a simple proof of kostant s theorem that u g is free over its center in this paper we present a simple proof of the fundamental result by b kostant which claims that the universal enveloping algebra of a reductive lie algebra g is free over its center we also indicate how this result allows to simplify the proof of another important result of b kostant the description of the algebra of functions on the nilpotent cone we use this technique to prove some generalizations of kostant s theorem we also deduce from it a way to check which subalgebras of g are centrally free

universal correlations and power law tails in financial covariance matrices abstract we investigate whether quantities such as the global spectral density or individual eigenvalues of financial covariance matrices can be best modelled by standard random matrix theory or rather by its generalisations displaying power law tails in order to generate individual eigenvalue distributions a chopping procedure is devised which produces a statistical ensemble of asset price covariances from a single instance of financial data sets local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets they are in good agreement with the universal tracy widom distribution and wigner surmise respectively this suggests a strong degree of robustness especially in the low lying sector of the spectra most relevant for portfolio selections conversely the global spectral density of a single covariance matrix as well as the average over all unfolded nearest neighbour spacing distributions deviate from standard gaussian random matrix predictions the data are in fair agreement with a recently introduced generalised random matrix model with correlations showing a power law decay

large deviations of the maximum eigenvalue in wishart random matrices we analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the wishart laguerre ensemble of positive definite random matrices we show that the probability that all the eigenvalues of a n n wishart matrix w xtx where x is a rectangular m n matrix with independent gaussian entries are smaller than the mean value n c decreases for large n as where 1 2 corresponds respectively to real and complex wishart matrices c n m 1 and x c is a rate sometimes also called large deviation function that we compute explicitly the result for the anti wishart case m n simply follows by exchanging m and n we also analytically determine the average spectral density of an ensemble of wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier numerical simulations are in excellent agreement with the analytical predictions

characters for projective modules in the bgg category o for general linear lie superalgebras abstract we determine the verma multiplicities and the characters of projective modules for atypical blocks in the bgg category o for the general linear lie superalgebras gl 2 2 and gl 3 1 we then explicitly determine the composition factor multiplicities of verma modules in the atypicality 2 block of gl 2 2

on completeness of quadratic systems abstract a dynamical system is called complete if every solution of it exists for all t r let k be the dimension of the vector space of quadratic systems the set of complete quadratic systems is shown to contain a vector subspace of dimension 2 k 3 we provide two proofs one by the gronwall lemma and the second by compactification that is capable of showing incompleteness as well characterization of a vector subspace of complete quadratic systems is provided the celebrated lorenz system for all real ranges of its parameters is shown to belong to this subspace we also provide a sufficient condition for a system to be incomplete

smallest eigenvalue distribution of the fixed trace laguerre beta ensemble in this paper we study the entanglement of the reduced density matrix of a bipartite quantum system in a random pure state it transpires that this involves the computation of the smallest eigenvalue distribution of the fixed trace laguerre ensemble of n n random matrices we showed that for finite n the smallest eigenvalue distribution may be expressed in terms of jack polynomials furthermore based on the exact results we found a limiting distribution when the smallest eigenvalue is suitably scaled with n followed by a large n limit our results turn out to be the same as the smallest eigenvalue distribution of the classical laguerre ensembles without the fixed trace constraint this suggests in a broad sense the global constraint does not influence local correlations at least in the large n limit consequently we have solved an open problem the determination of the smallest eigenvalue distribution of the reduced density matrix obtained by tracing out the environmental degrees of freedom for a bipartite quantum system of unequal dimensions

exact minimum eigenvalue distribution of an entangled random pure state a recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved in fact the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems they also provide a rare exactly solvable case for the distribution of the minimum of a set of nstrongly correlated random variables for all values of n and not just for large n

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

on the theory of frobenius extensions and its application to lie superalgebras by using an approach to the theory of frobenius extensions that emphasizes notions related to associative forms we obtain results concerning the trace and corestriction mappings and transitivity these are employed to show that the extension of enveloping algebras determined by a subalgebra of a lie superalgebra is a frobenius extension and to study certain questions in representation theory

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

polynomial flows in the plane soit f r n r un champ vectoriel c 1 et on considere le systeme d equations differentielles x f x avec x 0 x 0 la solution x t x 0 depend de t et x 0 pour n 2 sur r et c on donne le f tel que le flot depende polynomialement de la condition initiale x 0

recursion for the smallest eigenvalue density of β wishart laguerre ensemble the statistics of the smallest eigenvalue of wishart laguerre ensemble is important from several perspectives the smallest eigenvalue density is typically expressible in terms of determinants or pfaffians these results are of utmost significance in understanding the spectral behavior of wishart laguerre ensembles and among other things unveil the underlying universality aspects in the asymptotic limits however obtaining exact and explicit expressions by expanding determinants or pfaffians becomes impractical if large dimension matrices are involved for the real matrices beta 1 edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices very recently an analogous recurrence scheme has been obtained for the complex matrices beta 2 in the present work we extend this to beta wishart laguerre ensembles for the case when exponent alpha in the associated laguerre weight function lambda alpha e beta lambda 2 is a non negative integer while beta is positive real this also gives access to the smallest eigenvalue density of fixed trace beta wishart laguerre ensemble as well as moments for both cases moreover comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly exact evaluations for large values of n the matrix dimension and alpha also enable us to compare with tracy widom density and large deviation results of katzav and castillo we also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the wigner smith matrix and are relevant to the problem of quantum chaotic scattering

on a class of complete polynomial vector fields in the plane abstract the class b n of polynomial vector fields of degree n which define global flows in r 2 is characterized provided the zeros of the vector fields at infinity are simple some analytical properties of b n are also established

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

large deviations of extreme eigenvalues of random matrices we calculate analytically the probability of large deviations from its mean of the largest smallest eigenvalue of random matrices belonging to the gaussian orthogonal unitary and symplectic ensembles in particular we show that the probability that all the eigenvalues of an n ifmmode times else texttimes fi n random matrix are positive negative decreases for large n as ensuremath sim mathrm exp ensuremath ensuremath beta ensuremath theta 0 n 2 where the parameter ensuremath beta characterizes the ensemble and the exponent ensuremath theta 0 mathrm ln 3 4 0 274 text 653 dots is universal we also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number ensuremath zeta thus generalizing the celebrated wigner semicircle law the density of states generically exhibits an inverse square root singularity at ensuremath zeta

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

statistical distribution of quantum entanglement for a random bipartite state we compute analytically the statistics of the renyi and von neumann entropies standard measures of entanglement for a random pure state in a large bipartite quantum system the full probability distribution is computed by first mapping the problem to a random matrix model and then using a coulomb gas method we identify three different regimes in the entropy distribution which correspond to two phase transitions in the associated coulomb gas the two critical points correspond to sudden changes in the shape of the coulomb charge density the appearance of an integrable singularity at the origin for the first critical point and the detachment of the rightmost charge largest eigenvalue from the sea of the other charges at the second critical point analytical results are verified by monte carlo numerical simulations a short account of part of these results appeared recently in nadal et al phys rev lett 104 110501 2010

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

rank 1 real wishart spiked model in this paper we consider n dimensional real wishart matrices y in the class input amssym w bbb r sigma m in which all but one eigenvalue of is 1 let the nontrivial eigenvalue of be 1 then as n m with m n 2 finite and nonzero the eigenvalue distribution of y will converge into the marchenko pastur distribution inside a bulk region when increases from 0 one starts to see a stray eigenvalue of y outside of the support of the marchenko pastur density as this stray eigenvalue leaves the bulk region a phase transition will occur in the largest eigenvalue distribution of the wishart matrix in this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs we will first establish the results that are valid for all n and m and will use them to carry out the asymptotic analysis in particular we have derived a contour integral formula for the harish chandra itzykson zuber integral when x and y are real symmetric and y is a rank 1 matrix this allows us to write down a fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques as a result we obtain an integral formula for the largest eigenvalue distribution in the large n limit characterized by painleve transcendents the approach used in this paper is very different from a recent paper by bloemenal and virag in which the largest eigenvalue distribution was obtained using a stochastic operator method in particular the painleve formula for the largest eigenvalue distribution obtained in this paper is new 2012 wiley periodicals inc

extreme statistics of complex random and quantum chaotic states complex random states have the statistical properties of the gaussian and circular unitary ensemble eigenstates of random matrix theory even though their components are correlated by the normalization constraint it is nevertheless possible to derive compact analytic formulas for their extreme values statistical properties for all dimensionalities the maximum intensity result slowly approaches the gumbel distribution even though the variables are bounded whereas the minimum intensity result rapidly approaches the weibull distribution since random matrix theory is conjectured to be applicable to chaotic quantum systems we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic the statistical behaviors are consistent with the finite n formulas

airy distribution function from the area under a brownian excursion to the maximal height of fluctuating interfaces the airy distribution function describes the probability distribution of the area under a brownian excursion over a unit interval surprisingly this function has appeared in a number of seemingly unrelated problems mostly in computer science and graph theory in this paper we show that this distribution function also appears in a rather well studied physical system namely the fluctuating interfaces we present an exact solution for the distribution p h m l of the maximal height h m measured with respect to the average spatial height in the steady state of a fluctuating interface in a one dimensional system of size l with both periodic and free boundary conditions for the periodic case we show that p h m l l 1 2f h m l 1 2 for all l 0 where the function f x is the airy distribution function this result is valid for both the edwards wilkinson ew and the kardar parisi zhang interfaces for the free boundary case the same scaling holds p h m l l 1 2f h m l 1 2 but the scaling function f x is different from that of the periodic case we compute this scaling function explicitly for the ew interface and call it the f airy distribution function numerical simulations are in excellent agreement with our analytical results our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables some of these results were announced in a recent letter s n majumdar and a comtet phys rev lett 92 225501 2004

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

the primitive spectrum of a basic classical lie superalgebra we prove conjecture 5 7 in coulembier and musson math j arxiv 1409 2532 describing all inclusions between primitive ideals for the general linear superalgebra in terms of the rm ext 1 quiver of simple highest weight modules for arbitrary basic classical lie superalgebras we formulate two types of kazhdan lusztig quasi orders on the dual of the cartan subalgebra where one corresponds to the above conjecture both orders can be seen as generalisations of the left kazhdan lusztig order on hecke algebras and are related to categorical braid group actions we prove that the primitive spectrum is always described by one of the orders obtaining for the first time a description of the inclusions we also prove that the two orders are identical if category mathcal o admits enough abstract kazhdan lusztig theories in particular they are identical for the general linear superalgebra concluding the proof of the conjecture

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

strongly typical representations of the basic classical lie superalgebras 1 1 in ps1 i penkov and v serganova show that the category of representations of a basic classical lie superalgebra g of type i with a fixed typical central character is equivalent to the category of representations of the even part go with a suitable central character a similar result for type ii was proven by i penkov in p for generic central characters the aim of this paper is to understand for which central characters such an equivalence holds we also study a simplest example when such an equivalence fails to exist but the corresponding category of representations of a lie superalgebra still has a good description

laplace operators of infinite dimensional lie algebras and theta functions abstract until recently the generalized casimir operator constructed by kac kac v g 1974 funct anal appl 8 68 70 has been the only known element of the center of a completion of the enveloping algebra of a kac moody algebra it has been conjectured deodhar v v gabber o kac v g 1982 adv math 45 92 116 however that the image of the harish chandra homomorphism contains all theta functions defined on the interior of the complexified tits cone and hence separates the orbits of the weyl group developing the ideas of feigin and fuchs feigin b l fuchs d b 1983 dokl akad nauk sssr 269 1057 1060 i prove this conjecture another application of this method is the chevalley type restriction theorem for simple finite dimensional lie superalgebras

entanglement of random vectors we analytically calculate the average value of the ith largest schmidt coefficient for random pure quantum states schmidt coefficients i e eigenvalues of the reduced density matrix are expressed in the limit of large hilbert space size and for arbitrary bipartite splitting as an implicit function of index i

bernstein gelfand gelfand resolutions for basic classical lie superalgebras abstract we study kostant cohomology and bernstein gelfand gelfand resolutions for finite dimensional representations of basic classical lie superalgebras for each choice of parabolic subalgebra and representation of such a lie superalgebra there is a natural definition of the boundary and coboundary operators which define co homology of the nilradical of the parabolic subalgebra we prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of generalised verma modules every such a resolution is then given by modules induced by these homology groups we also prove that if these homology groups are completely reducible a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the kostant quabla operator which is equivalent with disjointness of the boundary and coboundary operators then we use these results to derive very explicit conditions under which bgg resolutions exist which are particularly useful for the superalgebras of type i for the unitarisable representations of gl m n and osp 2 2 n we derive conditions on the parabolic subalgebra under which the bgg resolutions exist we also apply the obtained theory to construct specific examples of bgg resolutions for osp m 2 n finally we state some results for typical modules of gl m n and osp 2 2 n

a remark on the casimir elements of lie superalgebras and quantized lie superalgebras it is shown that the second order casimir operators c and cq of the lie superalgebra osp 1 2 and the quantized lie superalgebra ospq 1 2 respectively possess natural square roots

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

a reduction method for atypical representations of classical lie superalgebras abstract we introduce a reduction method for studying representations of classical lie superalgebras with atypical central character we show that the atypical quotient of universal enveloping algebra has a non trivial jacobson radical the factor by this radical has a new center which is calculated for sl 1 n and psl 2 2 using this center we obtain new character formulae generalization of borel weil bott and beilinson bernstein localization theorems

superstatistical generalizations of wishart laguerre ensembles of random matrices using beck and cohen s superstatistics we introduce in a systematic way a family of generalised wishart laguerre ensembles of random matrices with dyson index 1 2 and 4 the entries of the data matrix are gaussian random variables whose variancesfluctuate from one sample to another according to a certain probability density f and a single deformation parameter three superstatistical classes for f are usually considered 2 inverse 2 and log normal distributions while the first class already considered by two of the authors leads to a power law decay of the spectral density we here introduce and solve exactly a superposition of wishart laguerre ensembles with inverse 2 distribution the corresponding macroscopic spectral density is given by a deformation of the semi circle and marcenko pastur laws on a non compact support with exponential tails after discussing in detail the validity of wigner s surmise in the wishart laguerre class we introduce a generalised dependent surmise with stretched exponential tails which well approximates the individual level spacing distribution in the bulk the analytical results are in excellent agreement with numerical simulations to illustrate our findings we compare the 2 and inverse 2 class to empirical data from financial covariance matrices

chevalley s restriction theorem for reductive symmetric superpairs let g k be a reductive symmetric superpair of even type i e so that there exists an even cartan subspace a p the restriction map s p k s a w where w w g0 a is the weyl group is injective we determine its image explicitly in particular our theorem applies to the case of a symmetric superpair of group type i e k k k with the flip involution where k is a classical lie superalgebra with a non degenerate invariant even form equivalently a finite dimensional contragredient lie superalgebra thus we obtain a new proof of the generalisation of chevalley s restriction theorem due to sergeev and kac gorelik for general symmetric superpairs the invariants exhibit a new and surprising behaviour we illustrate this phenomenon by a detailed discussion in the example g c q 1 osp 2 2q c endowed with a special involution here the invariant algebra defines a singular algebraic curve

universal asymptotic statistics of maximal relative height in one dimensional solid on solid models we study the probability density function p h m l of the maximum relative height h m in a wide class of one dimensional solid on solid models of finite size l for all these lattice models in the large l limit a central limit argument shows that for periodic boundary conditions p h m l takes a universal scaling form p h m l ensuremath sim sqrt 12 w l ensuremath 1 f mathbf h m sqrt 12 w l mathbf with w l the width of the fluctuating interface and f x the airy distribution function for one instance of these models corresponding to the extremely anisotropic ising model in two dimensions this result is obtained by an exact computation using the transfer matrix technique valid for any lg0 these arguments and exact analytical calculations are supported by numerical simulations which show in addition that the subleading scaling function is also universal up to a nonuniversal amplitude and simply given by the derivative of the airy distribution function f ensuremath x

universal distributions for growth processes in 1 1 dimensions and random matrices we develop a scaling theory for kardar parisi zhang growth in one dimension by a detailed study of the polynuclear growth model in particular we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape these distribution functions are computed using the partition function of gaussian random matrices in a cosine potential

maximal height statistics for 1 f alpha signals numerical and analytical results are presented for the maximal relative height distribution of stationary periodic gaussian signals one dimensional interfaces displaying a 1 f alpha power spectrum for 0 1 regime of strong correlations and a highly accurate picture gallery of distribution functions can be constructed numerically analytical results can be obtained in the limit alpha infinity and for large alpha by perturbation expansion furthermore using path integral techniques we derive a trace formula for the distribution function valid for alpha 2n even integer from the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha 1 is found to be in agreement with simulations comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions