twisted dual group algebras equivariant deformations of c0 g abstract we consider a family of twisted fourier algebras a g of a locally compact group g which in the case of a abelian group g are the fourier transforms of the usual twisted group algebras of over cap g o the corresponding c algebras c g o are deformations of c 0 g which are equivariant in the sense that g still acts by left translation the main examples come from cocycles on the dual of an abelian subgroup h of g we prove that for such cocycles the twisted dual group algebras c g o are induced from the twisted group algebras c h o and we give detailed formulas for the multiplication on a g which extend to larger dense subalgebras of c 0 g and c b g we anticipate that these larger subalgebras will be useful for constructing deformations of homogeneous spaces c 0 g

deformations of compact quantum groups via rieffel s quantization it is shown that compact quantum groups containing torus subgroups can be deformed into new compact quantum groups under rieffel s quantization this is applied to showing that the two classes of compact quantum groupskqu andkq studied by levendorkii and soibelman are strict deformation quantization of each other and that the quantum groupsau m have many deformations

multi specialization and multi asymptotic expansions abstract in this paper we extend the notion of specialization functor to the case of several closed submanifolds satisfying some suitable conditions applying this functor to the sheaf of whitney holomorphic functions we construct different kinds of sheaves of multi asymptotically developable functions whose definitions are natural extensions of the definition of strongly asymptotically developable functions introduced by majima

rieffel deformation via crossed products we start from rieffel data a where a is a c algebra is an action of an abelian group on a and is a 2 cocycle on the dual group using landstad theory of crossed product we get a deformed c algebra a in the case of rn we obtain a very simple proof of invariance of k groups under the deformation in the general case we also get a very simple proof that nuclearity is preserved under the deformation we show how our approach leads to quantum groups and investigate their duality the general theory is illustrated by an example of the deformation of sl 2 c a description of it in terms of noncommutative coordinates is given

equivariant deformations of homogeneous spaces abstract c algebraic deformations of homogeneous spaces g are constructed by completing dense subspaces of c 0 g in a different multiplication and c norm these deformations are equivariant in the sense that they still carry a natural action of g by left translation the motivating examples are the noncommutative heisenberg manifolds of rieffel but the construction given here is based on the authors twisted dual group algebras which are equivariant deformations of c 0 g the procedure is general enough to give other recently studied families of noncommutative spaces and some interesting new examples of simple c algebras

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

matricial quantum gromov hausdorff distance abstract we develop a matricial version of rieffel s gromov hausdorff distance for compact quantum metric spaces within the setting of operator systems and unital c algebras our approach yields a metric space of isometric unital complete order isomorphism classes of metrized operator systems which in many cases exhibits the same convergence properties as those in the quantum metric setting as for example in rieffel s approximation of the sphere by matrix algebras using berezin quantization within the metric subspace of metrized unital c algebras we establish the convergence of sequences which are cauchy with respect to a larger leibniz distance and we also prove an analogue of the precompactness theorems of gromov and rieffel

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

proper actions fixed point algebras and naturality in nonabelian duality abstract suppose a locally compact group g acts freely and properly on a locally compact hausdorff space x and let be the induced action on c 0 x we consider a category in which the objects are c dynamical systems a g for which there is an equivariant homomorphism of c 0 x into the multiplier algebra m a rieffel has shown that such systems are proper and saturated and hence have a generalized fixed point algebra a which is morita equivalent to a r g we show that the assignment a a is functorial and that rieffel s morita equivalence is natural in a suitable sense we then use our results to prove a categorical version of landstad duality which characterizes crossed products by coactions and to prove that mansfield imprimitivity for crossed products by homogeneous spaces is natural

deformation quantization for actions of qpd abstract the main objective of this article is to develop the theory of deformation of c algebras endowed with a group action from the perspective of non formal equivariant quantization this program initiated in 4 aims to extend rieffel s deformation theory 27 for more general groups than r d in 4 we have constructed such a theory for a class of non abelian lie groups in the present article we study the somehow opposite situation of abelian but non lie groups more specifically we construct here a deformation theory of c algebras endowed with an action of a finite dimensional vector space over a non archimedean local field of characteristic different from 2 at the root of our construction stands the p adic version of the weyl quantization introduced by haran 12 and further extended by bechata 1 and unterberger 34

2 cocycles and twisting of kac algebras we describe the twisting construction with the help of 2 cocycles on hopf von neumann and george kac algebras we show that twisted kac algebras are again kac algebras using this construction we give a wide class of new quantizations of the heisenberg group and describe several series of non trivial finite dimensional hopf algebras kac algebras of dimensions 4n and as twisting of finite groups

leray s quantization of projective duality 2 review on correspondences for sheaves and d modules 4 2 1 notations 4 2 1 1 geometry 4 2 1 2 sheaves 4 2 1 3 d modules 5 2 1 4 e modules 6 2 2 correspondences for sheaves and d modules 6

inclusions irreductibles de facteurs et unitaires multiplicatifs ii resume from an irreducible depth 2 inclusion m 0 m 1 of factors with a faithful semifinite normal operator valued weight verifying a regularity condition we construct the canonical tower m i i n we prove then that both relative commutants m 2 m 0 and m 3 m 1 bear woronowicz algebra structures dual to each other as defined by masuda and nakagami moreover there exists an outer action of m 3 m 1 on m 1 such that m 0 is the fixed point algebra and m 2 is the crossed product the tower appears then as the tower of successive crossed products

metric aspects of noncommutative homogeneous spaces for a closed cocompact subgroup of a locally compact group g given a compact abelian subgroup k of g and a homomorphism k g satisfying certain conditions landstad and raeburn constructed equivariant noncommutative deformations c g of the homogeneous space g generalizing rieffel s construction of quantum heisenberg manifolds we show that when g is a lie group and g is connected given any norm on the lie algebra of g the seminorm on c g induced by the derivation map of the canonical g action defines a compact quantum metric furthermore it is shown that this compact quantum metric space depends on continuously with respect to quantum gromov hausdorff distances

mathieu moonshine and symmetry surfing mathieu moonshine the observation that the fourier coefficients of the elliptic genus on k3 can be interpreted as dimensions of representations of the mathieu group has been proven abstractly but a conceptual understanding in terms of a representation of the mathieu group on the bps states is missing some time ago taormina and wendland showed that such an action can be naturally defined on the lowest non trivial bps states using the idea of symmetry surfing i e by combining the symmetries of different k3 sigma models in this paper we find non trivial evidence that this construction can be generalized to all bps states

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

continuous fields of c algebras coming from group cocycles and actions recently i have been attempting to formulate a suitable c algebraic framework for the subject of deformation quantization 3 19 continuous fields of c algebras provide one of the key elements for this framework the main examples of deformation quantizations which i have constructed up to now in this c algebra framework come from letting either cocycles on groups or actions of groups vary it has thus become necessary to show that one obtains in this way fields of c aigebras that are indeed continuous since this material is of a general nature and can be useful in other situations 4 6 11 13 it has seemed appropriate to give a separate exposition of it in the present article section 1 of this article contains a review of the published results on continuous fields which we will need as well as a discussion of the fact that the approach which we will take involves treating upper and lower semi continuity separately in sect 2 we discuss the continuity of fields of c algebras which arise from varying cocycles on groups while in sect 3 we do the same for actions which vary

c bialgebra defined by the direct sum of cuntz algebras abstract let o denote the c algebra defined by the direct sum of cuntz algebras o n 1 n where we write o 1 as c for convenience we introduce a non degenerate homomorphism from o to o o which satisfies the coassociativity and a homomorphism e from o to c such that e id id id e furthermore we show the following i for the smallest unitization o of o there exists a unital extension e of the pair e on o such that o is a unital bialgebra with the unital counit e ii the pair o satisfies the cancellation law iii there exists a unital homomorphism from o to the multiplier algebra m o o of o o such that id id iv there is no antipode for o v there exists a unique haar state on o vi for a certain one parameter bialgebra automorphism group of o there exists a kms state on o

hopf images in locally compact quantum groups abstract this manuscript is devoted to the study of the concept of a generating subset a k a hopf image of a morphism in the setting of locally compact quantum groups the aim of this paper is to provide an accurate description of the hopf image of a given morphism we extend and unify the previously existing approaches for compact and discrete quantum groups and present some results that can shed light on some local perspective in the theory of quantum groups in particular we provide a characterization of fullness of hopf image in the language of partial actions as well as in representation theoretic terms in the spirit of representation c categories extending some known results to the broader setting of non compact quantum groups

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

traces on noncommutative homogeneous spaces abstract we study properties of c algebraic deformations of homogeneous spaces g which are equivariant in the sense that they preserve the natural action of g by left translation the center is shown to be isomorphic to c g g 0 for a certain subgroup g 0 of g and there is a 1 1 correspondence between normalised traces and probability measures on g g 0 this makes it possible to represent the deformed algebra as operators over l 2 g applications to k theory are also mentioned

radon transform and the cavalieri condition a cohomological approach we apply the theory of integral transforms for sheaves andd modules to the study of the real radon transform by identifying abstract adjunction formulas with the explicit integral formulas appearing in the literature we give in particular a cohomological interpretation of gelfand s cavalieri condition and of helgason s support theorem

quantum group of orientation preserving riemannian isometries we formulate a quantum group analogue of the group of orientation preserving riemannian isometries of a compact riemannian spin manifold more generally of a possibly r twisted and of compact type spectral triple the main advantage of this formulation which is directly in terms of the dirac operator is that it does not need the existence of any good laplacian as in our previous works on quantum isometry groups several interesting examples including those coming from rieffel type deformation as well as the equivariant spectral triples on su 2 and s c2 are discussed

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

integral transforms with exponential kernels and laplace transform let x z y be a correspondence of complex manifolds f g we study integral transforms associated to kernels exp o with p meromor phic on z acting on formal or moderate cohomologies our main application is the laplace transform in this case x is the projective compactification of the vector space v c y is its dual space z x x y and o z w z w we obtain the isomorphisms f 0 ov f n f ov thom f v thom far n ov this content downloaded from 207 46 13 59 on mon 13 jun 2016 05 54 06 utc all use subject to http about jstor org terms 972 masaki kashiwara and pierre schapira where f is a conic and rll constructible sheaf on v and fa is its fourier sato transform some applications are discussed rims kyoto university kyoto 606 01 japan institut de mathematiques universit9 paris vi case 82 4 pl jussieu 75252 paris france e mail address schapirafmath jussieu fr this content downloaded from 207 46 13 59 on mon 13 jun 2016 05 54 06 utc all use subject to http about jstor org terms

rieffel deformation of homogeneous spaces let g1 g be a closed subgroup of a locally compact group g and let x g g1 be the quotient space of left cosets let x c0 x x be the corresponding g c algebra where g c0 g suppose that is a closed abelian subgroup of g1 and let be a 2 cocycle on the dual group let g be the rieffel deformation of g using the results of the previous paper of the author we may construct g c algebra x the rieffel deformation of x on the other hand we may perform the rieffel deformation of the subgroup g1 obtaining the closed quantum subgroup g1 g which in turn by the results of s vaes leads to the g c algebra g g1 in this paper we show that g g1 x we also consider the case where g is not a subgroup of g1 for which we cannot construct the subgroup g1 then generically x cannot be identified with a quantum quotient what may be shown is that it is a g simple object in the category of g c algebras

much ado about mathieu abstract eguchi ooguri and tachikawa observed that the coefficients of the elliptic genus of type ii string theory on k3 surfaces appear to be dimensions of representations of the largest mathieu group subsequent work by several people established a candidate for the elliptic genus twisted by each element of m 24 in this paper we prove that the resulting sequence of class functions are true characters of m 24 proving the eguchi ooguri tachikawa mathieu moonshine conjecture the integrality of multiplicities is proved using a small generalisation of sturm s theorem while positivity involves a modification of a method of hooley for finding an effective bound on a family of selberg kloosterman zeta functions at s 3 4 we also prove the evenness property of the multiplicities as conjectured by several authors and use that to investigate the proposal of gaberdiel hohenegger volpato that mathieu moonshine lifts to the conway groups co 0 and co 1 we identify the role group cohomology plays in both mathieu moonshine and monstrous moonshine in particular this gives a cohomological interpretation for the non fricke elements in norton s generalised monstrous moonshine conjecture and gives a lower bound for h 3 b c o 1 c

order unit quantum gromov hausdorff distance we introduce a new distance distoq between compact quantum metric spaces we show that distoq is lipschitz equivalent to rieel s distance distq and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to distoq as applications we show that the continuity of a parameterized family of quantum metric spaces induced by ergodic actions of a fixed compact group is determined by the multiplicities of the actions generalizing rieel s work on noncommutative tori and integral coadjoint orbits of semisimple compact connected lie groups we also show that the deformations of connes and landi are continuous in the parameter

noncommutative geometry and reality we introduce the notion of real structure in our spectral geometry this notion is motivated by atiyah s kr theory and by tomita s involution j it allows us to remove two unpleasant features of the connes lott description of the standard model namely the use of bivector potentials and the asymmetry in the poincare duality and in the unimodularity condition

hopf galois systems abstract we introduce the concept of hopf galois system a reformulation of the notion of galois extension of the base field for a hopf algebra the main feature of our definition is a generalization of the antipode of an ordinary hopf algebra we present several examples which indicate that although our axiomatic is slightly more complicated than the classical one it is also more natural and easier to handle with the main application of hopf galois systems is the construction of monoidal equivalences between comodule categories

on projective representations for compact quantum groups we study actions of compact quantum groups on type i factors which may be interpreted as projective representations of compact quantum groups we generalize to this setting some of woronowicz s results concerning peter weyl theory for compact quantum groups the main new phenomenon is that for general compact quantum groups more precisely those which are not of kac type not all irreducible projective representations have to be finite dimensional as applications we consider the theory of projective representations for the compact quantum groups associated with group von neumann algebras of discrete groups and consider a certain non trivial projective representation for quantum su 2

on a certain approach to quantum homogeneous spaces we propose a definition of a quantum homogeneous space of a locally compact quantum group we show that classically it reduces to the notion of homogeneous spaces giving rise to an operator algebraic characterization of the transitive group actions on the quantum level our definition goes beyond the quotient case providing a framework which besides the vaes quotient of a locally compact quantum group by its closed quantum subgroup our main motivation is also compatible with generically non quotient quantum homogeneous spaces of a compact quantum group studied by p podle as well as the rieffel deformation of g homogeneous spaces finally our definition rules out the paradoxical examples of the non compact quantum homogeneous spaces of a compact quantum group

deformation of a kac algebra by an abelian subgroup some examples of quantum groups in literature arise as deformations of a locally compact group by a dual 2 cocycle we make this construction in the framework of kac algebras we show that these deformations are still kac algebras using this construction we give new quantizations of the heisenberg group from this point of view we analyse the dimension 8 non trivial example of kac and paljutkin and give a new example of non trivial dimension 12 semi simple hopf algebras a dimension 12 kac algebra

krein duality for compact quantum groups we define noncommutative krein algebras and extend the classical krein duality for compact groups to compact quantum groups

hodge elliptic genera and how they govern k3 theories the complex hodge elliptic genus and its conformal field theoretic counterpart were recently introduced by kachru and tripathy refining the traditional complex elliptic genus we construct a different so called chiral hodge elliptic genus which is expected to agree with the generic conformal field theoretic hodge elliptic genus in contrast to the complex hodge elliptic genus as originally defined for k3 surfaces x the chiral hodge elliptic genus is shown to be independent of all moduli moreover employing kapustin s results on infinite volume limits it is shown that it agrees with the generic conformal field theoretic hodge elliptic genus of k3 theories while the complex hodge elliptic genus does not this new invariant governs part of the field content of k3 theories supporting the idea that all their spaces of states have a common subspace which underlies the generic conformal field theoretic hodge elliptic genus and thereby the complex elliptic genus mathematically this space is modelled by the sheaf cohomology of the chiral de rham complex of x it decomposes into irreducible representations of the n 4 superconformal algebra such that the multiplicity spaces of all massive representations have precisely the dimensions required in order to furnish the representation of the mathieu group m 24 that is predicted by mathieu moonshine this is interpreted as evidence in favour of the ideas of symmetry surfing which have been proposed by taormina and the author along with the claim that the sheaf cohomology of the chiral de rham complex is a natural home for mathieu moonshine these investigations also imply that the generic chiral algebra of k3 theories is precisely the n 4 superconformal algebra at central charge c 6 if the usual predictions on infinite volume limits from string theory hold true

rieffel type discrete deformation of finite quantum groups we introduce a discrete deformation of rieffel type for finite quantum groups using this we give an example of a finite quantum group a of order 18 such that neither a nor its dual can be expressed as a crossed product of the form c g1 g2 with g1 and g2 ordinary finite groups we also give a deformation of finite groups of lie type by using their maximal abelian subgroups

radon and fourier transforms for d modules the fourier and radon hyperplane transforms are closely related and one such relation was established by brylinski 4 in the framework of holonomic d modules the integral kernel of the radon hyperplane transform is associated with the hypersurface scp p of pairs x y where x is a point in the n dimensional complex projective space p belonging to the hyperplane yap as it turns out a useful variant is obtained by considering the integral transform associated with the open complement u of s in p p in the first part of this paper we generalize brylinski s result in order to encompass this variant of the radon transform and also to treat arbitrary quasi coherent d modules as well as twisted abelian sheaves our proof is entirely geometrical and consists in a reduction to the onedimensional case by the use of homogeneous coordinates the second part of this paper applies the above result to the quantization of the radon transform in the sense of 7 first we deal with line bundles more precisely let p 1 4 p v be the projective space of lines in the vector spacev denote by 3 d r

c algebres de kac et algebres de kac s baaj and g skandalis have proved that to every kac algebra as studied by j m schwartz and the first author corresponds a canonical c kac algebra as studied by the second author this article proves the converse result so we have then a complete proof that the c version and the von neumann version of this theory are exactly equivalent as are in the commutative case thanks to a weil s theorem locally compact groups and measured groups with a left invariant measure

quantized gromov hausdorff distance abstract a quantized metric space is a matrix order unit space equipped with an operator space version of rieffel s lip norm we develop for quantized metric spaces an operator space version of quantum gromov hausdorff distance we show that two quantized metric spaces are completely isometric if and only if their quantized gromov hausdorff distance is zero we establish a completeness theorem as applications we show that a quantized metric space with 1 exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized gromov hausdorff distance and that matrix algebras converge naturally to the sphere for quantized gromov hausdorff distance

the primitive ideal space of two step nilpotent group c algebras abstract let n be a two step nilpotent locally compact second countable group having center z and quotient a n z we study the jacobson topology on the primitive ideal space prim c n of the group c algebra of n we are able to describe this topology in terms of convergence of subgroup representation pairs as used by the first author in an earlier work under appropriate conditions on n we are able to describe prim c n globally as the quotient of a principal bundle over modulo an equivalence relation determined entirely by the group structure we use this second result to compute the primitive ideal spaces of several examples including all finitely generated non torsion two step nilpotent discrete groups of rank less than or equal to five applications of our methods to more general central twisted crossed products are discussed

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

on cocycle twisting of compact quantum groups abstract we provide a class of examples of compact quantum groups and unitary 2 cocycles on them such that the twisted quantum groups are non compact but still locally compact quantum groups in the sense of kustermans and vaes this also gives examples of cocycle twists where the underlying c algebra of the quantum group changes

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

compact quantum metric spaces and ergodic actions of compact quantum groups we show that for any co amenable compact quantum group a c g there exists a unique compact hausdorff topology on the set ea g of isomorphism classes of ergodic actions of g such that the following holds for any continuous field of ergodic actions of g over a locally compact hausdorff space t the map t ea g sending each t in t to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of in each fibre is continuous over t for every equivalence class of irreducible unitary representations of g generalizations for arbitrary compact quantum groups are also obtained in the case g is a compact group the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of landstad and wassermann podle spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum su 2 group we also introduce a notion of regularity for quantum metrics on g and show how to construct a quantum metric from any ergodic action of g starting from a regular quantum metric on g furthermore we introduce a quantum gromov hausdorff distance between ergodic actions of g when g is separable and show that it induces the above topology

on a morita equivalence between the duals of quantum su 2 and quantum e 2 abstract let su q 2 and e q 2 be woronowicz s q deformations of respectively the compact lie group su 2 and the non trivial double cover of the lie group e 2 of euclidean transformations of the plane we prove that in some sense their duals are morita equivalent locally compact quantum groups in more concrete terms we prove that the von neumann algebraic quantum groups l su q 2 and l e q 2 are unitary cocycle deformations of each other

residual bar partial cohomology and the complex radon transform on subvarieties of mathbb c p n we show that the complex radon transform realizes an isomorphism between the quotient space of residual bar partial cohomologies of a locally complete intersection algebraic subvariety in a linearly concave domain of mathbb c p n and the space of holomorphic solutions of the associated homogeneous system of differential equations with constant coefficients in the dual domain in mathbb c p n

noncommutative finite dimensional manifolds ii moduli space and structure of noncommutative 3 spheres this paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3 spheres we develop the notion of central quadratic form for quadratic algebras and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart it allows to construct a morphism from an involutive quadratic algebra to a c algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form we apply the general theory in the case of noncommutative 3 spheres and show that the above morphism corresponds to a natural ramified covering by a noncommutative 3 dimensional nilmanifold we then compute the jacobian of the ramified covering and obtain the answer as the product of a period of an elliptic integral by a rational function we describe the real and complex moduli spaces of noncommutative 3 spheres relate the real one to root systems and the complex one to the orbits of a birational cubic automorphism of three dimensional projective space we classify the algebras and establish duality relations between them

non compact quantum groups associated with abelian subgroups letg be a lie group for any abelian subalgebra mathfrak h of the lie algebra g ofg and any r in mathfrak h lambda mathfrak h the difference of the left and right translates ofr gives a compatible poisson bracket ong we show how to construct the corresponding quantum group in thec algebra setting the main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onc algebras

cohomology and coquasi bialgebra extensions associated to a matched pair of bialgebras abstract by using braid diagrams we explicitly reconstruct the cohomology associated to a matched pair of cocommutative bialgebras in order to give a method of constructing coquasi bialgebras which generalize bialgebras and classifying them up to monoidal equivalence of their comodule categories an alternative homological proof is given for schauenburg s generalized kac sequence involving the abelian group opext h k of bialgebra extensions we define an abelian group opext h k of coquasi bialgebra extensions associated to a singer pair h k of bialgebras and prove a variant of schauenburg s sequence which involves the group it is also proved that there is a natural isomorphism opext h 1 k opext h 2 k that preserves monoidal equivalence classes if h 1 k and h 2 k arise from such matched pairs that are shifts of each other

deformation of c algebras by cocycles on locally compact quantum groups given a c algebra a with a left action of a locally compact quantum group g on it and a unitary 2 cocycle on g we define a deformation a of a the construction behaves well under certain additional technical assumptions on the most important of which is regularity meaning that c0 g g is isomorphic to the algebra of compact operators on some hilbert space in particular then a is stably isomorphic to the iterated twisted crossed product g op g a also in good situations the c algebra a carries a left action of the deformed quantum group g and we have an isomorphism g a g a when g is a genuine locally compact group we show that the action of g on c0 g cr g is always integrable stronger assumptions of properness and saturation of the action imply regularity as an example we make a preliminary analysis of the cocycles on the duals of some solvable lie groups recently constructed by bieliavsky et al and discuss the relation of our construction to that of bieliavsky and gayral

the 290 fixed point sublattices of the leech lattice abstract we determine the orbits of fixed point sublattices of the leech lattice with respect to the action of the conway group co 0 there are 290 such orbits detailed information about these lattices the corresponding coinvariant lattices and the stabilizing subgroups is tabulated in several tables

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

k groups of c algebras deformed by actions of rd abstract we show that the topological k groups of a c algebra deformed by an action of r d are isomorphic to those of the original c a1gebra we do this by exhibiting the deformed algebra as a suitable generalized fixed point algebra this also shows that the deformed algebra is nuclear if the original one is

the radon penrose correspondence ii line bundles and simple d modules on a complex manifoldxof dimension 3 we show that coherent dx modules which are simple all overp xare classified by pic x the family of holomorphic line bundles onx as a corollary using the penrose transform we obtain that on the complex minkowski space m simple dm modules along the characteristic variety of the wave equation are classified by half integers the so called helicity

θ deformations as compact quantum metric spaces let m be a compact spin manifold with a smooth action of the n torus connes and landi constructed deformations m of m parameterized by n n real skew symmetric matrices the m s together with the canonical dirac operator d open image in new window on m are an isospectral deformation of m the dirac operator d defines a lipschitz seminorm on c m which defines a metric on the state space of c m we show that when m is connected this metric induces the weak topology this means that m is a compact quantum metric space in the sense of rieffel