an invariant action for noncommutative gravity in four dimensions two main problems face the construction of noncommutative actions for gravity with star products the complex metric and finding an invariant measure the only gauge groups that could be used with star products are the unitary groups i propose an invariant gravitational action in d 4 dimensions based on the constrained gauge group u 2 2 broken to u 1 1 u 1 1 no metric is used thus giving a naturally invariant measure this action is generalized to the noncommutative case by replacing ordinary products with star products the four dimensional noncommutative action is studied and the deformed action to first order in deformation parameter is computed

cellular networks as models for planck scale physics starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the planck scale one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of our ordinary continuum physics and mathematics we base our own approach on what we call cellular networks consisting of cells nodes interacting with each other via bonds figuring as elementary interactions according to a certain local law geometrically our dynamical networks are living on graphs hence a substantial amount of the investigation is devoted to the development of various versions of discrete functional analysis and geometry on such almost random webs another important topic we address is a suitable concept of intrinsic fractal dimension on erratic structures of this kind in the course of the investigation we make comments concerning both different and related approaches to quantum gravity as say the spin network framework it may perhaps be said that certain parts of our programme seem to be a realization of ideas sketched by smolin some time ago

bergman kernel from path integral we rederive the expansion of the bergman kernel on kahler manifolds developed by tian yau zelditch lu and catlin using path integral and perturbation theory and generalize it to supersymmetric quantum mechanics one physics interpretation of this result is as an expansion of the projector of wave functions on the lowest landau level in the special case that the magnetic field is proportional to the kahler form this is relevant for the quantum hall effect in curved space and for its higher dimensional generalizations other applications include the theory of coherent states the study of balanced metrics noncommutative field theory and a conjecture on metrics in black hole backgrounds discussed in 24 we give a short overview of these various topics from a conceptual point of view this expansion is noteworthy as it is a geometric expansion somewhat similar to the dewitt seeley gilkey et al short time expansion for the heat kernel but in this case describing the long time limit without depending on supersymmetry

riemannian geometry of noncommutative surfaces a riemannian geometry of noncommutative n dimensional surfaces is developed as a first step toward the construction of a consistent noncommutative gravitational theory historically as well riemannian geometry was recognized to be the underlying structure of einstein s theory of general relativity and led to further developments of the latter the notions of metric and connections on such noncommutative surfaces are introduced and it is shown that the connections are metric compatible giving rise to the corresponding riemann curvature the latter also satisfies the noncommutative analog of the first and second bianchi identities as examples noncommutative analogs of the sphere torus and hyperboloid are studied in detail the problem of covariance under appropriately defined general coordinate transformations is also discussed and commented on as compared to other treatments

on the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

differential calculi and linear connections a method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra as an example the generalized quantum plane is studied it is found that there is a strong correlation but not a one to one correspondence between the module structure of the 1 forms and the metric torsion free connections on it in the commutative limit the connection remains as a shadow of the algebraic structure of the 1 forms

on the noncommutative riemannian geometry of glq n a recently proposed definition of a linear connection in noncommutative geometry based on a generalized permutation is used to construct linear connections on glq n restrictions on the generalized permutation arising from the stability of linear connections under involution are discussed candidates for generalized permutations on glq n are found it is shown that for a given generalized permutation there exists one and only one associated linear connection properties of the linear connection are discussed in particular its bicovariance torsion and commutative limit

equality of ordinary and symbolic powers of stanley reisner ideals abstract this paper studies properties of simplicial complexes with the equality i m i m for a given m 2 the main results are combinatorial characterizations of such complexes in the two dimensional case it turns out that there exists only a finite number of complexes with this property and that these complexes can be described completely as a consequence we are able to determine all complexes for which i m is cohen macaulay for some m 2 in particular there are complexes with i 2 i 2 or i 3 i 3 but i m i m for all m 4 and that if i m i m for some m 4 then i m i m for all m 1 similarly there are complexes for which i 2 is cohen macaulay but i m is not cohen macaulay for all m 3 and if i m is cohen macaulay for some m 3 then i is a complete intersection

linear connections on graphs in recent years discrete spaces such as graphs have attracted much attention as models for physical spacetime or as models for testing the spirit of noncommutative geometry in this work we construct the differential algebras for graphs by extending the work of dimakis et al and discuss linear connections and curvatures on graphs especially we calculate connections and curvatures explicitly for the general nonzero torsion case there is a metric but no metric compatible connection in general except the complete symmetric graph with two vertices

noncommutative differential geometry of matrix algebras the noncommutative differential geometry of the algebra mn c of complex n n matrices is investigated the role of the algebra of differential forms is played by the graded differential algebra c sl n c mn c mn c sl n c sl n c acting by inner derivations on mn c a canonical symplectic structure is exhibited for mn c for which the poisson bracket is to within a factor i the commutator also a canonical riemannian structure is described for mn c finally the analog of the maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge

on the cohen macaulayness of the conormal module of an ideal abstract in the present paper we investigate a question stemming from a long standing conjecture of vasconcelos given a generically complete intersection perfect ideal i in a regular local ring r when is it true that the cohen macaulayness of i i 2 or r i 2 implies that r i is gorenstein this property is known to hold for licci ideals and essentially squarefree monomial ideals we show that a positive answer actually holds for every monomial ideal we then give a positive answer for several special classes of ideals and provide application to algebroid curves with low multiplicity we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings for which the answer is negative and use them to show the sharpness of our main result as they lie in the first class of ideals not covered by it the homogeneous examples have been found thanks to the help of j c migliore as a by product we exhibit several classes of cohen macaulay ideals whose square is not cohen macaulay our methods work both in the homogeneous and in the local settings

nonsymmetric gravitation theory in noncommutative geometry we propose a reformulation of the nonsymmetric gravitation theory ngt in the context of noncommutative geometry it is shown that all of the moffat s added terms in his new version of the ngt have a geometric origin

pre torsors and equivalences abstract properties of most general non commutative torsors or a b torsors are analysed starting with pre torsors it is shown that they are equivalent to a certain class of galois extensions of algebras by corings it is shown that a class of faithfully flat pre torsors induces equivalences between categories of comodules of associated corings it is then proven that a b torsors correspond to monoidal functors and under some additional conditions equivalences between categories of comodules of bialgebroids

linear connections on extended space time a modification of kaluza klein theory is proposed which is general enough to admit an arbitrary finite noncommutative internal geometry it is shown that the existence of a nontrivial extension to the total geometry of a linear connection on space time places severe restrictions on the structure of the noncommutative factor a counter example is given

an l2 cohomology construction of negative spin mass zero equations for u p q abstract in this paper a geometric construction is given of the ladder representations of g u p q which correspond to negative spin solutions of the massless field equations the construction is based on the unitary model of the ladder representations in a bargmann segal fock space of square integrable 0 q forms the representations are constructed as holomorphic sections of certain vector bundles over g k and the construction is implemented via an integral transform analogous to the penrose transform of mathematical physics finally the resulting differential equations are exhibited explicitly over g k

the laplacian on homogeneous spaces the solution of the eigenvalue problem of the laplacian on a general homogeneous space g h is given here g is a compact semisimple lie group h is a closed subgroup of g and the rank of h is equal to the rank of g it is shown that the multiplicity of the lowest eigenvalue of the laplacian on g h is just the degeneracy of the lowest landau level for a particle moving on g h in the presence of the background gauge field moreover the eigenspace of the lowest eigenvalue of the laplacian on g h is up to a sign equal to the g equivariant index of the kostant s dirac operator on g h

groupoid approach to noncommutative quantization of gravity we propose a new scheme for quantizing gravity based on a noncommutative geometry our geometry corresponds to a noncommutative algebra a gc g c of smooth compactly supported complex functions with convolution as multiplication on the groupoid g e being the semidirect product of a structured space e of constant dimension or a smooth manifold and a group in the classical case e is the total space of the frame bundle and is the lorentz group the differential geometry is developed in terms of a z a submodule v of derivations of a and a noncommutative counterpart of einstein s equation is defined a pair a where is a subset of derivations of a satisfying the noncommutative einstein s equation is called an einstein pair we introduce the representation of a in a suitable hilbert space by completing a with respect to the corresponding norm change it into a c algebra and perform quantization with the help of the standard c algebraic method hermitian elements of this algebra are interp

geometrical tools for quantum euclidean spaces we apply one of the formalisms of noncommutative geometry to n q the quantum space covariant under the quantum group so q n over n q there are two so q n covariant differential calculi for each we find a frame a metric and two torsion free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature this generalizes results found in a previous article for the case of 3 q as in the case n 3 one has to slightly enlarge the algebra n q for n odd one needs only one new generator whereas for n even one needs two as in the particular case n 3 there is a conformal ambiguity in the natural metrics on the differential calculi over n q while in our previous article the frame was found by hand here we disclose the crucial role of the quantum group covariance and exploit it in the construction as an intermediate step we find a homomorphism from the cross product of n q with u q so n into n q an interesting result in itself

metrics on the real quantum plane using the frame formalism we determine some possible metrics and metric compatible connections on the noncommutative differential geometry of the real quantum plane by definition a metric maps the tensor product of two 1 forms into a function on the quantum plane it is symmetric in a modified sense namely in the definition of symmetry one has to replace the permutator map with a deformed map fulfilling some suitable conditions correspondingly also the definition of the hermitian conjugate of the tensor product of two 1 forms is modified but reduces to the standard one if coincides with the permutator the metric is real with respect to such modified structure

the chern simons action in noncommutative geometry a general definition of chern simons actions in noncommutative geometry is proposed and illustrated in several examples these examples are based on space times which are products of even dimensional riemannian spin manifolds by a discrete two point set if the algebras of operators describing the noncommutative spaces are generated by functions over such space times with values in certain clifford algebras the chern simons actions turn out to be the actions of topological gravity on the even dimensional spin manifolds by constraining the space of field configurations in these examples in an appropriate manner one is able to extract dynamical actions from chern simons actions

d3 0 d2 0 differential calculi on certain noncommutative super spaces in this paper we construct a covariant differential calculus on a quantum plane with two parametric quantum group as a symmetry group the two cases d2 0 and d3 0 are completely established we also construct differential calculi n 2 and n 3 nilpotent on super quantum spaces with one and two parametric symmetry quantum supergroup

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

curvature and geometric modules of noncommutative spheres and tori when considered as submanifolds of euclidean space the riemannian geometry of the round sphere and the clifford torus may be formulated in terms of poisson algebraic expressions involving the embedding coordinates and a central object is the projection operator projecting tangent vectors in the ambient space onto the tangent space of the submanifold in this note we point out that there exist noncommutative analogues of these projection operators which implies a very natural definition of noncommutative tangent spaces as particular projective modules these modules carry an induced connection from euclidean space and we compute its scalar curvature

on finite 4d quantum field theory in non commutative geometry the truncated 4 dimensional spheres4 and the action of the self interacting scalar field on it are constructed the path integral quantization is performed while simultaneously keeping theso 5 symmetry and the finite number of degrees of freedom the usual field theory uv divergences are manifestly absent

generalized hilbert coefficients and northcott s inequality abstract let r be a cohen macaulay local ring of dimension d with infinite residue field let i be an r ideal that has analytic spread l i d satisfies the g d condition and the weak artin nagata property a n d 2 we provide a formula relating the length i n 1 j i n to the difference p i n h i n where j is a general minimal reduction of i p i n and h i n are respectively the generalized hilbert samuel polynomial and the generalized hilbert samuel function we then use it to establish formulas to compute the generalized hilbert coefficients of i as an application we extend northcott s inequality to non m primary ideals furthermore when equality holds we prove that the ideal i enjoys nice properties indeed if this is the case then the reduction number of i is at most one and the associated graded ring of i is cohen macaulay we also recover results of g colome nin c polini b ulrich and y xie on the positivity of the generalized first hilbert coefficient j 1 i our work extends that of s huckaba c huneke and a ooishi to ideals that are not necessarily m primary

supersymmetric quantum theory and non commutative geometry classical differential geometry can be encoded in spectral data such as connes spectral triples involving supersymmetry algebras in this paper we formulate non commutative geometry in terms of supersymmetric spectral data this leads to generalizations of connes non commutative spin geometry encompassing non commutative riemannian symplectic complex hermitian and hyper kahler geometry a general framework for non commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples in particular the non commutative torus and the non commutative 3 sphere are studied in some detail

non commutative geometry of finite groups a finite set can be supplied with a group structure which can then be used to select classes of differential calculi on it via the notions of left right and bicovariance a corresponding framework has been developed by woronowicz more generally for hopf algebras including quantum groups a differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and moreover for the formulation of field theories and dynamics on finite sets associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures for a covariant calculus there are notions of invariance for linear connections and tensors all these concepts are explored for finite groups and illustrated with examples some results are formulated more generally for arbitrary associative hopf algebras in particular the problem of extension of a connection on a bimodule over an associative algebra to tensor products is investigated leading to the class of extensible connections it is shown that invariance properties of an extensible connection on a bimodule over a hopf algebra are carried over to the extension furthermore an invariance property of a connection is also shared by a dual connection which exists on the dual bimodule as defined in this work

koszul homology and the structure of low codimension cohen macaulay ideals the relationship between the properties of the koszul homology modules of two ideals connected by linkage is studied if the ideal i is either i a cohen macaulay ideal of codimension 3 or ii a gorenstein ideal of codimension 4 the one dimensional koszul module carries considerable information on the struct ural nature of the linkage class of i in case i or on the conormal module of i in case ii emphasis is given to the verification of the properties by computation introduction the aim of this paper is to examine the structure of cohen macaulay ideals of low i e 3 and 4 codimension as it is expressed by the koszul homology modules more specifically given a perfect ideal i of a gorenstein nng r we are interested in how i its numerical data e g betti numbers and degrees of its generating syzygies in the case of a homogeneous ideal ii properties of the full linkage class of i and iii whether i is suitably generated e g generated by a d sequence are mirrored on the module and the algebra structures of the homology of the koszul complex build on a generating set of i the man result theorem 2 4 is the proof that for a perfect ideal i of codimension threes the condition that the 1 dimensional koszul homology module h1 i r be cohen macaulay is an invanant of the full linkage class of i the framework for this result is provided by several facts first the classical descnptions of cohen macaulay ideals of codimension two and of gorenstein ideals of heit ht three as belonging to the linkage classes of complete intersections a be ga w then a theorem of huneke hu1 asserting that the property hi i r is cohen macaulay for i ms is an invariant of the eren linkage class of i finally in case of ideals of polynomial rings data on the koszul homology has the advantage of being in principle known at the outset that is hypotheses on them can be venfied once the ideal has been specified by a set of generators in the last section we indicate how symbolic computation packages may be used to obtain this informa tion to contrast already in the case of codimension four gorenstein ideals the depths of the koszul homology modules of directly linked ideals may differ sharply furthermore in codimension three we show that this invanance does not extend to the next koszul module to further enhance the role of h1 i r as a carrier of

quantization of the action of u k l on r2 k l abstract the natural action of u k l on c k l leaves invariant a real skew non degenerate bilinear form b which turns c k l into a symplectic manifold m the polarization f of m defined by the complex structure of c k l is non positive if l is the prequantization complex line bundle carried by m then u k l acts on the space u of square integrable l f forms on m leaving invariant the natural non degenerate but non definite inner product on u the polarization f also defines a closed densely defined covariant differential on u which is u k l invariant let denote orthocomplementation with respect to it is shown that the restriction of to the u k l stable subspace ker im is semi definite and that the unitary representation of uk l on the hilbert space h arising from by dividing out null vectors is unitarily equivalent to the representation of u k l obtained from the tensor product of the metap ectic and det 1 2 representations of mu k l the double cover of u k l

singular unitary representations and indefinite harmonic theory abstract square integrable harmonic spaces are defined and studied in a homogeneous indefinite metric setting in the process dolbeault cohomologies are unitarized and singlar unitary representations are obtained and studied

jacobian criteria for complete intersections the graded case letp be a positively graded polynomial ring over a fieldk of characteristic zero leti be a homogeneous ideal ofp and setr p i the paper investigates the homological properties of somer modules canonically associated withr among them the module r k of kahler differentials and the conormal modulei i 2 it is shown that a subexponential bound on the betti numbers of any of these modules implies thati is generated by ap regular sequence in particular the finiteness of the projective dimension of the conormal module impliesr is a complete intersection similarly the finiteness of the projective dimension of the differential module impliesr is a reduced complete intersection this provides strong converses to some well known properties of complete intersections and establishes special cases of conjectures of vasconcelos the proofs of these results make extensive use of differential graded homological algebra the crucial step is to show that any homomorphism of complexes from the minimal cotangent complexl r k of andre and quillen into the minimal free resolution of the irrelevant maximal ideal m ofr which extends the euler map r k is a split embedding of gradedr modules

on curvature in noncommutative geometry a general definition of a bimodule connection in noncommutative geometry has been recently proposed for a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra the corresponding curvatures are also compared

gravity in non commutative geometry we study general relativity in the framework of non commutative differential geometry as a prerequisite we develop the basic notions of non commutative riemannian geometry including analogues of riemannian metric curvature and scalar curvature this enables us to introduce a generalized einstein hilbert action for non commutative riemannian spaces as an example we study a space time which is the product of a four dimensional manifold by a two point space using the tools of non commutative riemannian geometry and derive its generalized einstein hilbert action in the simplest situation where the riemannian metric is taken to be the same on the two copies of the manifold one obtains a model of a scalar field coupled to einstein gravity this field is geometrically interpreted as describing the distance between the two points in the internal space

noncommutative geometry of super jordanian osph 2 1 covariant quantum space extending a recently proposed procedure of construction of various elements of differential geometry on noncommutative algebras we obtain these structures on noncommutative superalgebras as an example a quantum superspace covariant under the action of super jordanian osph 2 1 is studied it is shown that there exist a two parameter family of torsionless connections and the curvature computed from this family of connections is bilinear it is also shown that the connections are not compatible with the metric

gradient type differential operators and unitary highest weight representations of su p q abstract let c p q be equipped with a hermitian form of signature p q and let su p q denote the subgroup of the corresponding invariance group u p q consisting of matrices with determinant 1 to certain highest weights we associate a first order group invariant linear differential operator d whose kernel contains a unitary highest weight representation with highest weight the fock model realization of unitary highest weight representations of u p q is the fundamental tool used to implement this construction the operator d is shown to be equivalent to an operator d which acts on hol g k h the space of holomorphic vector valued functions defined on g k we identify a set 1 of highest weights such that ker d is a proper subspace of hol g k h and show that those in 1 correspond to points occurring at the far right of the discrete set in the classification scheme of enright howe and wallach first order differential equations arising from this proper containment are explicitly derived from the operator d we illustrate the fundamental nature of these first order equations by deriving from them a system which completely determines the irreducible spaces for ladder representations of su p q

quantum hall effect in higher dimensions matrix models and fuzzy geometry we give a brief review of the quantum hall effect in higher dimensions and its relation to fuzzy spaces for a quantum hall system the lowest landau level dynamics is given by a one dimensional matrix action whose large n limit produces an effective action describing the gauge interactions of a higher dimensional quantum hall droplet the bulk action is a chern simons type term whose anomaly is exactly cancelled by the boundary action given in terms of a chiral gauged wess zumino witten theory suitably generalized to higher dimensions we argue that the gauge fields in the chern simons action can be understood as parametrizing the different ways in which the large n limit of the matrix theory is taken the possible relevance of these ideas to fuzzy gravity is explained other applications are also briefly discussed

an cohomology construction of unitary highest weight modules for in this paper a geometric construction is given of all unitary highest weight modules of g u p q the construction is based on the unitary model of the kth tensor power of the metaplectic representation in a bargmannsegal fock space of square integrable differential forms the representations are constructed as holomorphic sections of certain vector bundles over g k and the construction is implemented via an integral transform analogous to the penrose transform of mathematical physics 0 introduction a major area of research in the representation theory of lie groups is the explicit analysis of naturally occurring unitary representations one aspect of this problem is that of producing geometric constructions of the unitary representations of a semisimple lie group g here we consider unitary highest weight representations of g u p q these representations have been classified algebraically in various ways see ehw j1 a and have been constructed as spaces of vector valued functions on the generalized unit disk d ds most singular highest weight representations appear in the l 2 cohomology construction of rsw z our approach starts with the kth tensor power of the metaplectic representation of u p q in the l cohomology space zo k cnxk as defined by blattner and rawnsley br the reductive dual pair u p q u k acts on this space results of kashiwara and vergne kv decompose this space into a sum of irreducible representations ua indexed by a set a c u k of representations of u k all unitary highest weight modules of u p q appear in this way for some k j2 ep each irreducible g u p q module corresponds to a vector bundle over g k as follows corresponding to a particular representation a of g each element c of g k determines a subspace vs c received by the editors october 31 1989 results presented in the special session on geometric methods in representation theory at the american mathematical society meeting in los angeles califormia november 1987 1980 mathematics subject classification 1985 revision primary 22e45

induced qcd from the noncommutative geometry of a supermanifold the noncommutative geometry of a two leaf parisi sourlas supermanifold in connes formalism using different k cycles over the grassmann algebra valued functions on the supermanifold is studied we find that the curvature of the trivial noncommutative vector bundle defines in the simplest case the super yang mills action coupled to a scalar field by considering a modified dirac operator and a suitable limit of its parameters we then obtain an action that turns out to be the continuum limit of the induced qcd in the kazakov migdal model

brs symmetry in connes non commutative geometry we extend the brs and anti brs symmetry to the two point space of connes non commutative model building scheme the constraint relations are derived and the quantum lagrangian constructed we find that the quantum lagrangian can be written as a functional of the curvature for symmetric gauges with the brs anti brs auxiliary field finding a geometrical interpretation as the extension of the higgs scalar

discrete kaluza klein from scalar fluctuations in noncommutative geometry we compute the metric associated with noncommutative spaces described by a tensor product of spectral triples well known results of the two sheets model distance on a sheet distance between the sheets are extended to any product of two spectral triples the distance between different points on different fibers is investigated when one of the triples describes a manifold one finds a pythagorean theorem as soon as the direct sum of the internal states viewed as projections commutes with the internal dirac operator scalar fluctuations yield a discrete kaluza klein model in which the extra component of the metric is given by the internal part of the geometry in the standard model this extra component comes from the higgs field

unitary structures on cohomology let cp 4 be endowed with a hermitian form h of signature p q let m be the manifold of r dimensional subspaces of cp p q on which h is positive defi nite and let e be the determinant bundle of the tautological bundle on mr we show starting from the oscillator representation of su p q that there is an invariant subspace of hr p r mr 9 e p k which defines a unitary representation of su p q for w ei mp gr r w is the subvariety of r dimensional subspaces of w integration over gr r w associates to an r p r cohomology class a a function p a on mp we show that this map is injective and provides an intertwin ing operator with representations of su p q on spaces of holomorphic functions on siegel space

metrics and pairs of left and right connections on bimodules properties of metrics and pairs consisting of left and right connections are studied on the bimodules of differential 1 forms those bimodules are obtained from the derivation based calculus of an algebra of matrix valued functions and an slq 2 c covariant calculus of the quantum plane at a generic q and the cubic root of unity it is shown that in the aforementioned examples giving up the middle linearity of metrics significantly enlarges the space of metrics a metric compatibility condition for the pairs of left and right connections is defined also a compatibility condition between a left and right connection is discussed consequences entailed by reducing to the center of a bimodule the domain of those conditions are investigated in detail alternative ways of relating left and right connections are considered

quantum k theory i the chern character we construct a cocycle on an infinite dimensional generalization of ap summable fredholm module our framework is related to connes cyclic cohomology and is motivated by our work on index theory on infinite dimensional manifolds thep summability condition is characteristic of dimensiono p we replace this assumption by the requirement that there exists an underlying heat kernel which is trace class then we use the heat kernel to regularize states in dimension independent fashion our cocycle may be interpreted as an infinite dimensional chern character

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

perfect fluid theory and its extensions we review the canonical theory for perfect fluids in euler s and lagrange s formulations the theory is related to a description of extended structures in higher dimensions internal symmetry and supersymmetry degrees of freedom are incorporated additional miscellaneous subjects that are covered include physical topics concerning quantization as well as mathematical issues of volume preserving diffeomorphisms and representations of chern simons terms vortex or magnetic helicity

cohen macaulayness of powers of two dimensional squarefree monomial ideals abstract two dimensional squarefree monomial ideals can be seen as the stanley reisner ideals of graphs the main results of this paper are combinatorial characterizations for the cohen macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph