projective techniques and functional integration for gauge theories a general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact hausdorff spaces the procedure is then applied to gauge theories to carry out integration over the non linear infinite dimensional spaces of connections modulo gauge transformations this method of evaluating functional integrals can be used either in the euclidean path integral approach or the lorentzian canonical approach a number of measures discussed are diffeomorphism invariant and therefore of interest to the connection dynamics version of quantum general relativity the account is pedagogical in particular prior knowledge of projective techniques is not assumed

fuzzy sets and residuated logic abstract the properties of a set theory rzf defined in a logic weaker than intuitionistic logic are investigated using natural models rzf is intended to act as a framework for fuzzy set theory more general and applicable than heyting valued structures such as topoi the theory rzf contains an interpretation of classical set theory and is consistent relative to this theory it is also a suitable theory for the expression of attributes of uncertainty and vagueness since a certain subclass of the sets definable in this theory are crisp or two valued and behave classically whereas sets in general obey only residuated logic of which the ukasiewicz semantics on 0 1 the real interval is a particular realisation

construction of idempotent 2 cocycles abstract let f g g l be a weak 2 cocycle where l k is a finite galois field extension with galois group g and a f l k f be the associated weak crossed product k algebra we associate a partition of g to a f and construct idempotent 2 cocycles of a f

the defektsatz for central simple algebras let q be a central simple algebra finite dimensional over its center f and let v be a valuation ring of f the v has en extension to q i e there exists a dubrovin valuation ring b of q satisfying v f b generally the number of extensions of v to q is not finite and therefore the so called intersection property of dubrovin valuation rings b 1 b n is introduced this property is defined in terms of the prime ideals and the valuation overrings of the intersection b b n it is shown that there exists a uniquely determined natural number n depending only on v and having the following property if b 1 b k are extensions of v having the intersection property then k n and k n holds if and only if b 1 b k is integral over v

representations of low rank orthosymplectic superalgebras by superfield techniques representations of lie superalgebras may be realised as functions superfields on graded manifolds suitably chosen onset spaces the technique is illustrated for osp 1 2 and osp 2 2 with a review of well known results and applied in the cases of osp 3 2 and osp 4 2 to construct some classes of irreducible representations

representations of osp m 2n and young supertableaux young supertableaux are introduced for finite dimensional representations of orthosymplectic superalgebras and used to study their content

k automorphisms of a weak crossed product f algebra over a galois extension k f abstract we compute the k automorphism group of a k k f a f of a weak crossed product algebra a f for a weak 2 cocycle f over a galois extension k f with galois group g the k automorphism group of a k decomposes into its unipotent part and the reductive part h automorphisms in h are computed via their restriction to semi simple part of a k there is a strong relationship between h and the lower subtractive relation induced by f on g we introduce a subgroup of h namely which contains interesting combinatorial information of we also present a duality on lower subtractive relations which simplifies the computation of the automorphism group for the weak bruhat order of a coxeter system which is an important example of a lower subtractive relation it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph

a note on kripke semantics for residuated logic abstract residuated logic is a suitable logical system for approximate reasoning and for the development of fuzzy mathematics residuated propositional logic rpc is shown to be a fragment of a well known substructural logic denoted by h bck using results for h bck kriple style semantics is described for rpc for which soundness and completeness theorems hold these results answer a question previously posed for rpc and situate rpc in the currently active research field of substructural logics

analytic bethe ansatz and functional equations associated with any simple root systems of the lie superalgebra sl r 1 s 1 abstract the lie superalgebra sl r 1 s 1 admits several inequivalent choices of simple root systems we have carried out analytic bethe ansatz for any simple root systems of sl r 1 s 1 we present transfer matrix eigenvalue formulae in dressed vacuum form which are expressed as the young supertableaux with some semistandard like conditions these formulae have determinant expressions which can be viewed as quantum analogue of jacobi trudi and giambelli formulae for sl r 1 s 1 we also propose a class of transfer matrix functional relations which is specialization of hirota bilinear difference equation using the particle hole transformation relations among the bethe ansatz equations for various kinds of simple root systems are discussed

the brauer monoid of a field in this article we introduce an invariant of a field f the brauer monoid the elements of this monoid are based on certain f central finite dimensional algebras the strongly primary algebras over f which are more general than central simple f algebras under a suitable equivalence relation and product the classes of these algebras form a monoid denoted m f the group of invertible elements of this monoid is the brauer group of f this article is based on ideas first introduced by sweedler in 6 and later developed and extended by sweedler larsen and the present author in 2 3 1 in these articles a cohomology theory was introduced which generalizes the usual galois and amitsur cohomology theories in section 2 of this article it is shown that these new cohomology monoids play the same role for the brauer monoid as the usual cohomology groups play for the brauer group that is a given cohomology monoid m gal k f k for k a finite galois extension of f is isomorphic to the submonoid of elements of m f that are split by k in a suitable sense in the first section we define strongly primary algebras examine their properties and construct the monoid in the second section we show the connection with cohomology in the third section we begin a dissection of the brauer monoid of f and show in particular that it is the union of the groups m f where for each idempotent e in m f we let m f xem f ixe e and xy e for some yea f f as in 2 we classify the possible idempotents and show that associated to each one is a finite graph moreover we state a result proved in section 4 that determines m f in the case where the associated graph is a tree at the end of section 3 we present some examples we work throughout over a fixed base field f unlabelled tensors are over

fuzzy sets and toposes towards higher order fuzzy logic abstract a series of paper which have appeared in fuzzy sets and systems over the last three years are reviewed in an expository manner the claims and counter claims made in them are dealt with some of the problems of representing knowledge in the first order predicate calculus are discussed briefly and a case is made for exposing the relationship between topoi and fuzzy mathematics with a view to making a true fuzzy logic both consistent complete and respectable the paper is an attempt to present the controversy to the non specialist and especially to practitioners in expert systems and artificial intelligence

on dubrovin valuation rings in crossed product algebras let f be a field and let v be a valuation ring in f if a is a central simple f algebra then v can be extended to a dubrovin valuation ring in a in this paper we consider the structure of dubrovin valuation rings with center v in crossed product algebras k f g f where k f is a finite galois extension with galois group g unramified over v and f is a normalized two cocycle in the case where v is indecomposed in k we introduce a family of orders naturally associated to f examine their basic properties and determine which of these orders is dubrovin in the case where v is decomposed we determine the structure in the case of certain special discrete finite rank valuations

integral semihereditary orders inside a bezout maximal order in this paper we show that over a commutative valuation ringv a semihereditaryv orderhin a finite dimensional simple artinian ring is a finite intersection of bezout maximal v orders iff eitherhis bezout orhis contained in a bezoutv order andj v is a principal ideal ofv

maximal orders over valuation rings abstract in this paper we study maximal orders over commutative valuation rings in central simple algebras we are particularly interested in maximal orders which are either bezout or semihereditary we construct a class of bezout maximal orders and a class of semihereditary maximal orders and show that for any valuation ring v resp v with value group z m any bezout resp semihereditary maximal order over v belongs to the class constructed furthermore we classify all maximal orders in m2 f over a valuation ring with value group z m and in mn f given a mild defectless assumption

integral semihereditary orders extremality and henselization abstract in this paper we study integral semihereditary orders over a valuation ring in a finite dimensional simple artinian ring in the first section we prove that such orders are extremal consequently in a central division algebra admitting a total valuation ring the intersection of all the conjugates of the total valuation ring is the unique integral semihereditary order over the center of the total valuation ring in the second section we characterize up to conjugacy integral semihereditary orders over a henselian valuation ring in the last section we show that an integral order r over an arbitrary valuation ring v is semihereditary iff its henselization r v v h where v h is the henselization of v is a semihereditary v h order in this case there is an inclusion preserving bijective correspondence between semihereditary v orders inside r and semihereditary v h orders inside r v v h

young diagrams supercharacters of osp m n and modification rules the supercharacter of osp m n associated with an arbitrary young diagram is defined the distinction is made between osp m n standard and non standard supercharacters the corresponding modification rule which may be used to express a non standard supercharacter in terms of standard supercharacters is presented exemplified and proved this rule involves the removal of n 2 1 boundary strips from the young diagram in the case n 0 the rule reduces to the well known rule appropriate to o m for a non standard supercharacter corresponding to a typical irreducible representation of osp m n the modification yields a single typical standard supercharacter on the other hand for a non standard supercharacter corresponding to an atypical irreducible representation the rule yields a linear combination of atypical standard supercharacters

functional integration on spaces of connections suppose thatgis a compact connected lie group andp mis a smooth principalg bundle we define a cylinder function on the space a of smooth connections onpto be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves inm completing the algebra of cylinder functions in the sup norm we obtain a commutative c algebra fun a let a generalized measure on a be a bounded linear functional on fun a we construct a generalized measure 0on a that is invariant under all automorphisms of the bundlep not necessarily fixing the basem this result extends previous work which assumedmwas real analytic and used only piecewise analytic curves in the definition of cylinder functions as before any graph withnedges embedded inmdetermines a c subalgebra of fun a isomorphic toc gn and the generalized measure 0 fun a c restricts to the linear functional onc gn given by integration against normalized haar measure ongn our result implies that the group g of gauge transformations acts as unitary operators onl2 a the hilbert space completion of fun a in the norm f 2 0 f 2 1 2 using spin networks we construct explicit functions spanning the subspacel2 a g l2 a consisting of vectors invariant under the action of g

semiclassical limits of extended racah coefficients we explore the geometry and asymptotics of extended racah coefficients the extension is shown to have a simple relationship to the racah coefficients for the positive discrete unitary representation series of su 1 1 which is explicitly defined moreover it is found that this extension may be geometrically identified with two types of lorentzian tetrahedra for which all the faces are timelike the asymptotic formulas derived for the extension are found to have a similar form to the standard ponzano regge asymptotic formulas for the su 2 6j symbol and so should be viable for use in a state sum for three dimensional lorentzian quantum gravity

representations of supergroups an explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations the rules of supersymmetrization are characterized by extended young supertableaux this constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all casimir operators can be calculated we have applied this approach so far to the supergroups su n m osp n 2m p n for which we have obtained all the representations constructible as direct products of the fundamental defining representations an argument is presented toward the irreducibility of all these representations

crossed product algebras defined by separable extensions abstract we generalize the classical construction of crossed product algebras defined by finite galois field extensions to finite separable field extensions by studying properties of rings graded by groupoids we are able to calculate the jacobson radical of these algebras we use this to determine when the analogous construction of crossed product orders yield azumaya maximal or hereditary orders in a local situation thereby we generalize results by haile larson and sweedler

geometry and representations of the quantum supergroup ospq 1 2n the quantum supergroup ospq 1 2n is studied systematically a haar functional is constructed and an algebraic version of the peter weyl theory is extended to this quantum supergroup quantum homogeneous superspaces and quantum homogeneous supervector bundles are defined following the strategy of connes theory parabolic induction is developed by employing the quantum homogeneous supervector bundles quantum frobenius reciprocity and a generalized borel weil theorem are established for the induced representations

the haar integral for lie supergroups each supermanifold can be considered as equivalent to a certain family of real manifolds the haar integral of a lie supergroup is defined using this equivalence a useful general formula is derived together with explicit construction methods of the simple lie supergroups of types spl n m el osp n 2r el and b n el and the extended poincare group

analytic bethe ansatz related to a one parameter family of finite dimensional representations of the lie superalgebra as is well known the type 1 lie superalgebra admits a one parameter family of finite dimensional irreducible representations we have carried out an analytic bethe ansatz related to this family of representations we present formulae which are deformations of previously proposed determinant formulae labelled by a young superdiagram these formulae will provide a transfer matrix eigenvalue in a dressed vacuum form related to the solutions of a graded yang baxter equation which depend not only on the spectral parameter but also on a non additive continuous parameter a class of transfer matrix functional relations among these formulae is briefly mentioned

on semi hereditary orders integral over a commutative valuation ring in this paper we study semi hereditary v orders in a simple artinian ring q with finite dimension over its center where v is a commutative valuation ring let s be a minimal v overring of a semihereditary v order r in section 1 we investigate some relations between all maximal ideals of s and r and characterize the commutativities of idempotent maximal ideals of r in terms of orders of ideals in section 2 we show that there is a bijection between the set of all v overrings of r and the set of all idempotent ideals which are finitely generated as left ideals any element in the latter set is characterized by four different types of cycles in section 3 we discuss the principalness of the jacobson radical j r some results in sections 1 3 are used to derive the exact numbers of all semi hereditary maximal v orders containing r and of all v overrings of r and to study the nilpotency of j r modulo j v r some invariant properties of semi hereditary v orders are also given

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

semiclassical mechanics of the wigner 6j symbol the semiclassical mechanics of the wigner 6j symbol is examined from the standpoint of wkb theory for multidimensional integrable systems to explore the geometrical issues surrounding the ponzano regge formula the relations among the methods of roberts and others for deriving the ponzano regge formula are discussed and a new approach based on the recoupling of four angular momenta is presented a generalization of the yutsis type of spin network is developed for this purpose special attention is devoted to symplectic reduction the reduced phase space of the 6j symbol the 2 sphere of kapovich and millson and the reduction of poisson bracket expressions for semiclassical amplitudes general principles for the semiclassical study of arbitrary spin networks are laid down some of these were used in our recent derivation of the asymptotic formula for the wigner 9j symbol

quantization of diffeomorphism invariant theories of connections with local degrees of freedom quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions this provides in particular a quantization of the husain kucha model the main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity in the riemannian case i e signature the approach appears to contain all the necessary ingredients already in the lorentzian case it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections

fuzzy real numbers as dedekind cuts with respect to a multiple valued logic abstract viewing fuzzy set theory as a model of set with respect to a given multiple valued logic we establish the dedekind macneille completion of a partially ordered l valued set contrary to other constructions in fuzzy set theory the completion is unique up to an order isomorphism specializing this situation to the set of all crisp rational numbers we obtain in a unique way the fuzzy dedekind real numbers

on weak crossed products frobenius algebras and the weak bruhat ordering abstract the weak crossed product algebra was studied first by haile larson and sweedler amer j math 105 1983 689 they gave interesting properties for such an algebra one of which is that if a f is a weak crossed product induced by a weak 2 cocycle f defined on a galois group g gal k f and h is the inertial subgroup of g then a f has a wedderburn splitting that is a f b j where j is the radical of a f and b is a k h central simple algebra the purpose of this paper is to give the necessary and sufficient condition for a weak crossed product to be frobenius and to describe an algorithm for constructing lower subtractive graphs from a finite group g and a generating set s a special case of this construction is the so called weak bruhat ordering on a coxeter group g s we show that the nilcoxeter algebra associated to g s is a special case of the restricted algebra associated to a lower subtractive graph

operator algebraic construction of gauge theories and jones actions of thompson s groups using ideas from jones lattice gauge theory and loop quantum gravity we construct 1 1 dimensional gauge theories on a spacetime cylinder given a separable compact group g we construct localized time zero fields on the spatial torus as a net of c algebras together with an action of the gauge group that is an infinite product of g over the dyadic rationals and using a recent machinery of jones an action of thompson s group t as a replacement of the spatial diffeomorphism group adding a family of probability measures on the unitary dual of g we construct a state and obtain a net of von neumann algebras carrying a state preserving gauge group action for abelian g we provide a very explicit description of our algebras for a single measure on the dual of g we have a state preserving action of thompson s group and semi finite von neumann algebras for g mathbf s the circle group together with a certain family of heat kernel states providing the measures we obtain hyperfinite type iii factors with a normal faithful state providing a nontrivial time evolution via tomita takesaki theory kms condition in the latter case we additionally have a non singular action of the group of rotations with dyadic angles as a subgroup of thompson s group t for geometrically motivated choices of families of heat kernel states

job search and employment abstract a multistage search method based on fuzzy preference is used to analyze the processes of job search an employment decision based on fuzzy preference results in an exact choice the incorporation of fuzzy interval preference into employment decision however may lead to multiple choices which provides a theoretical justification for the existence of fuzzy supply of labor thus fuzziness in addition to randomness offers an explanation to wage dispersion

weakly azumaya algebras abstract we define a class of algebras weakly azumaya algebras which includes both azumaya algebras and weak crossed products cf haile 1982 j algebra 74 270 279 1983 j algebra 91 521 539 and haile et al 1983 amer j math 105 no 3 689 814 just as with azumaya algebras these have a rank function whose values at localizations of the center are always a square after a general description of these algebras we specialize to the case where the center is a field f the jacobson radical need not be 0 and we prove a wedderburn principal theorem for these algebras our class is closed under extension of scalars and under tensor products and yields an interesting monoid which generalizes the brauer group our monoid is a union of groups called stalks in each of which the unit element is represented by an algebra called an idempotent algebra the ideal structure of members of the same stalk is the same a given stalk is not torsion but the kernel of the restriction map to the algebraic closure is torsion modulo the idempotent algebras at the end we consider low dimensional examples in detail

spin network states in gauge theory abstract given a real analytic manifoldm a compact connected lie groupgand a principalg bundlep m there is a canonical generalized measure on the space a g of smooth connections onpmodulo gauge transformations this allows one to define a hilbert spacel2 a g here we construct a set of vectors spanningl2 a g these vectors are described in terms of spin networks graphs embedded inm with oriented edges labelled by irreducible unitary representations ofgand with vertices labelled by intertwining operators from the tensor product of representations labelling the incoming edges to the tensor product of representations labelling the outgoing edges we also describe an orthonormal basis of spin network states associated to any fixed graph we conclude with a discussion of spin networks in the loop representation of quantum gravity and give a category theoretic interpretation of the spin network states

homological dimension of skew group rings and crossed products abstract in this paper we study the homological dimension of skew group rings and crossed products a sufficient condition for r g a crossed product to have finite right global dimension is given in terms of crossed products over simple artinian factors of r if r is right fbn and left coherent and g is finite some necessary conditions and sufficient conditions for r g a skew group ring of a finite group over a local or semilocal right noetherian ring to have finite right global dimension are also given then in particular if r is commutative noetherian and g is finite we obtain some equivalent conditions for r g a skew group ring to have finite global dimension using work of aljadeff e aljadeff serre s extension theorem for crossed products j london math soc 44 1991 47 54 these results are extended to polycyclic by finite groups

skew group algebras in the representation theory of artin algebras then the corresponding crossed product algebra i g or i x g for short has an elements cgieg gi a li i addition is componentwise and multiplication is given by g a g and g g2 y g g2 m in this paper we assume that the values of y lie in the center z a of i hence the action of g on i is given by a group homomorphism g p aut i and 3 can be left out in the special case that y is the trivial map we write 1g instead of i g and the elements as c c li gi ag is then called a skew group ring there is a lot of literature on skew group algebras and crossed product algebras and on the relationship with the ring ac whose elements are those elements of a left fixed by g much work has been done on which properties of li are inherited by n g or ac some of the work on the relationship between these rings has its roots in trying to develop a galois theory for noncommutative rings we refer to 3 7 13 15 19 21 23 25 27 281 and their references in this paper we study these constructions when i is an artin algebra and g a finite group of order n such that n is invertible in 1 under these assumptions the construction preserves central properties of interest in the

the logic of tied implications part 2 syntax an implication operator a is said to be tied if there is a binary operation t that ties a that is the identity a a a b z a t a b z holds for all a b z we aim at the construction of a complete predicate logic for prelinear tied adjointness algebras we realize this in three steps in the first step we establish a propositional calculus adjtpc complete for the class of all tied adjointness algebras on partially ordered sets without prelinearity and ignoring the lattice operations for that we supply a hilbert system based on seven axioms and one deduction rule modus ponens in the second and third steps we extend adjtpc to propositional and predicate calculi complete for prelinear tied adjointness algebras we apply a duality principle due to morsi in all three calculi through which we manage to cut down the number of proofs

the clebsch gordan coefficients and isoscalar factors of the graded unitary group su m n it is pointed out that the clebsch gordan coefficients cgc of the permutation group are the coefficients which couple irreducible bases irb of su m n and su p q to the irb of su mp nq mp np and the outer product reduction coefficients orc of the permutation group are the coefficients which couple the irb of su m n and su p q into the irb of su m p n q the orc of permutation groups with due modification in signs are the cgc for the special gel fand basis of su m n the isoscalar factor isf for the permutation group chain s f contains implies s f1 s f2 is the isf for the graded unitary group chain su mp nq mq np contains implies su m n su p q and the outer product isf for the group chain s f contains implies s f1 s f2 is the isf for su mp nq mq np contains implies su m n x su p q all these coefficients can be calculated easily and some are already available

valuation rings in finite dimensional division algebras a subring b of a division algebra d is called a valuation ring of d if x or x 1 is contained in b for every x in d x 0 such a ring is called an extension of the valuation ring b of k the centre of d if b k b let d be a division algebra finite dimensional over its centre k d k n2 b a valuation ring of k and b bi iei the set of all extension of b in d theorem 1 b possesses at most n extensions in d i e b n theorem 2 any two extensions of b in d are conjugate in d theorems 3 and 4 the set t of elements in d which are integral over b is a subring of d if and only if b 1 in this case t bi bi in b

a small set of axioms for residuated logic we show that a set of three axioms together with modus ponens suffices to build up a formal system for the class r of all quadruples l t j in which l is a partially ordered set with a top element and t is a triangular norm on l that has a residuated implication j we show that this r is not a variety

patching and weak approximation in isometry groups let r be a semilocal principal ideal domain two algebraic objects over r in which scalar extension makes sense e g quadratic spaces are said to be of the same genus if they become isomorphic after extending scalars to all completions of r and its fraction field we prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over not necessarily commutative r orders is always a finite power of 2 and under further assumptions e g that the order is hereditary this number is 1 the same result is also shown for related objects e g systems of sesquilinear forms a key ingredient in the proof is a weak approximation theorem for groups of isometries which is valid over any topological base field and even over semilocal base rings

physical properties of quantum field theory measures well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory the difference of typical configurations of free massive scalar field theories with different masses is studied we apply the same methods to study the ashtekar lewandowski al measure on spaces of connections in particular we prove that the diffeomorphism group acts ergodically with respect to the al measure on the ashtekar isham space of quantum connections modulo gauge transformations we also prove that a typical with respect to the al measure quantum connection restricted to a piecewise analytic curve leads to a parallel transport discontinuous at every point of the curve

superunitary reduction of the orthosymplectic supertableaux the supertableaux of the orthosymplectic groups osp 2 2p and osp 2 1 2p are decomposed in sums of supertableaux of the superunitary group su p

introduction to supersymmetric spin networks in this article we study the construction of supersymmetric spin networks which has a direct interpretation in context of the representation theory of the superalgebra in particular we analyze a special kind of spin network associated with superalgebra osp 1 2n it turns out that the set of corresponding spin network states forms an orthogonal basis of the hilbert space l2 a g and this argument holds even in the q deformed case the osp n 2 spin networks are also discussed briefly we expect they could provide useful techniques to quantum supergravity and gauge field theories from the point of nonperturbative view

character expansions for the orthogonal and symplectic groups formulas for the expansion of arbitrary invariant group functions in terms of the characters for the sp 2n so 2n 1 and so 2n groups are derived using a combinatorial method the method is similar to one used by balantekin to expand group functions over the characters of the u n group all three expansions have been checked for all n by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions an expansion of the exponential of the traces of group elements appearing in the finite volume gauge field partition functions is worked out for the orthogonal and symplectic groups

hereditary crossed product orders over discrete valuation rings in this paper we consider weak crossed product orders af sx with coefficients in the integral closure s of a discrete valuation ring r in a tamely ramified galois extension of the field of fractions of r in the first section we compute the jacobson radical of af when s is local and we give a characterization of the hereditarity of the order in terms of the cocycle values in the second section we prove again in the local case that every in the inertia group for s r must belong to g f 1 is a unit of s in the final section we compute the jacobson radical in the general case s is semilocal and show how the hereditarity of af can be determined locally under an additional hypothesis

the inverse loop transform the loop transform in quantum gauge field theory can be recognized as the fourier transform or characteristic functional of a measure on the space of generalized connections modulo gauge transformations since this space is a compact hausdorff space conversely we know from the riesz markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular borel measure on the moduli space in the present article we show how one can compute the finite joint distributions of a given characteristic functional that is we derive the inverse loop transform

dimension and character formulas for lie supergroups a character formula is derived for lie supergroups the basic technique is that of symmetrization and antisymmetrization associated with young tableaux generalized to supergroups we rewrite the characters of the ordinary lie groups u n o n and sp 2n in terms of traces in the fundamental representation it is then shown that by simply replacing traces with supertraces the characters of certain representations for u n m and osp n 2m are obtained dimension formulas are derived by calculating the characters of a special diagonal supergroup element with 1 and 1 eigenvalues formulas for the eigenvalues of the quadratic casimir operators are given as applications the decomposition of a representation into representations of subgroups is discussed examples are given for the lie supergroup su 6 4 which has physical applications as a dynamical supersymmetry in nuclei

a length operator for canonical quantum gravity we construct an operator that measures the length of a curve in four dimensional lorentzian vacuum quantum gravity we work in a representation in which an su 2 connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined does not suffer from factor ordering singularities and does not require any renormalization we show that the length operator admits self adjoint extensions and compute part of its spectrum which like its companions the volume and area operators already constructed in the literature is purely discrete and roughly quantized in units of the planck length the length operator contains full and direct information about all the components of the metric tensor which facilitates the construction of so called weave states which approximate a given classical three geometry

invariant integration on classical and quantum lie supergroups invariant integrals on hopf superalgebras in particular the classical and quantum lie supergroups are studied the uniqueness up to scalar multiples of a left integral is proved and a z2 graded version of maschke s theorem is discussed a construction of left integrals is developed for classical and quantum lie supergroups applied to several classes of examples the construction yields the left integrals in explicit form

formal frameworks for approximate reasoning abstract this paper discusses applications of a formal set theoretic foundation for fuzzy set theory 23 to problems of approximate and uncertain reasoning and also looks at other formalisms such as the classical alternative set theory 39 and its intuitionistic version 21 20 for uncertain reasoning formal definitions of these frameworks will be given many suggestions for semantic operations on values of formulae involving intermediate truth have been made 41 40 contain a survey of these there are two main problems how to reconcile the use of the linear order 0 1 as the value space when in reality degrees of truth are often incomparable and secondly within this simplified range of values to select operations which are both logically coherent with each other and intuitively reasonable we will derive general semantic constraints on operations which will motivate the formal frameworks that we define

a springer theorem for hermitian forms throughout this paper we let d be a central f division algebra with involution such that f d f d d is a henselian valued field by 11 the valuation on f extends uniquely to a valuation on d we denote this valuation by v moreover we assume that the characteristic of the residue field open image in new window is not 2 if the valuation on f is discrete then any quadratic form q can be written as q q1 q2 where is a uniformizer and qi are unit forms springer s theorem states that q is isotropic if and only if at least one of the residue forms open image in new window and open image in new window is isotropic in this paper we generalize this result to hermitian forms in section 4 we use the connection between involutions on algebras and hermitian forms to prove an analog of the springer theorem for involutions

a new invariant for c over r almost invertible cohomology theory and the classification of idempotent cohomology classes and algebras by partially ordered sets with a galois group action

classification of finite dimensional unitary irreps for uq gl m n all finite dimensional irreducible unitary representations of the quantum supergroup uq gl m n are classified in terms of their highest weights for generic q 0

bezout orders and henselization abstract let b be a bezout maximal order over a valuation ring v in a central simple f algebra s if fh vh is the henselization of f v then the main result of this paper is that b vvh is a maximal vh order in s ffh furthermore we show that this maximal order is semihereditary but not bezout in general and that certain numerical invariants of b considered by wadsworth and grater readily show up in the structure of b v vh

the hilbert space costratification for the orbit type strata of su 2 lattice gauge theory we construct the hilbert space costratification of g su 2 quantum gauge theory on a finite spatial lattice in the hamiltonian approach we build on previous work f furstenberg g rudolph and m schmidt j geom phys 119 66 81 2017 where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture in this picture each element of the classical stratification corresponds to the zero locus of a finite subset pi of the algebra r of g invariant representative functions on gcn viewing the invariants as multiplication operators p i on the hilbert space h the union of their images defines a subspace of h whose orthogonal complement h is the element of the costratification corresponding to to construct h one has to determine the images of the p i explicitly to accomplish this goal we construct an orthonormal basis in h and determine the multiplication law for the basis elements that is we determine the structure constants of r in this basis this part of our analysis applies to any compact lie group g for g su 2 the above procedure boils down to a problem in combinatorics of angular momentum theory using this theory we obtain the union of the images of the operators p i as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of wigner s 3nj symbols the latter are further expressed in terms of 9j symbols using these techniques we are also able to reduce the eigenvalue problem for the hamiltonian of this theory to a problem in linear algebra we construct the hilbert space costratification of g su 2 quantum gauge theory on a finite spatial lattice in the hamiltonian approach we build on previous work f furstenberg g rudolph and m schmidt j geom phys 119 66 81 2017 where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture in this picture each element of the classical stratification corresponds to the zero locus of a finite subset pi of the algebra r of g invariant representative functions on gcn viewing the invariants as multiplication operators p i on the hilbert space h the union of their images defines a subspace of h whose orthogonal complement h is the element of the costratification corresponding to to construct h one has to determine the images of the p i explicitly to accomplish this goal we construct an orthonormal basis in h and determine the multiplication law for the basis elements that is we determine the structure constants of r in thi

kronecker product of sp 2n representations using generalised young tableaux using generalised young tableaux the authors obtain an explicit formula for the reduction of the kronecker product of irreducible representations of the symplectic groups this extends a previous work devoted to the case of orthogonal groups

weak crossed product orders over valuation rings abstract let f be a field let v be a valuation ring of f of arbitrary krull dimension rank let k be a finite galois extension of f with group g and let s be the integral closure of v in k let f g g k 0 be a normalized two cocycle such that f g g s 0 but we do not require that f should take values in the group of multiplicative units of s one can construct a crossed product v order a f g s x with multiplication given by x s x s f x for s s g we characterize semihereditary and dubrovin crossed product orders under mild valuation theoretic assumptions placed on the nature of the extension k f

tableau method for residuated logic abstract residuated logic is a generalization of intuitionistic logic which does not assume the idempotence of the conjunction operator such generalized conjunction operators have proved important in expert systems in the area of approximate reasoning and in some areas of theoretical computer science here we generalize the intuitionistic tableau procedure and prove that this generalized tableau method is sound for the semantics the class of residuated algebras of residuated propositional calculus rpc since the axioms of rpc are complete for the semantics we may conclude that whenever a formula 0 is tableau provable it is deducible in rpc we present two different approaches for constructing residuated algebras which give us countermodels for some formulas which are not tableau provable the first uses the fact that the theory of residuated algebras is equational to construct quotients of free algebras the second uses finite algebras we end by discussing a number of open questions

a remarkable connection between the representations of the lie superalgebras osp 1 2n and the lie algebras o 2n 1 we show that there is a one to one correspondence between the graded representations of osp 1 2n and the non spinorial representations of o 2n 1 the clebsch gordan series for osp 1 2n reduce to the corresponding series for o 2n 1 and the properly defined casimir operators of order at least up to four have the same eigenvalues

propositional calculus under adjointness we develop a formal system for the class of all implications a and conjunctions k on partially ordered sets l with top elements 1 such that a and k are related by adjointness and they satisfy the neutrality principle that is 1 is their left identity element we call the resulting logic propositional calculus under adjointness abbreviated adjpc most algebraic theorems on those l a and k are inequalities in the posets l of truth values in consequence we have to find means for abstracting inequalities within syntax which must be free from partial truth values this is achieved by employing a further implication like adjoint h of a and k whereby the partial order of l coincides with the binary relation h 1 accordingly the semantical domain for adjpc should become the class of all such quintuples l a k h which we call adjointness algebras our axiom scheme for adjpc features seven axioms however it may be the case that no finite set of axioms can complete adjpc if inference is carried out by means of modus ponens mp alone this is because adjpc is too general it lacks some basic theorems of the more restricted logics such as residuated logic and intuitionistic logic as a result we have to adopt four inference rules for adjpc namely mp and three bits of the substitution theorem we deduce enough theorems and inferences in adjpc to delineate its basic structure this enables us to establish the completeness of adjpc for the semantical domain of adjointness algebras by means of a quotient algebra structure a lindenbaum type of algebra we also show how certain models can help disprove some incorrect inferences in adjpc we end by developing complete syntax with fewer axioms and inference rules for the smaller semantical domain of all adjointness algebras whose implications satisfy the exchange principle

representations of the superalgebra f4 and young supertableaux the authors present a method for constructing typical as well as atypical finite dimensional representations of the superalgebra f4 for such a purpose young supertableaux are introduced