a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

sugawara operators and kac kazhdan conjecture soit g une algebre de lie de kac moody affine de type x l 1 soit l le module de poids le plus eleve unique irreductible pour chaque poids de kac kazhdan on a chl exp re 1 exp 1

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

lp strong convergence of the averaging principle for slow fast spdes with jumps abstract the averaging principle is an important method to extract effective macroscopic dynamic from complex systems with slow component and fast component this paper concerns the l p strong convergence of the averaging principle for two time scales stochastic partial differential equations spdes driven by wiener processes and poisson jumps to achieve this a key step is to show the existence for an invariant measure with exponentially ergodic property for the fast equation where the dissipative conditions are needed furthermore it is shown that under suitable assumptions the slow component l p strongly converges to the solution of the averaged equation the rate of the convergence is also obtained as a byproduct

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

averaging for fundamental solutions of parabolic equations abstract herein an averaging theory for the solutions to cauchy initial value problems of arbitrary order e dependent parabolic partial differential equations is developed indeed by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the fundamental solution of an averaged parabolic equation we bring forth a novel approach to comparing x derivatives of t u e x t k 2p a k x t e k x u e x t f e x t u e x 0 e x on r d 0 t to like derivatives of t u x t k 2p a 0 k x k x u e x t f x t u x x as e 0 under general regularity conditions and our basic hypothesis that t 0 a k x s e a 0 k x ds e 0 0 for each x t i e pointwise the flexibility afforded by studing fundamental vis a vis specific solutions of these equations not only permits e dependent cauchy data and provides a unified method of treating all x derivatives of u e up to order 2 p 1 but also proves an invaluable tool when considering related problems of stochastic averaging our development was motivated by and retains a strong resemblance to the classical theory of parabolic partial differential equations however it will turn out that the classical conditions under which fundamental solutions are known to exist are somewhat unsuitable for our purposes and a modified set of conditions must be used

solvable symmetric poisson algebras and their derived lengths abstract let l be a lie algebra over a field of positive characteristic and let s l and s l denote respectively the symmetric poisson algebra and the truncated symmetric poisson algebra of l as a natural continuation of the work by monteiro alves and petrogradsky in 9 we investigate the structure of l when s l or s l is solvable we first disprove a conjecture stated in 9 about solvability of s l and s l in characteristic 2 next the derived lengths of s l are studied in particular we provide bounds for the derived lengths of s l establish when s l is metabelian and characterize truncated symmetric poisson algebras of minimal derived length

polynomial identities of matrices with the transpose involution the low degrees in this paper we investigate polynomial identities of minimal degree for the algebra of n x n matrices over a field where n 5 and is the transpose involution we first present some basic generators and then proceed to show that all other minimal degree identities can be derived from those

automorphisms and ideals of the weyl algebra let a 1 be the first weyl algebra and let g be its automorphism group we study the natural action of g on the space of isomorphism classes of right ideals of a 1 equivalently of finitely generated rank 1 torsion free right a 1 modules we show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety our results are strikingly similar to those for the commutative algebra of polynomials in two variables however we do not know of any general principle that would allow us to predict this in advance as a key step in the proof we obtain a new description of the bispectral involution of w1 we also make some comments on the group g from the viewpoint of shafarevich s theory of infinite dimensional algebraic groups

vector quantization of image subbands a survey subband and wavelet decompositions are powerful tools in image coding because of their decorrelating effects on image pixels the concentration of energy in a few coefficients their multirate multiresolution framework and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system vector quantization vq provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces since 1988 a growing body of research has examined the use of vq for subband wavelet transform coefficients we present a survey of these methods

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

a law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter consider the following random ordinary differential equation dot x epsilon tau f x epsilon tau tau epsilon omega text subject to x epsilon 0 x 0 where f x t omega t geq 0 are stochastic processes indexed by x in mathfrak r d and the dependence on x is sufficiently regular to ensure that the equation has a unique solution x epsilon tau omega over the interval 0 leq tau leq 1 for each epsilon 0 under rather general conditions one can associate with the preceding equation a nonrandom averaged equation dot x 0 tau overline f x 0 tau text subject to x 0 0 x 0 such that lim epsilon rightarrow 0 sup 0 leq tau leq 1 e x epsilon tau x 0 tau 0 in this article we show that as epsilon rightarrow 0 the random function x epsilon cdot x 0 cdot sqrt 2 epsilon log log epsilon 1 almost surely converges to and clusters throughout a compact set k of c lbrack 0 1 rbrack

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

principal nilpotent pairs in a semisimple lie algebra 1 this is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple lie algebra that enjoy quite remarkable properties and which are expected to play a major role in representation theory the properties of these pairs and their role is similar to those of the principal nilpotents each principal nilpotent pair gives rise to a harmonic polynomial on the cartesian square of the cartan subalgebra that transforms under an irreducible representation of the weyl group in the special case of n the conjugacy classes of principal nilpotent pairs and the irreducible representations of the symmetric group s n are both parametrised in a compatible way by young diagrams in general our theory provides a natural generalization to arbitrary weyl groups of the classical construction of simple s n modules in terms of young s symmetrisers first results towards a complete classification of all principal nilpotent pairs in a simple lie algebra are presented at the end of this paper in an appendix written by a elashvili and d panyushev

on adaptive vector transform quantization for speech coding adaptive vector transform quantization avtq as a coding system is discussed the optimal bit assignment is derived based on vector quantization asymptotic theory for different pdfs probability density functions of the transform coefficients strategies for shaping the quantization noise spectrum and for adapting the bit assignment to the changes in the speech statistics are discussed a good estimate of the efficiency of any coding system is given by the system coding gain over scalar pcm pulse code modulation based on the optimal bit allocation the coding gain of the vector transform quantization vtq system operating on a stationary input signal is derived the vtq coding gain demonstrates a significant advantage of vector quantization over scalar quantization within the framework of transform coding system simulation results are presented for a first order gauss markov process and for typical speech waveforms the results of fixed and adaptive systems are compared for speech input also the avtq results are compared to known scalar speech coding systems

image compression by vector quantization of multiresolution decompositions abstract an image compression scheme is introduced which involves a multiresolution decomposition derived from the wavelet transform the use of this transformation in image coding is motivated by its similarity to the processing which occurs at the cortical level of the visual system and by the fact that subband coding in general allows the coder to be better matched to the signal statistics in this work each subband is coded separately by vector quantization this work differs from previous wavelet transform coding schemes involving vector quantization in that the quantization parameters for each subband are selected by a nonlinear optimization procedure which involves constraints on the overall bit rate and the encoding complexity the multiplicative visual model is reviewed and a motivation for cascading this system with the wavelet transform to give a more complete description of human vision is provided empirical results are presented which demostrate the effectiveness of the technique for compressing monochromatic images signal to noise ratio measurements are shown which demostrate the system performance for various selections of bit rate and complexity

slices for biparabolic coadjoint actions in type a abstract let q be a truncated biparabolic subalgebra of a simple lie algebra in previous work it was shown that the invariant subalgebra y q of the algebra r q of regular functions on q is polynomial in most cases this includes all cases if g s l n and so provides 2 2 n 1 mostly non isomorphic lie algebras q for which y q is polynomial in general the structure of the generators and even their number have no known simple expression one may recall that the set of companion matrices forms a slice to the regular co adjoint orbits in g g s l n the present work is a quite remarkable and far reaching generalisation of this companion slice precisely one gives an element y q and a subspace v q such that restriction of functions gives an algebra isomorphism of y q onto r y v the construction is a combinatorial procedure based on the data specifying q unlike the semisimple case there can be many equivalence classes of such pairs as a consequence it is found that the nilfibre of the geometric quotient map may have many irreducible components containing a regular element namely the closure of the co adjoint orbit defined by such y q an example shows it may also have a component with no regular elements

strong convergence in stochastic averaging principle for two time scales stochastic partial differential equations abstract the theory of stochastic averaging principle provides an effective approach for the qualitative analysis of stochastic systems with different time scales and is relatively mature for stochastic ordinary differential equations in this paper we study the averaging principle for a class of stochastic partial differential equations with two separated time scales driven by scalar noises under suitable assumptions it is shown that the slow component strongly converges to the solution of the corresponding averaged equation

invariant theory in exterior algebras and amitsur levitzki type theorems abstract this article discusses invariant theories in some exterior algebras which are closely related to amitsur levitzki type theorems first we consider the exterior algebra on the vector space of square matrices of size n and look at the invariants under conjugations we see that the algebra of these invariants is isomorphic to the exterior algebra on an n dimensional vector space moreover we give a cayley hamilton type theorem for these invariants the anticommutative version of the cayley hamilton theorem this cayley hamilton type theorem can also be regarded as a refinement of the amitsur levitzki theorem we discuss two more amitsur levitzki type theorems related to invariant theories in exterior algebras one is a famous amitsur levitzki type theorem due to kostant and rowen and this is related to o v invariants in 2 v the other is a new amitsur levitzki type theorem and this is related to gl v invariants in 2 v s 2 v

standard identities for skew symmetric matrices abstract let n be a positive even integer and let k n f denote the subspace of skew symmetric matrices of mn f the full matrix algebra with coefficients in a field f a theorem of kostant states that k n f satisfies the 2 n 2 fold standard identity s 2 n 2 in this paper we refine this result by showing that s 2 n 2 may be written non trivially as the sum of two polynomial identities of k n f

a lie algebra generalization of the amitsur levitski theorem 0 1 if r is any algebra over a field f and xi e r i l k let iix 1 y 1 e r denote the alternating sum of products z sg 0 x x over all permutations 0 of l k one says that r satisfies the k fold standard identity if ix xk 0 for any choice xi e r of course if pc r is any subspace of r one can restrict one s attention to p and say that p satisfies the k fold standard identity if 1 x1 x 0 for any choice xi f p now the amitsur levitski theorem asserts that the n x n matrix algebra m n f satisfies the 2n fold standard identity for a survey and history of this theorem see 9 1 see also 17 i it is of course enough to consider the case where f c some time ago see 3 we came upon a proof of the amitsur levitski theorem from the point of lie algebra cohomology the cohomology of m n c as a lie algebra is an exterior algebra with primitive generators having degrees 1 2 2n 1 the failure of m n cc to have a primitive class in degree 2n 1 leads to the trace relation tr x x 0 for any xi e m n c for i i 2n 1 but this quickly implies the amitsur levitski theorem in fact the argument can be pushed further if d g m n cc denotes the lie algebra of all n x n skew symmetric matrices one knows that d has no primitive class in degree 2n 1 in case n is even this led to a new identity in 3 1 for iz even

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

subband coding of images using vector quantization a novel two dimensional subband coding technique is presented that can be applied to images as well as speech a frequency band decomposition of the image is carried out by means of 2d separable quadrature mirror filters which split the image spectrum into 16 equal rate subbands these 16 parallel subband signals are regarded as a 16 dimensional vector source and coded as such using vector quantization in the asymptotic case of high bit rates a theoretical analysis yields that a lower bound to the gain is attainable by choosing this approach over scalar quantization of each subband with an optimal bit allocation it is shown that vector quantization in this scheme has several advantages over coding the subbands separately experimental results are given and it is shown the scheme has a performance that is comparable to that of more complex coding techniques

lie structure of truncated symmetric poisson algebras abstract the paper naturally continues series of works on identical relations of group rings enveloping algebras and other related algebraic structures let l be a lie algebra over a field of characteristic p 0 consider its symmetric algebra s l n 0 u n u n 1 which is isomorphic to a polynomial ring it also has a structure of a poisson algebra where the lie product is traditionally denoted by this bracket naturally induces the structure of a poisson algebra on the ring s l s l x p x l which we call a truncated symmetric poisson algebra we study lie identical relations of s l namely we determine necessary and sufficient conditions for l under which s l is lie nilpotent strongly lie nilpotent solvable and strongly solvable where we assume that p 2 to specify the solvability we compute the strong lie nilpotency class of s l also we prove that the lie nilpotency class coincides with the strong lie nilpotency class in case p 3 shestakov proved that the symmetric algebra s l of an arbitrary lie algebra l satisfies the identity x y z 0 if and only if l is abelian we extend this result for the strong lie nilpotency and the strong solvability of s l we show that the solvability of s l and s l in case char k 2 is different from other characteristics namely we construct examples of such algebras which are solvable but not strongly solvable we use delta sets for lie algebras and the theory of identical relations of poisson algebras also we study filtrations in poisson algebras and prove results on products of terms of the lower central series for poisson algebras

averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients in this paper we consider the parabolic equation with random coefficients d t u varepsilon t x sum limits ij a ij left frac t varepsilon user2 x omega right d x i x j u varepsilon t user2 x sum limits i b i left frac t varepsilon user2 x omega right d x i u varepsilon t user2 x we show that ue t x converges to the solution uo t x of the averaging equation d t u 0 t x sum limits ij e left a ij left frac t varepsilon user2 x omega right right d x i x j u 0 t user2 x sum limits i e left b i left frac t varepsilon user2 x omega right right d x i u 0 t user2 x also the fluctuation process y varepsilon t x left equiv left u varepsilon t x u 0 t user2 x right sqrt varepsilon right converges weakly to a generalized ornstein uhlenbeck process on l

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

invariance principles for parabolic equations with random coefficients a general hilbert space based stochastic averaging theory is brought forth herein for arbitrary order parabolic equations with possibly long range dependent random coefficients we use regularity conditions on tue t x 0 k 2pak t e x kxue t x ue 0 x x 1 which are slightly stronger than those required to prove pathwise existence and uniqueness for 1 equation 1 can be obtained from the singularly perturbed system ve x 0 k 2peak x kxve x ve 0 x x 2 through time change next we impose on the coefficients of 1 a pointwise inxandt weak law of large numbers and a weak invariance principleeh te 10ak x s a0k x ds k 2p k k 2p 3 inc 0 t h1 h1being a separable hilbert space of functions andh 0 1 denoting a constant h 1 2 allows for long range time dependence then under these extraordinarily general conditions we infer the weak invariance principleeh 1 ue u uis the non random e homogeneous solution of tu t x 0 k 2pa0k x kxu t x u 0 x x 4 and mildly satisfies the linear stochastic partial differential equation ty t x k 2pa0k x kxy t x dt k 2p k dt x kxu t x 5

invariance principles for sums of banach space valued random elements and empirical processes almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable banach space it is then shown that empirical processes readily fit into this general framework thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed