacetime are constructed assuming a particular kind of net considered by bisognano and wichmann the properties of such nets are discussed within the general framework of algebraic quantum field theory with an emphasis on the relation between the geometry of spacetime and the structure of the nets an intersection property for the local algebras is stated and proved the structure of subnets of such nets is discussed a number of particular possible features of the nets are shown to be hereditary in the sense that the subnets have the same features possible further extensions of the domain of definition of these nets are discussed briefly

an analogue of the kac wakimoto formula and black hole conditional entropy a local formula for the dimension of a superselection sector in quantum field theory is obtained as vacuum expectation value of the exponential of the proper hamiltonian in the particular case of a chiral conformal theory this provides a local analogue of a global formula obtained by kac and wakimoto within the context of representations of certain affine lie algebras our formula is model independent and its version in general quantum field theory applies to black hole thermodynamics the relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors where the conditional entropy is defined as the connes stormer entropy associated with the dhr localized endomorphism representing the sector the constant of proportionality is half of the hawking temperature as a consequence the relative free energy is quantized proportionally to the logarithm of a rational number in particular it is equal to a linear function of the logarithm of an integer once the initial state or the final state is taken fixed

extensions of automorphisms and gauge symmetries we characterize the automorphisms of ac algebra open image in new window which extend to automorphisms of the crossed product open image in new window by a compact group dual the case where the inclusion open image in new window is equipped with a group of automorphisms commuting with the dual action is also treated these results are applied to the analysis of broken gauge symmetries in quantum field theory to draw conclusions on the structure of the degenerate vacua on the field algebra

fuzzy guaranteed cost sampled data control of nonlinear systems coupled with a scalar reaction diffusion process a fuzzy guaranteed cost sampled data control problem is addressed in this paper for a class of nonlinear coupled systems of an n dimensional lumped parameter system modeled by ordinary differential equation ode with a scalar reaction diffusion process represented by parabolic partial differential equation pde a takagi sugeno t s fuzzy coupled ode pde model is initially proposed to accurately represent the nonlinear coupled ode pde system based on the t s fuzzy coupled ode pde model a fuzzy sampled data controller is subsequently developed via the lyapunov s direct method to not only locally exponentially stabilize the nonlinear coupled system but also provide an upper bound of the given cost function moreover a suboptimal fuzzy sampled data control design is also addressed in the sense of minimizing the upper bound of the cost function the main contribution of this study lies in that a new parameterized linear matrix inequality lmi technique is proposed to reduce the conservativeness of the method and an lmi based fuzzy suboptimal sampled data control design is developed for the nonlinear coupled ode pde system based on this parameterized lmi technique simulation results on the sampled data control of hypersonic rocket car are provided to illustrate the effectiveness and merit of the design method

galois objects with normal bases for free hopf algebras of prime degree suppose r is a commutative ring and h a commutative cocommutative hopf algebra which is a free r module of prime rank the set of isomorphism classes of r algebras which are galois h objects forms an abelian group see chase and sweedler a in this paper we show that when h is an algebra over a special subring of the p adic integers containing a full set of p 1 st roots of unity then the subring consisting of those galois objects which are isomorphic to h as h module h hom h r is isomorphic to ue r u r where the notation u r means units of r congruent to 1 modx and 5 is a p 1 st root of an element b of r which will be seen to characterize the hopf algebra h the paper is organised as follows we briefly review notation related to hopf algebras integrals and galois objects in section 0 in section 1 we present the description of hopf algebras of order p given by tate and oort in their paper group schemes of prime order s such a hopf algebra h is associated with a pair of constants a b in fact hz r x x ax and h x r x xp bx where ab wi some unit of r in section 2 we describe the h module structure of h and the h module structure of a galois h object s and we state what it means for s to have a normal basis s z h as h module in section 3 we assume r contains a p l st root of 6 we show rz pz c h from which it follows that s contains a kummer order c modp s with s r and sis si i

noether s theorem for hopf orders in group algebras let k be a local field with valuation ring r of residue character istic p containing a primitive pth root of unity fp we state an analog to noether s theorem for modules over i hopf algebras and use induction tech niques to deduce a criterion for this analog to hold we then construct a family of noncommutative hopf algebras which satisfy the criterion if l k is a galois extension of number fields with group g and rings of integers s r then in a natural way l is a a g module and s is an rg module the normal basis theorem asserts that l kg as kg modu es the analogous result does not generally hold for s and rg even after localizing r in 1931 e noether n showed that locally for all primes p of r s rg as left 5g modules if and only if the extension l k is tame that is for every prime p of r the ramification index of any prime p of s lying over p is relatively prime to the characteristic of the residue field r p in fact the following are equivalent 1 s rg locally as 5g modules 2 l k is tame 3 the trace map s r is surjective 4 5 is a projective 5g module in 1986 l childs and s hurley c h generalized the notions of tameness and local normal basis to modules where h is an 5 hopf algebra their definitions specialized to those above when it rg and they obtained in par ticular the result that when h is commutative s h locally as modules if and only if s is tame waterhouse w notes that the childs hurley notions of tameness do not imply the existence of local normal basis when h is noncommutative even for group rings over fields the counterexample is not entirely satisfactory in that any natural number theoretic example would have normal basis over the quotient field in accordance with a result of kreimer and cook k c note that the childs hurley results assume commutativity of h yet noether s theorem holds for arbitrary finite groups accordingly there seems reason to believe that analogous results should hold for at least some non commutative h other than group algebras we say that noether s theorem

maximal tame extensions over hopf orders in rings of integers of p adic number fields abstract let k k be a cyclic totally ramified kummer extension of degree pn with galois group g let o and o be the rings of integers in k and k respectively for n 1 f bertrandias and m j ferton determined the ring end o g o of o g endomorphisms of o and l n childs constructed the maximal order s in o whose ring of o g endomorphisms is a hopf order a hopf order whose linear dual is a larson order in kg is called a dual larson order in this paper the generators of a dual larson order are given we determine end o g o and obtain a maximal dual larson order contained in end o g o we construct a hopf galois extension s in o whose ring h s of o g endomorphisms is a dual larson order we affirm an order s in o with a dual larson order h s is a tame h s extension and show that s is maximal in such orders s

relative galois module structure of rings of integers and elliptic functions iii in this paper we consider the problem of describing the galois module structure of rings of integers of abelian extensions of a quadratic imaginary number field k in certain cases by use of elliptic functions we are able to exhibit specific generators of such rings of integers as galois modules the general problem of finding galois integral generators by means of evaluation of transcendental functions has been considered by a number of mathematicians our results represent the first major advance outside the cyclotomic case we recall that the principal cyclotomic results are the hilbert speiser theorem and more generally the following result of leopoldt cf le theorem 1 let e denote an abelian extension of q let f gal e q

convergence of local charges and continuity properties of w inclusions the local generators of symmetry transformations which have recently been constructed from a quantum field theoretical version of noether s theorem are shown to converge to the global ones as the volume tends to the whole space the proof relies on the continuous volume dependence of the universal localizing maps which are associated to the local splitw inclusions

monogenic hopf algebras and local galois module theory let k be an unramified extension of qp and denote the ring of integers of k by r ok let h be an r hopf algebra with monogenic dual h we realize h as the kernel of an isogeny of one dimensional formal groups this allows us to give a complete list of fields l for which l k is h k hopf galois and s ol is a free h module

half sided modular inclusion and the construction of the poincare group in this note we will give a construction of the poincare group out of the modular groups of the wedge algebras provided the groups act on the algebra of every double cone like the associated lorentz boosts this construction will use the concept of half sided modular inclusions instead of the first and second cohomology of the poincare group as used by brunetti guido and longo by our method we obtain directly the poincare group and not its covering group

classification of subsystems for graded local nets with trivial superselection structure we classify haag dual poincare covariant subsystems open image in new window of a graded local net open image in new window on 4d minkowski spacetime which satisfies standard assumptions and has trivial superselection structure the result applies to the canonical field net open image in new window of a net open image in new window of local observables satisfying natural assumptions as a consequence provided that it has no nontrivial internal symmetries such an observable net open image in new window is generated by the abstract versions of the local energy momentum tensor density and the observable local gauge currents which appear in the algebraic formulation of the quantum noether theorem moreover for a net open image in new window of local observables as above we also classify the poincare covariant local extensions open image in new window which preserve the dynamics

the valuative condition and r hopf algebra orders in kc p 3 let k be a finite extension of qp endowed with the p adic valuation and let r be its ring of integers in this paper we give a complete classification of r hopf algebra orders in the group ring kcp3 for an arbitrary hopf order h we show that either the group scheme sph or the group scheme sph with h the linear dual can be viewed as the middle term of the baer product of a distinguished extension with a generically trivial extension using this characterization we then compute algebra generators for either h or h

monogenic hopf orders and associated orders of valuation rings abstract let r be a complete discrete valuation ring of mixed characteristic with perfect residue field and let h be a finite local commutative r hopf algebra we consider when there exists a finite extension of the field of fractions of r whose valuation ring is a galois h object if this occurs then h is monogenic conversely if h is also cocommutative and h is monogenic then there exists a valuation ring which is a galois h object to prove this result we represent h as the kernel of an isogeny of a special type between formal groups over r we deduce that if a is a finite abelian r hopf algebra such that both a and its dual are local then a is the associated order of a valuation ring if and only if the dual of a is monogenic

half sided modular inclusions of von neumann algebras letn m be von neumann algebras on a hilbert spaceh a common cyclic and separating vector denote m n resp jm jn the associated modular operators and conjugations assume m it n it n fort 0 we call such an inclusion half sided modular then we prove the existence of a oneparameter unitary groupu a onh a r with generator frac 1 2 pi ln delta cal n ln delta m mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x 0 and relations 1 itu a it it nit u a nit u e 2 ta for alla t r 2 jmjn u 2 3 nit u 1 it u 1 for allt r 4 n u 1 u 1

local aspects of superselection rules we study a theory of short range forces in terms of local observable quantities among the superselection structure determined by the algebra of all local observables to each additive independent charge we associate local observables having a meaning analogous to the regularized integrals of charge density fields over a finite volume among other assumptions we require that parastatistics are absent from the theories considered

superselection theory for subsystems an inclusion of observable nets satisfying duality induces an inclusion of canonical field nets any bose net intermediate between the observable net and the field net and satisfying duality is the fixed point net of the field net under a compact group this compact group is its canonical gauge group if the occurrence of sectors with infinite statistics can be ruled out for the observable net and its vacuum hilbert space is separable

integral hopf galois structures on degree p2 extensions of p adic fields abstract let l k be a totally ramified normal extension of p adic fields of degree p2 we investigate the behavior of the valuation ring o l in the various hopf galois structures on l k specifically we determine when o l is hopf galois with respect to a hopf order in the corresponding hopf algebra when this occurs o l is necessarily a free module over this hopf order we also determine which hopf orders can arise in this way for cyclic extensions l k of degree p2 l n childs has shown under certain restrictions on the ramification numbers that if o l is hopf galois with respect to a hopf order in one of the hopf galois structures on l k then the same is true in all p hopf galois structures on l k we show that this no longer holds if the ramification conditions are relaxed or if elementary abelian extensions of degree of p2 are considered we illustrate our results with a special family of kummer extensions and with certain extensions arising from lubin tate formal groups

tame realisable classes over hopf orders abstract letobe the ring of algebraic integers in a number fieldkand letgbe a finite abelian group mcculloh has characterised the classes in the locally free classgroup cl og which are realisable by rings of integers of tame normal extensions ofkwith galois groupg we extend this to certain extensions which in general need be neither tame nor normal by replacingogwith a hopf order a and introducing the strongly a tame orders as an analogue of tame rings of integers we also describe the classes realised by principal homogeneous spaces over the dual of a

tameness and local normal bases for objects of finite hopf algebras let r be a commutative ring s an r algebra h a hopf ralgebra both finitely generated and projective as r modules and suppose s is an h object so that h homr h r acts on s via a measuring let i be the space of left integrals of h we say s has normal basis if s h as h modules and s has local normal bases if sp hp as hp modules for all prime ideals p of r when r is a perfect field h is commutative and cocommutative and certain obvious necessary conditions on s hold then s has normal basis if and only if is r sh if r is a domain with quotient field k h is cocommutative and l s or k has normal basis as h 0 k module then s has local normal bases if and only if is r sh suppose k is a number field with ring of integers r l is a finite galois extension of k with galois group g and s is the integral closure of r in l then g acts as a group of automorphisms of s relative galois module theory seeks to understand s as an rg module via this action the most basic question is to inquire whether s is locally isomorphic to rg that is for all primes p of r sp has a local normal basis as a free rp module this question was answered by emmy noether who showed that s has a local normal basis at every prime p of r if and only if l k is tamely ramified here tamely ramified means that at each prime ideal p of r the ramification index of any prime p of s lying over p is relatively prime to the residue field characteristic it is well known that this latter condition is equivalent to the surjectivity of the trace map tr s r tr s ezacg s the purpose of this paper is to formulate and prove an extension of noether s theorem to objects of finite cocommutative hopf algebras assume now only that r is a commutative ring h is a hopf r algebra which is a finitely generated projective r module and h homr h r the dual hopf algebra a commutative r algebra s is an h object if s is a right h comodule via a map a s s 0 h which is an r algebra homomorphism if s is an h object then a induces a measuring a h s s in the sense of sweedler 21 galois module theory in this setting is the study of s as an h module via the measuring a the trivial example is s h itself with a a the comultiplication on h so we say that the h object has normal basis if s h as h modules and s has local normal bases if for all primes p of r sp hp as h modules since rgf rg as rg modules for g any finite group this notion of normal basis when specialized to h rg is equivalent to the classical notion received by the editors may 7 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 13b05 secondary 16a24 1986 american mathematical society 0002 9947 86 1 00 25 per page

on systems of local operators and the duality condition systems of von neumann algebras of local operators are considered and a particular procedure for the construction of such a system from any given set of intrinsically local operators is described the condition of duality is discussed with particular emphasis on a special form of this condition which appears in previous work by bisognano and wichmann subsystems generated by any set of local operators in an original irreducible system of local operators are considered some properties of the irreducible vacuum representation of such a subsystem are derived assuming the special form of the duality condition for the original system it is shown that this special condition also holds for the subsystem it is furthermore shown that weak additivity holds for the reduced system if it holds for the original system

unitary positive energy representations of scalar bilocal quantum fields the superselection sectors of two classes of scalar bilocal quantum fields in d 4 dimensions are explicitly determined by working out the constraints imposed by unitarity the resulting classification in terms of the dual of the respective gauge groups u n and o n confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory but the approach using standard lie algebra methods rather than abstract duality theory is complementary the result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension d 2 in models with global conformal invariance another remarkable outcome is the observation that with an appropriate choice of the hamiltonian a lie algebra embedded into the associative algebra of observables completely fixes the representation theory

on the causal structure of minkowski spacetime the causal structure of minkowski spacetime m is discussed in terms of the notions of causal complementation and causal completion these geometric notions are relevant for quantum field theory and the theory of the klein gordon equation particular attention is given to closed convex causally complete subsets of m and the properties of such sets are discussed the study of such sets is motivated by potential applications to the theory of local nets of von neumann algebras the notion of the envelope of uniqueness of a subset of m familiar from the theory of the wave equation is discussed and some results about the relation of this envelope to the causal completion of the set are presented