quantization of poisson lie groups and applications letg be a connected poisson lie group we discuss aspects of the question of drinfel d can g be quantized and give some answers wheng is semisimple a case where the answer isyes we introduce quantizable poisson subalgebras ofc g related to harmonic analysis ong they are a generalization of f r t models of quantum groups and provide new examples of quantized poisson algebras

bidual renormings of banach spaces abstract given a separable banach space x with no isomorphic copies of l 1 and a separable subspace y of its bidual we provide a sufficient condition on y to ensure that x admits an equivalent norm such that the restriction to y of the corresponding bidual norm is midpoint locally uniformly rotund this result applies to the separable subspaces of the bidual of a banach space with a shrinking unconditional schauder basis and to the bidual of the james space

text mining by tsallis entropy abstract long range correlations between the elements of natural languages enable them to convey very complex information complex structure of human language as a manifestation of natural languages motivates us to apply nonextensive statistical mechanics in text mining tsallis entropy appropriately ranks the terms relevance to document subject taking advantage of their spatial correlation length we apply this statistical concept as a new powerful word ranking metric in order to extract keywords of a single document we carry out an experimental evaluation which shows capability of the presented method in keyword extraction we find that tsallis entropy has reliable word ranking performance at the same level of the best previous ranking methods

deformations of lie group and lie algebra representations a complete study of deformations of lie group and lie algebra representations including differentiability and integrability results is given adapted results are given in the semisimple case a notion of induced deformations is introduced various examples are given including deformations of indecomposable representations

a growth model that generates an n tuple zipf law abstract the n tuple zipf law widely exists in real symbol sequences such as dna and language based on the n tuple zipf analysis on a vast amount of empirical data we propose a model that can generate sequences with n tuple zipf features in simple words it is an iterative copy and paste process repeatedly select a random subsequence from the current sequence and attach it to the end numerical results of our model show that the n tuple zipf law exists in our model generated data we give detailed analytical derivations of our model and get two estimation equations the zipf exponent and the minimal length of n tuple for zipf law to appear our model can also reproduce the symmetry breaking process of atgc number differences in dna data

strictly convex norms and topology we introduce a new topological property called and the corresponding class of topological spaces which includes spaces with g diagonals and gruenhage spaces using we characterize those banach spaces which admit equivalent strictly convex norms and give an internal topological characterization of those scattered compact spaces k for which the dual banach space c k admits an equivalent strictly convex dual norm we establish some relationships between and other topological concepts and the position of several well known examples in this context for instance we show that c k admits an equivalent strictly convex dual norm where k is kunen s compact s space also under additional axioms we provide examples of compact scattered non gruenhage spaces of cardinality 1 having

on a classical renorming construction of v klee we further develop a classical geometric construction of v klee and show typically that if x is a nonreflexive banach space with separable dual then x admits an equivalent norm which is frechet differentiable locally uniformly rotund its dual norm is uniformly g teaux differentiable the weak and the norm topologies coincide on the sphere of x and yet is not rotund this proves a stronger form of a conjecture of v klee

distributions of triplets in genetic sequences abstract distributions of triplets in some genetic sequences are examined and found to be well described by a two parameter markov process with a sparse transition matrix the standard deviations of all the relevant parameters are not large indicating that most sequences gather in a small region in the parameter space different sequences have very similar values of the entropy calculated directly from the data and the two parameters characterizing the markov process fitting the sequence no relevance with taxonomy or coding noncoding is clearly observed

continuous functions on totally ordered spaces that are compact in their order topologies abstract the totally ordered spaces that are compact and connected in their order topologies are characterized as images of lexicographic cubes the banach space of continuous functions on the lexicographic product of compact totally ordered spaces whose spaces of continuous functions have locally uniformly convex norms has an equivalent locally uniformly convex norm if and only if the product is countable the banach space of continuous functions on a compact totally ordered space always has an equivalent kadec norm those compact totally ordered spaces for which the banach space of continuous functions has an equivalent locally uniformly convex norm are characterized in terms of the bounded decreasing interval functions that can be defined on the intervals of the totally ordered spaces some examples are discussed in detail

topological properties of banach spaces weakly compact sets in banach spaces are fragmentable we introduce the weaker notion of the a fragmentability of a subset of a banach space a notion that has a far wider range of applicability we prove that all weakly cech analytic subsets of a banach space are a fragmented this implies that all banach spaces having kadec norms are a fragmented however l is not fragmented

some nonlinear maps and renormings of banach spaces we consider two classes of nonlinear maps between normed spaces which are relevant for the study of banach spaces that admit equivalent locally uniformly rotund and midpoint locally uniformly rotund norms it turns out to be a useful characterization of such maps in terms of optimization topology and probability using these characterizations we obtain some renorming results

symmetric pairs and gorelik elements abstract the second author gave a formula for the elements of the enveloping algebra of a lie superalgebra defined by gorelik under an appropriate unimodularity assumption we show that this formula is a particular case of a formula for the jacobian of the exponential map of a symmetric superspace

the minimal primitive spectrum of the enveloping algebra of the lie superalgebra osp 1 2l abstract a well known theorem of duflo the annihilation theorem claims that the annihilator of a verma module in the enveloping algebra of a complex semisimple lie algebra is centrally generated for the lie superalgebra osp 1 2 l this result does not hold in this article we introduce a correct analogue of the centre for which the annihilation theorem does hold in the case osp 1 2 l this substitute of the centre is the centralizer of the even part of the enveloping algebra this algebra shares some nice properties with the centre as a consequence of the annihilation theorem we obtain the description of the minimal primitive spectrum of the enveloping algebra of the lie superalgebra osp 1 2 l we also deduce a criterium for a osp 1 2 l verma module to be a direct sum of sp 2 l verma modules

hilbert generated spaces we classify several classes of the subspaces of banach spaces x for which there is a bounded linear operator from a hilbert space onto a dense subset in x dually we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in hilbert spaces or in c0 spaces in their weak topology the existence of such embeddings is characterized by the existence of certain uniformly g teaux smooth norms

alur dual renormings of banach spaces we give a covering type characterization for the class of dual banach spaces with an equivalent alur dual norm

on casimir s ghost we define on the universal enveloping superalgebra of a nonstandard adjoint action endowing it with a module structure this allows in particular to construct a bosonic operator which anticommutes with all the fermionic generators and which appears to be the square root of a certain casimir operator

smoothing of bump functions abstract let x be a separable banach space with a schauder basis admitting a continuous bump which depends locally on finitely many coordinates then x admits also a c smooth bump which depends locally on finitely many coordinates

benford s law and first letter of words a universal first letter law fll is derived and described it predicts the percentages of first letters for words in novels the fll is akin to benford s law bl of first digits which predicts

keyword extraction by nonextensivity measure the presence of a long range correlation in the spatial distribution of a relevant word type in spite of random occurrences of an irrelevant word type is an important feature of human written texts we classify the correlation between the occurrences of words by nonextensive statistical mechanics for the word ranking process in particular we look at the nonextensivity parameter as an alternative metric to measure the spatial correlation in the text from which the words may be ranked in terms of this measure finally we compare different methods for keyword extraction

on binormality in non separable banach spaces we study binormality a separation property of the norm and weak topologies of a banach space we show that every banach space which belongs to a p class is binormal we also show that the asplundness of a banach space is equivalent to a related separation property of its dual space

primitive ideals twisting functors and star actions for classical lie superalgebras we study three related topics in representation theory of classical lie superalgebras the first one is classification of primitive ideals i e annihilator ideals of simple modules and inclusions

on trees and dual rotund norms abstract we give a necessary and sufficient condition for the existence of an equivalent dual rotund norm on c 0 l 1 where is a tree the condition is expressed succinctly in terms of the embeddability of into a particular totally ordered set and compares very well with the analogous situation for local uniform rotundity this resolves an open problem from haydon s work in asplund spaces trees and renorming theory

quantum groups and deformation quantization explicit approaches and implicit aspects deformation quantization which gives a development of quantum mechanics independent of the operator algebra formulation and quantum groups which arose from the inverse scattering method and a study of yang baxter equations share a common idea abstracted earlier in algebraic deformation theory that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations i e quantizations in deformation quantization the algebraic object is the algebra of observables functions on symplectic phase space whose infinitesimal deformation is the poisson bracket and global deformation a star product in quantum groups it is a hopf algebra generally either of functions on a lie group or often its dual in the topological vector space sense as we briefly explain a completed universal enveloping algebra of a lie algebra with for infinitesimal a matrix satisfying the modified classical yang baxter equation mcybe frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract one success here comes from universal deformation formulas udfs expressions built from a lie algebra which deform any algebra on which the lie algebra operates as derivations the most famous of these is the moyal product a special case of a class in which the lie algebra is abelian another comes from recognition that the belavin drinfel d solutions to the mcybe are in fact infinitesimal deformations for which in the case of the special linear groups it is possible to give explicit formulas for the corresponding quantum yang baxter equations this review paper discusses necessarily in brief these and related topics including twisting as a form of udf and finding formulas for preferred deformations of hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation

new class of level statistics in correlated disordered chains we study the properties of the level statistics of 1d disordered systems with long range spatial correlations we find a threshold value in the degree of correlations below which in the limit of large system size the level statistics follows a poisson distribution as expected for 1d uncorrelateddisordered systems and above which the level statistics is described by a new class of distribution functions at the threshold we find that with increasing system size the standard deviation of the function describing the level statistics converges to the standard deviation of the poissonian distribution as a power law above the threshold we find that the level statistics is characterized by different functional forms for different degrees of correlations doi 10 1103 physrevlett 93 176804

symmetrizable quantum affine superalgebras and their representations aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable cartan matrices are studied the irreducible integrable highest weight representations are classified and shown to be deformations of their classical counterparts it is also shown that jimbo type quantum affine superalgebras can be obtained by deforming universal enveloping algebras of ordinary i e nongraded affine algebras supplemented by certain parity operators

polyhedral norms on non separable banach spaces abstract we prove the existence of equivalent polyhedral norms on a number of classes of non separable spaces the majority of which being of the form c k in particular we obtain a complete characterization of those trees t such that c 0 t admits an equivalent polyhedral norm

on weakly locally uniformly rotund banach spaces abstract we show that every normed space e with a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm after obtaining a discrete network of the unit sphere s e for the weak topology we deduce that the space e must have a countable cover by sets of small local diameter which in turn implies the renorming conclusion this solves a question posed by deville godefroy haydon and zizler for a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topology despite the fact that it may not have the kadec property moreover banach spaces having a countable cover by sets of small local diameter coincide with the descriptive banach spaces studied by hansell so we present here some new characterizations of them

approaching word length distribution via level spectra abstract treating a text after the removal of paragraphs and punctuations as a spectrum of blanks the distributions of the length of words of ten languages are analyzed using models from the statistical theory of spectra it is found that the ten languages can be classified into two families one with words that follow a wigner like distribution while the words of the other obey a poisson like distribution

probability distribution of intersymbol distances in random symbolic sequences applications to improving detection of keywords in texts and of amino acid clustering in proteins symbolic sequences have been extensively investigated in the past few years within the framework of statistical physics paradigmatic examples of such sequences are written texts and deoxyribonucleic acid dna and protein sequences in these examples the spatial distribution of a given symbol a word a dna motif an amino acid is a key property usually related to the symbol importance in the sequence the more uneven and far from random the symbol distribution the higher the relevance of the symbol to the sequence thus many techniques of analysis measure in some way the deviation of the symbol spatial distribution with respect to the random expectation the problem is then to know the spatial distribution corresponding to randomness which is typically considered to be either the geometric or the exponential distribution however these distributions are only valid for very large symbolic sequences and for many occurrences of the analyzed symbol here we obtain analytically the exact randomly expected spatial distribution valid for any sequence length and any symbol frequency and we study its main properties the knowledge of the distribution allows us to define a measure able to properly quantify the deviation from randomness of the symbol distribution especially for short sequences and low symbol frequency we apply the measure to the problem of keyword detection in written texts and to study amino acid clustering in protein sequences in texts we show how the results improve with respect to previous methods when short texts are analyzed in proteins which are typically short we show how the measure quantifies unambiguously the amino acid clustering and characterize its spatial distribution

weak locally uniformly rotund norms and descriptive compact spaces we prove that a dual banach space x has an equivalent w lur norm if and only if the weak topology has a isolated network we give sufficient conditions for the existence of equivalent norms with various kinds of differentiability

the annihilation theorem for the completely reducible lie superalgebras a well known theorem of duflo claims that the annihilator of a verma module in the enveloping algebra of a complex semisimple lie algebra is generated by its intersection with the centre for a lie superalgebra this result fails to be true for instance in the case of the orthosymplectic lie superalgebra osp 1 2 pinczon gave in pi an example of a verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra more generally musson produced in mu1 a family of such singular verma modules for osp 1 2l cases in this article we give a necessary and sufficient condition on the highest weight of a osp 1 2l verma module for its annihilator to be generated by its intersection with the centre this answers a question of musson the classical proof of the duflo theorem is based on a deep result of kostant which uses some delicate algebraic geometry reasonings unfortunately these arguments can not be reproduced in the quantum and super cases this obstruction forced joseph and letzter in their work on the quantum case see jl to find an alternativeapproach to the duflo theorem following their ideas we compute the factorization of the parthasarathy ranga rao varadarajan prv determinants comparing it with the factorization of shapovalov determinants we find unlike to the classical and quantum cases that the prv determinant contains some extrafactors the set of zeroes of these extrafactors is precisely the set of highest weights of verma modules whose annihilators are not generated by their intersection with the centre we also find an analogue of hesselink formula see he giving the multiplicity of every simple finite dimensional module in the graded component of the harmonic space in the symmetric algebra

on the center of the enveloping algebra of a classical simple lie superalgebra 0 1 let g be a semisimple lie algebra with cartan subalgebra h weyl group w and enveloping algebra u g several results illustrate the important role played by the center z g of u g first there is the harish chandra homomorphism which shows that spec z g h rw and describes the action of z g on highest weight modules next we w x recall the separation of variables theorem of kostant ko d 8 2 4 which states that there is an ad g invariant subspace k of u g such that the multiplication map k m z g a u g is an isomorphism of ad g modules w kostant s theorem is a key ingredient in the proof of a result of duflo d x 8 4 3 stating that the annihilator of a verma module is generated by its intersection with z g a version of the harish chandra homomorphism has been given by kac for basic classical simple lie superalgebras we recall the details in subsection 1 1 the proof of kostant s theorem depends heavily on the fact that every finite dimensional g module is completely reducible by contrast if g is a simple lie superalgebra such that every finite dimensional g module is completely reducible then either g is a simple lie algebra or g is an w orthosymplectic lie superalgebra g s osp 1 2 r for some r g 1 sch x theorem 1 p 239 in this paper we show that for g s osp 1 2 r we again have u g k m z g for k ad invariant in addition if e is any finite dimensional

locally uniformly convex norms in banach spaces and their duals abstract it is shown that a banach space with locally uniformly convex dual admits an equivalent norm that is itself locally uniformly convex

the hidden group structure of quantum groups strong duality rigidity and preferred deformations a notion of well behaved hopf algebra is introduced reflexivity for strong duality between hopf algebras of drinfeld type and their duals algebras of coefficients of compact semi simple groups is proved a hidden classical group structure is clearly indicated for all generic models of quantum groups moyal product like deformations are naturally found for all frt models on coefficients andc functions strong rigidity hbi2 0 under deformations in the category of bialgebras is proved and consequences are deduced

rank distributions of words in correlated symbolic systems and the zipf law the binary many step markov chain with the step like memory function is considered as a model for the analysis of rank distributions of words in correlated stochastic symbolic systems we prove that this distribution obeys the power law with the exponent of the order of unity in the case of rather strong persistent correlations the zipf law is shown to be valid for the rank distribution of words with lengths about and shorter than the correlation length in the markov sequence a self similarity in the rank distribution with respect to the decimation procedure is observed

level statistics of words finding keywords in literary texts and symbolic sequences using a generalization of the level statistics analysis of quantum disordered systems we present an approach able to extract automatically keywords in literary texts our approach takes into account not only the frequencies of the words present in the text but also their spatial distribution along the text and is based on the fact that relevant words are significantly clustered i e they self attract each other while irrelevant words are distributed randomly in the text since a reference corpus is not needed our approach is especially suitable for single documents for which no a priori information is available in addition we show that our method works also in generic symbolic sequences continuous texts without spaces thus suggesting its general applicability

sequential normal compactness versus topological normal compactness in variational analysis abstract we study relationships between two normal compactness properties of sets in banach spaces that play an essential role in many aspects of variational analysis and its applications particularly in calculus rules for generalized differentiation necessary optimality and suboptimality conditions for optimization problems etc both properties automatically hold in finite dimensional spaces and reveal principal features of the infinite dimensional variational theory similar formulations of these properties involve the weak convergence of sequences and nets respectively containing generalized normal cones in duals to banach spaces we prove that these properties agree for a large class of banach spaces that include weakly compactly generated spaces we also show that they are always different in banach spaces whose unit dual ball is not weak sequentially compact moreover the sequential and topological normal compactness properties may not coincide even in non separable asplund spaces that admit an equivalent c smooth norm

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

deformations of the metaplectic representations we describe all possible natural deformations of the metaplectic representation of the lie superalgebras osp 1 2n and of its natural extension and show in particular that in most cases for the metaplectic representation itself it is rigid

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

a formalism for studying long range correlations in many alphabets sequences abstract we formulate a mean field like theory of long range correlated l alphabets sequences which are actually systems with l 1 independent parameters depending on the values of these parameters the variance on the average number of any given symbol in the sequence shows a linear or a superlinear dependence on the total length of the sequence we present exact solution to the four alphabets and three alphabets sequences we also demonstrate that a mapping of the given sequence into a smaller alphabets sequence namely a coarse graining process does not necessarily imply that long range correlations found in the latter would correspond to those of the former

lie superalgebras clifford algebras induced modules and nilpotent orbits let g be a classical simple lie superalgebra to every nilpotent orbit o in g0 we associate a clifford algebra over the field of rational functions on o we find the rank k o of the bilinear form defining this clifford algebra and deduce a lower bound on the multiplicity of a u g module with o or an orbital subvariety of o as associated variety in some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction the invariant k o is in many cases equal to the odd dimension of the orbit g o where g is a lie supergroup with lie superalgebra g

some remarks on smooth renormings of banach spaces abstract we prove that in every separable banach space x with a schauder basis and a c k smooth norm it is possible to approximate uniformly on bounded sets every equivalent norm with a c k smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the schauder basis our result solves a problem from the recent monograph of guirao montesinos and zizler

random texts exhibit zipf s law like word frequency distribution it is shown that the distribution of word frequencies for randomly generated texts is very similar to zipf s law observed in natural languages such as english the facts that the frequency of occurrence of a word is almost an inverse power law function of its rank and the exponent of this inverse power law is very close to 1 are largely due to the transformation from the word s length to its rank which stretches an exponential function to a power law function

the dependence of frequency distributions on multiple meanings of words codes and signs the dependence of the frequency distributions due to multiple meanings of words in a text is investigated by deleting letters by coding the words with fewer letters the number of meanings per code

centralizers of the superalgebra mathfrak osp 1 2 o s p 1 2 the brauer algebra as a quotient of the bannai ito algebra

average locally uniform rotundity and a class of nonlinear maps abstract we consider some topological characterizations of dual banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of slicely continuous maps

randomness versus specifics for word frequency distributions the text length dependence of real word frequency distributions can be connected to the general properties of a random book it is pointed out that this finding has strong implications when decidi

identifying characteristic scales in the human genome received 1 june 2006 published 16 march 2007the scale free long range correlations detected in dna sequences contrast with characteristic lengths ofgenomic elements being particularly incompatible with the isochores long homogeneous dna segments bycomputing the local behavior of the scaling exponent of detrended uctuation analysis dfa we discrimi nate between sequences with and without true scaling and we nd that no single scaling exists in the humangenome instead human chromosomes show a common compositional structure with two characteristic scales the large one corresponding to the isochores and the other to small and medium scale genomic elements doi 10 1103 physreve 75 032903 pacs number s 87 15 cc 87 14 gg 87 15 aa

uniformly gâteaux smooth approximations on c0 γ abstract every lipschitz mapping from c 0 into a banach space y can be uniformly approximated by lipschitz mappings that are simultaneously uniformly g teaux smooth and c frechet smooth

beyond the zipf mandelbrot law in quantitative linguistics abstract in this paper the zipf mandelbrot law is revisited in the context of linguistics despite its widespread popularity the zipf mandelbrot law can only describe the statistical behaviour of a rather restricted fraction of the total number of words contained in some given corpus in particular we focus our attention on the important deviations that become statistically relevant as larger corpora are considered and that ultimately could be understood as salient features of the underlying complex process of language generation finally it is shown that all the different observed regimes can be accurately encompassed within a single mathematical framework recently introduced by c tsallis

network properties of written human language we investigate the nature of written human language within the framework of complex network theory in particular we analyze the topology of orwell s 1984 focusing on the local properties of the network such as the properties of the nearest neighbors and the clustering coefficient we find a composite power law behavior for both the average nearest neighbor s degree and average clustering coefficient as a function of the vertex degree this implies the existence of different functional classes of vertices furthermore we find that the second order vertex correlations are an essential component of the network architecture to model our empirical results we extend a previously introduced model for language due to dorogovtsev and mendes we propose an accelerated growing network model that contains three growth mechanisms linear preferential attachment local preferential attachment and the random growth of a predetermined small finite subset of initial vertices we find that with these elementary stochastic rules we are able to produce a network showing syntacticlike structures

size dependent word frequencies and translational invariance of books it is shown that a real novel shares many characteristic features with a null model in which the words are randomly distributed throughout the text such a common feature is a certain translational

analysis of symbolic sequences using the jensen shannon divergence we study statistical properties of the jensen shannon divergence d which quantifies the difference between probability distributions and which has been widely applied to analyses of symbolic sequences we present three interpretations of d in the framework of statistical physics information theory and mathematical statistics and obtain approximations of the mean the variance and the probability distribution of d in random uncorrelated sequences we present a segmentation method based on d that is able to segment a nonstationary symbolic sequence into stationary subsequences and apply this method to dna sequences which are known to be nonstationary on a wide range of different length scales

gruenhage compacta and strictly convex dual norms abstract we prove that if k is a gruenhage compact space then c k admits an equivalent strictly convex dual norm as a corollary we show that if x is a banach space and x span k where k is a gruenhage compact in the w topology and is equivalent to a coarser w lower semicontinuous norm on x then x admits an equivalent strictly convex dual norm we give a partial converse to the first result by showing that if is a tree then c 0 admits an equivalent strictly convex dual norm if and only if is a gruenhage space finally we present some stability properties satisfied by gruenhage spaces in particular gruenhage spaces are stable under perfect images

on the goldie quotient ring of the enveloping algebra of a classical simple lie superalgebra abstract if g is a classical simple lie superalgebra g p n the enveloping algebra u g is a prime ring and hence has a simple artinian ring of quotients q u g by goldie s theorem we show that if g has type i then q u g is a matrix ring over q u g 0 on the other hand if g osp 1 2r then by extending the center of u g we obtain a prime ring whose goldie quotient ring is a matrix ring over the quotient division ring of a weyl algebra this is an analog of a result of gelfand and kirillov