the z2 z2 graded general linear lie superalgebra we present a novel realization of the z2 z2 graded lie superalgebra gl m1 m2 n1 n2 inside an algebraic extension of the enveloping algebra of the z2 graded lie superalgebra gl m n with m m1 m2 and n n1 n2 a consequence of this realization is that the representations of gl m n lift up to representations of gl m1 m2 n1 n2 with matrix elements differing only by a sign which we are able to characterize concisely we present a novel realization of the z2 z2 graded lie superalgebra gl m1 m2 n1 n2 inside an algebraic extension of the enveloping algebra of the z2 graded lie superalgebra gl m n with m m1 m2 and n n1 n2 a consequence of this realization is that the representations of gl m n lift up to representations of gl m1 m2 n1 n2 with matrix elements differing only by a sign which we are able to characterize concisely

reduced wigner coefficients for lie superalgebra gl m n corresponding to unitary representations and beyond in this paper fundamental wigner coefficients are determined algebraically by considering the eigenvalues of certain generalized casimir invariants here this method is applied in the context of both type 1 and type 2 unitary representations of the lie superalgebra gl m n extensions to the non unitary case are investigated a symmetry relation between two classes of wigner coefficients is given in terms of a ratio of dimensions

application of generating function techniques to lie superalgebras the construction of generating functions for multiplicities of irreducible representations of lie superalgebras from generating functions for characters is examined and applied in order to obtain polynomial tensors and branching rules problems arising because of the existence of typical and atypical representations are discussed in detail the techniques are applied to osp 1 2 spl 1 2 osp 3 2 and osp 4 2

solutions of the boundary yang baxter equation for arbitrary spin we use boundary quantum group symmetry to obtain recursion formulae which determine nondiagonal solutions of the boundary yang baxter equation reflection equation of the xxz type for any spin j

character formula for infinite dimensional unitarizable modules of the general linear superalgebra the fock space of m p bosonic and n q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra glm p n q its tensor powers decompose into direct sums of infinite dimensional irreducible highest weight glm p n q modules we obtain an explicit decomposition of any tensor power of this fock space into irreducibles and develop a character formula for the irreducible glm p n q modules arising in this way

on the eigenvalue problem for arbitrary odd elements of the lie superalgebra and applications in a wigner quantum mechanical model with a solution in terms of the lie superalgebra one is faced with determining the eigenvalues and eigenvectors for an arbitrary self adjoint odd element of in any unitary irreducible representation w we show that the eigenvalue problem can be solved by the decomposition of w with respect to the branching the eigenvector problem is much harder since the gel fand zetlin basis of w is involved and the explicit actions of generators on this basis are fairly complicated using properties of the gel fand zetlin basis we manage to present a solution for this problem as well our solution is illustrated for two special classes of unitary representations the so called fock representations and the ladder representations

a narrow band pattern matching model of vowel perception the purpose of this paper is to propose and evaluate a new model of vowel perception which assumes that vowel identity is recognized by a template matching process involving the comparison of narrow band input spectra with a set of smoothed spectral shape templates that are learned through ordinary exposure to speech in the present simulation of this process the input spectra are computed over a sufficiently long window to resolve individual harmonics of voiced speech prior to template creation and pattern matching the narrow band spectra are amplitude equalized by a spectrum level normalization process and the information bearing spectral peaks are enhanced by a flooring procedure that zeroes out spectral values below a threshold function consisting of a center weighted running average of spectral amplitudes templates for each vowel category are created simply by averaging the narrow band spectra of like vowels spoken by a panel of talkers in the present implementation separate templates are used for men women and children the pattern matching is implemented with a simple city block distance measure given by the sum of the channel by channel differences between the narrow band input spectrum level equalized and floored and each vowel template spectral movement is taken into account by computing the distance measure at several points throughout the course of the vowel the input spectrum is assigned to the vowel template that results in the smallest difference accumulated over the sequence of spectral slices the model was evaluated using a large database consisting of 12 vowels in hvd context spoken by 45 men 48 women and 46 children the narrow band model classified vowels in this database with a degree of accuracy 91 4 approaching that of human listeners

representations of low rank orthosymplectic superalgebras by superfield techniques representations of lie superalgebras may be realised as functions superfields on graded manifolds suitably chosen onset spaces the technique is illustrated for osp 1 2 and osp 2 2 with a review of well known results and applied in the cases of osp 3 2 and osp 4 2 to construct some classes of irreducible representations

unitarity and complete reducibility of certain modules over quantized affine lie algebras let uq g denote the quantized affine lie algebra and uq g 1 the quantized nontwisted affine lie algebra let ofin be the category defined in sec iii it is shown that when the deformation parameter q is not a root of unit all integrable representations of uq g in the category ofin are completely reducible and that every integrable irreducible highest weight module over uq g 1 corresponding to q 0 is equivalent to a unitary module

the quantum group structure of 2d gravity and minimal models i on the unit circle an infinite family of chiral operators is constructed whose exchange algebra is given by the universalr matrix of the quantum groupsl 2 q this establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group the method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofsl 2 q the formulae so derived which generalize an earlier particular case worked out by babelon are remarkably compact and may be entirely written in terms of q deformed factorials and binomial coefficients

lie superalgebras and the multiplet structure of the genetic code ii branching schemes continuing our attempt to explain the degeneracy of the genetic code using basic classical lie superalgebras we present the branching schemes for the typical codon representations typical 64 dimensional irreducible representations of basic classical lie superalgebras and find three schemes that do reproduce the degeneracies of the standard code based on the orthosymplectic algebra osp 5 2 and differing only in details of the symmetry breaking pattern during the last step

spherical functions on homogeneous superspaces homogeneous superspaces arising from the general linear supergroup are studied within a hopf algebraic framework spherical functions on homogeneous superspaces are introduced and the structures of the superalgebras of the spherical functions on classes of homogeneous superspaces are described explicitly

representations of osp 2 1 and the metaplectic representation shift operator techniques are used to treat the irreducible representations of the superalgebra osp 2 1 apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact su 2 algebra the case when the star conditions on the even part are those satisfied by the noncompact su 1 1 algebra is also treated in this case no gradestar representations arise and the star representations are found to consist of the direct sum of two discrete series representations of su 1 1 one of these representations can be realized in terms of functions of a single complex variable and turns out to be a simple example of a metaplectic representation

extension of the mathfrak o mathfrak s mathfrak p m n sim mathfrak s mathfrak o m n correspondence to the infinite dimensional chiral spinors and self dual tensors the spinor representations of the orthosymplectic lie superalgebras are considered and constructed these are infinite dimensional irreducible representations of which the superdimension coincides with the dimension of the spinor representation of next we consider the self dual tensor representations of and their generalizations these are also infinite dimensional and correspond to the highest irreducible component of the pth power of the spinor representation we determine the character of these representations and deduce a superdimension formula from this it follows that also for these representations the correspondence holds

the relative phonetic contributions of a cochlear implant and residual acoustic hearing to bimodal speech perception the addition of low passed lp speech or even a tone following the fundamental frequency f0 of speech has been shown to benefit speech recognition for cochlear implant ci users with residual acoustic hearing the mechanisms underlying this benefit are still unclear in this study eight bimodal subjects ci users with acoustic hearing in the non implanted ear and eight simulated bimodal subjects using vocoded and lp speech were tested on vowel and consonant recognition to determine the relative contributions of acoustic and phonetic cues including f0 to the bimodal benefit several listening conditions were tested ci vocoder lp tf0 env ci vocoder lp ci vocoder tf0 env compared with ci vocoder performance lp significantly enhanced both consonant and vowel perception whereas a tone following the f0 contour of target speech and modulated with an amplitude envelope of the maximum frequency of the f0 contour tf0 env enhanced only consonant perception information transfer analysis revealed a dual mechanism in the bimodal benefit the tone representing f0 provided voicing and manner information whereas lp provided additional manner place and vowel formant information the data in actual bimodal subjects also showed that the degree of the bimodal benefit depended on the cutoff and slope of residual acoustic hearing

classification of all star irreps of gl m n all the finite dimensional star irreps of gl m n are classified in terms of their highest weights thereby completing the classification of all finite dimensional star irreps of the basic classical lie superalgebras the lowest weights of such irreps are determined explicitly and it is shown that the contravariant and covariant tensor irreps of gl m n are star irreps of type 1 and 2 respectively this explains the applicability of the young diagram method to these two classes of representations however it is shown that one parameter families of typical star irreps occur that are intrinsically different from the tensorial irreps

difference equations in statistical mechanics i cluster statistics models a review and some new results are presented for several cluster statistics models solutions of which can be reduced to difference equations mathematical techniques suitable for solving these equations are surveyed

the general linear supergroup and its hopf superalgebra of regular functions abstract the structure of the hopf superalgebra b of regular functions on the general linear supergroup is developed and applied to study the representation theory of the supergroup it is shown that the general linear supergroup can be reconstructed from b in a way reminiscent of the tannaka krein theory a borel weil type realization is also obtained for the irreducible integrable representations as a side result on the structure of b schur superalgebras are introduced and are shown to be semi simple over the complex field with the simple ideals determined explicitly

classification of all star and grade star irreps of gl n 1 a method for inducing nondegenerate forms on irreducible gl m n modules that implies some general results on star and grade star representations is investigated these results are applied to obtain a complete classification in terms of highest weights of the irreducible star and grade star representations of gl n 1 it is demonstrated that while gl n 1 admits a large class of star representations the irreducible grade star representations are comparatively rare moreover for n 2 all grade star irreducible representations are also star representations and for n 2 are atypical the superalgebra gl 2 1 proves to be a special case and admits a two parameter family of four dimensional typical grade star irreducible representations that are not star representations in particular typical grade star irreducible representations of gl n 1 exist only for n 1 2

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

representations of supergroups an explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations the rules of supersymmetrization are characterized by extended young supertableaux this constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all casimir operators can be calculated we have applied this approach so far to the supergroups su n m osp n 2m p n for which we have obtained all the representations constructible as direct products of the fundamental defining representations an argument is presented toward the irreducibility of all these representations

matrix elements for type 1 unitary irreducible representations of the lie superalgebra gl m n using our recent results on eigenvalues of invariants associated to the lie superalgebra gl m n we use characteristic identities to derive explicit matrix element formulae for all gl m n generators particularly non elementary generators on finite dimensional type 1 unitary irreducible representations we compare our results with existing works that deal with only subsets of the class of type 1 unitary representations all of which only present explicit matrix elements for elementary generators our work therefore provides an important extension to existing methods and thus highlights the strength of our techniques which exploit the characteristic identities

the quantum strip liouville theory for open strings the quantum group structure of 2d gravity recently put forward by one of us j l g is used to study quantum gravity on the strip the boundary conditions previously studied by a neveu and this author become easy to implement when one introduces the universal family of chiral operators associated withu q sl 2 a general formula for inverse powers of the metric tensor operator is thereby derived it contains a new universal matrixa acting in representation space which obeys identities involving ther matrix the clebsch gordon coefficients and the co products ofu q sl 2 the physical meaning of these identities is to ensure that these powers of the metric are local and closed by fusion

the first fundamental theorem of invariant theory for the orthosymplectic super group we give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of atiyah bott and patodi to the supergroup context we use methods from super algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space in this way super schur weyl brauer duality is established between the orthosymplectic supergroup of superdimension m 2n and the brauer algebra with parameter m 2n the result may be interpreted either in terms of the group scheme osp v over mathbb c where v is a finite dimensional super space or as a statement about the orthosymplectic lie supergroup over the infinite dimensional grassmann algebra lambda we take the latter point of view here and also state a corresponding theorem for the orthosymplectic lie superalgebra which involves an extra invariant generator the super pfaffian

u cohomology formula for unitarizable modules over general linear superalgebras abstract the u cohomology groups with coefficients in infinite dimensional irreducible unitarizable highest weight modules are introduced for the general linear superalgebra the corresponding cohomology groups associated with a large class of infinite dimensional irreducible unitarizable representations known as the oscillator representations are computed explicitly results lead to a more detailed understanding of the character formula for such modules derived in an earlier publication using other methods

analytic bethe ansatz related to a one parameter family of finite dimensional representations of the lie superalgebra as is well known the type 1 lie superalgebra admits a one parameter family of finite dimensional irreducible representations we have carried out an analytic bethe ansatz related to this family of representations we present formulae which are deformations of previously proposed determinant formulae labelled by a young superdiagram these formulae will provide a transfer matrix eigenvalue in a dressed vacuum form related to the solutions of a graded yang baxter equation which depend not only on the spectral parameter but also on a non additive continuous parameter a class of transfer matrix functional relations among these formulae is briefly mentioned

indices of representations of simple superalgebras indices and anomaly numbers for representations of basic classical lie superalgebras are defined and their explicit expressions are derived in terms of kac dynkin labels useful properties of indices and anomalies are determined and several examples are given a similar analysis is made for superindices and superanomalies and it is demonstrated how all these objects form a helpful tool in decomposing tensor products or in constructing branching rules for representations

structure of atypical representations of the lie superalgebras sl m n in contrast to lie algebras lie superalgebras contain typical and atypical irreducible representations character formulae for typical representations of classical lie superalgebras have been known for a long time but the structure of atypical representations remained unsolved the author constructs the character formula for atypical representations of the special linear lie superalgebras sl m n

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

constant solutions of reflection equations and quantum groups to the yang baxter equation an additional relation can be added this is the reflection equation that appears in various places with or without spectral parameter e g in factorizable scattering on a half line integrable lattice models with nonperiodic boundary conditions noncommutative differential geometry on quantum groups etc two forms of spectral parameter independent reflection equations are studied chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations for a variety of known solutions of the yang baxter equation the constant solutions of the reflection equations are given various quadratic algebras defined by the reflection equations are also given explicitly

reduced matrix elements of the orthosymplectic lie superalgebra we utilize characteristic identities to construct eigenvalue formulae for invariants and reduced matrix elements corresponding to irreducible representations of in presenting these results we further develop our programme of constructive representation theory via characteristic identities

invariant integration on orthosymplectic and unitary supergroups the orthosymplectic supergroup osp m 2n and unitary supergroup u p q are studied following a new approach that starts from harish chandra pairs and links the sheaf theoretical supermanifold approach of berezin and others with the differential geometry approach of rogers and others the matrix elements of the fundamental representation of the lie supergroup are expressed in terms of functions on the product supermanifold with the underlying lie group and n the odd dimension of this product supermanifold is isomorphic to the supermanifold of this leads to a new expression for the standard generators of the corresponding lie superalgebra as invariant derivations on using these results a new and transparent formula for the invariant integrals on osp m 2n and u p q is obtained

classical a n w geometry by analyzing theextrinsic geometry of two dimensional surfaces chirally embedded incp n thecp n w surface 1 we give exact treatments in various aspects of the classical w geometry in the conformal gauge first the basis of tangent and normal vectors are defined at regular points of the surface such that their infinitesimal displacements are given by connections which coincide with the vector potentials of the conformal a n toda lax pair since the latter is known to be intrinsically related with the w symmetries this gives the geometrical meaning of thea n w algebra second w surfaces are put in one to one correspondence with solutions of the conformally reduced wznw model which is such that the toda fields give the cartan part in the gauss decomposition of its solutions third the additional variables of the toda hierarchy are used as coordinates ofcp n this allows us to show that w transformations may be extended as particular diffeomorphisms of this target space higher dimensional generalizations of the wznw equations are derived and related with the zakharov shabat equations of the toda hierarchy fourth singular points are studied from a global viewpoint using our earlier observation 1 that w surfaces may be regarded as instantons the global indices of the w geometry which are written in terms of the toda fields are shown to be the instanton numbers for associated mappings of w surfaces into the grassmannians the relation with the singularities of w surface is derived by combining the toda equations with the gauss bonnet theorem

representations of n 2 extended supergravity and unitarity conditions in osp n 4 the structure of unitary irreducible representations of the lie superalgebra osp 2 4 the algebra of n 2 extended supergravity is investigated in detail in particular four new classes of shortened multiplets are found and the complete unitarity conditions are given the shortened multiplets are shown to correspond to atypical infinite dimensional representations finally unitarity conditions for osp n 4 3 n 8 are constructed

invariants and reduced matrix elements associated with the lie superalgebra gl m n we construct explicit formulae for the eigenvalues of certain invariants of the lie superalgebra gl m n using characteristic identities we discuss how such eigenvalues are related to reduced wigner coefficients and the reduced matrix elements of generators and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra

fuzzy information transmission analysis for continuous speech features feature information transmission analysis fita estimates information transmitted by an acoustic feature by assigning tokens to categories according to the feature under investigation and comparing within category to between category confusions fita was initially developed for categorical features e g voicing for which the category assignments arise from the feature definition when used with continuous features e g formants it may happen that pairs of tokens in different categories are more similar than pairs of tokens in the same category the estimated transmitted information may be sensitive to category boundary location and the selected number of categories this paper proposes a fuzzy approach to fita that provides a smoother transition between categories and compares its sensitivity to grouping parameters with that of the traditional approach the fuzzy fita was found to be sufficiently robust to boundary location to allow automation of category boundary selection traditional and fuzzy fita were found to be sensitive to the number of categories this is inherent to the mechanism of isolating a feature by dividing tokens into categories so that transmitted information values calculated using different numbers of categories should not be compared four categories are recommended for continuous features when twelve tokens are used

operator coproduct realization of quantum group transformations in two dimensional gravity i a simple connection between the universalr matrix ofuq sl 2 for spins 1 2 andj and the required form of the coproduct action of the hilbert space generators of the quantum group symmetry is put forward this leads us to an explicit operator realization of the coproduct action on the covariant operators it allows us to derive the expected quantum group covariance of the fusion and braiding matrices although it is of a new type the generators depend upon worldsheet variables and obey a new central extension of theuq sl 2 algebra realized by what we call fixed point commutation relations this is explained by showing on a general ground that the link between the algebra of field transformations and that of the coproduct generators is much weaker than previously thought the central charges of our extendeduq sl 2 algebra which includes the liouville zero mode momentum in a non trivial way are related to virasoro descendants of unity we also show how our approach can be used to derive the hopf algebra structure of the extended quantum group symmetry open image in new window related to the presence of both of the screening charges of 2d gravity

solving the strongly coupled 2d gravity 1 unitary truncation and quantum group structure strongly compled gravity theories with virasoro central charges equal to 7 13 and 19 are shown to enjoy striking properties at these values the subset of chiral operators with real virasoro weights acting on a subspacehphys is shown to be closed by fusion and braiding and to leave this subspace invariant moreover the representation of the virasoro algebra becomes unitary when it is restricted tohphys strongly coupled 2d gravity withcgrav 7 13 or 19 may thus be naturally truncated obtaining a consistent conformal theory this result is similar to the truncation that occurs forc 1 6 p p 2 pp withp andp integers where only a finite number of primary fields remains as is well known in rational theories the proof of this unitary truncation theorem already summarized in a recent letter is fully described here

dimension and character formulas for lie supergroups a character formula is derived for lie supergroups the basic technique is that of symmetrization and antisymmetrization associated with young tableaux generalized to supergroups we rewrite the characters of the ordinary lie groups u n o n and sp 2n in terms of traces in the fundamental representation it is then shown that by simply replacing traces with supertraces the characters of certain representations for u n m and osp n 2m are obtained dimension formulas are derived by calculating the characters of a special diagonal supergroup element with 1 and 1 eigenvalues formulas for the eigenvalues of the quadratic casimir operators are given as applications the decomposition of a representation into representations of subgroups is discussed examples are given for the lie supergroup su 6 4 which has physical applications as a dynamical supersymmetry in nuclei

use of a compound approach to derive auditory filter wide frequency importance functions for vowels and consonants speech recognition in noise presumably relies on the number and spectral location of available auditory filter outputs containing a relatively undistorted view of local target signal properties the purpose of the present study was to estimate the relative weight of each of the 30 auditory filter wide bands between 80 and 7563 hz because previous approaches were not compatible with this goal a technique was developed similar to the hole approach the weight of a given band was assessed by comparing intelligibility in two conditions differing in only one aspect the presence or absence of the band of interest in contrast to the hole approach however random gaps were also created in the spectrum these gaps were introduced to render the auditory system more sensitive to the removal of a single band and their location was randomized to provide a general view of the weight of each band i e irrespective of the location of information elsewhere in the spectrum frequency weighting functions derived using this technique confirmed the main contribution of the 400 2500 hz frequency region however they revealed a complex microstructure contrasting with the bell curve shape typically reported

regular subalgebras of lie superalgebras and extended dynkin diagrams using the method of extended cartan matrices and extended dynkin diagrams a classification of maximal regular semisimple subalgebras of the basic classical lie superalgebras is obtained especially in the case of exceptional lie superalgebras some curious inclusion relations are discovered

explicit construction of the exceptional superalgebras f 4 and g 3 a general method for constructing the extension of an ordinary lie algebra a0 to a superalgebra a0 a1 is given once one knows in which representation of a0 the odd generators a1 are explicit matrix representations for the superalgebras f 4 and g 3 and for ordinary algebras e8 f4 and g2 are presented

unitary highest weight representations of quantum general linear superalgebra fock space realisations of unitary highest weight representations of compact and noncompact real forms of the quantum general linear superalgebra uq glm n are constructed we decompose all tensor powers of the fock spaces into direct sums of irreducible unitary submodules for compact or noncompact real forms of the quantum superalgebra obtaining multiplicities of the irreducibles and also explicit formulae for their highest weight vectors a generalised howe duality of type uq glm n uq glk is also established which is used in the study of unitary representations and should also be of independent interest

principal five dimensional subalgebras of lie superalgebras the analog of sl 2 for lie superalgebras is osp 1 2 a five dimensional superalgebra all basic classical lie superalgebras l that contain a principal five dimensional osp 1 2 subalgebra are classified moreover the decomposition of the standard representation and of the adjoint representation of l into irreducible components of the principal osp 1 2 subalgebra is given

generalised verma modules for the orthosymplectic lie superalgebra ospk 2 abstract the composition factors with their multiplicities and the submodule lattices are determined for generalised verma modules over the orthosymplectic lie superalgebra osp k 2 the results enable us to obtain explicit formulae for the formal characters and dimensions of the finite dimensional irreducible modules applying these results we also compute the first and second cohomology groups of the lie superalgebra with coefficients in finite dimensional kac modules and irreducible modules

irreducible representations of the exceptional lie superalgebras d 2 1 α the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the exceptional lie superalgebrasd 2 1 for all cases the star or grade star conditions for the algebra are investigated among the finite dimensional representations there are no star and only a few grade star representations but an infinite class of infinite dimensional star representations is found explicit expressions are given for the doublet representation of d 2 1 the one missing label problem d 2 1 su 2 su 2 su 2 is discussed in detail and solved explicitly

the quantum group structure of 2d gravity and minimal models ii the genus zero chiral bootstrap the chiral operator algebra of the quantum group covariant operators of vertex type is completely worked out by making use of the operator approach suggested by the liouville theory where the quantum group symmetry is explicit this completes earlier articles along the same line the relationship between the quantum group invariant of irf type and quantum group covariant of vertex type chiral operator algebras is fully clarified and connected with the transition to the shadow world for quantum group symbols the corresponding 3 j symbol dressing is shown to reduce to the simpler transformation of babelon and one of the authors j l g in a suitable infinite limit defined by analytic continuation the above two types of operators are found to coincide when applied to states with liouville momenta going to in a suitable way the introduction of quantum group covariant operators in the three dimensional picture gives a generalization of the quantum group version of discrete three dimensional gravity that includes tetrahedra associated with 3 j symbols and universalr matrix elements altogether the present work and a previous parallel article gives the concrete realization of moore and seiberg s scheme that describes the chiral operator algebra of two dimensional gravity and minimal models

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

diagram and superfield techniques in the classical superalgebras introduces the concept of graded permutation group in the analysis of tensor operators in the classical superalgebras for u m n and su m n irreducible tensor representations correspond to classes of young tableaux with definite graded symmetry type diagram techniques are given for kronecker products dimensions and branching rules the tensor techniques are complemented by the introduction of a superfluid formalism in which u m n and su m n act on polynomial functions over the appropriate superspace such superfluids may admit constraints a general superfield interpolates between the classes of young tableaux which correspond to particular types of constraint the tensor and superfield techniques are illustrated with case studies of su 2 1 and su m 1

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

structure and representations of the quantum general linear supergroup the structure and representations of the quantum general linear supergroup gl q m n are studied systematically by investigating the hopf superalgebra g q of its representative functions g q is factorized into g q pi g q bar pi and a peter weyl basis is constructed for each factor parabolic induction for the quantum supergroup is developed the underlying geometry of induced representations is discussed and an analog of frobenius reciprocity is obtained a quantum borel weil theorem is proven for the irreducible covariant and contravariant tensorial representations and explicit realizations are given for classes of irredducible tensorial representatins in terms of sections of quantum super vector bundles over quantum projective superspaces

extensions of inhomogeneous polynomial representations for sl m 1 n we extend an inhomogeneous polynomial representation of special linear lie superalgebras sl m 1 n to an inhomogeneous representation on the tensor space of any irreducible tensor representation of the general linear lie superalgebras gl m n with this polynomial space via shen s mixed product we find a sufficient and necessary condition for irreducibility of these inhomogeneous representations and get the jordan holder series of those which are not irreducible furthermore we point out that all these irreducible modules are lowest weight modules their lowest weights and character formulas are obtained

the first fundamental theorem of invariant theory for the orthosymplectic supergroup we give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of atiyah bott and patodi to the supergroup context we use methods from super algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space in this way super schur weyl brauer duality is established between the orthosymplectic supergroup of superdimension m 2n and the brauer algebra with parameter m 2n the result may be interpreted either in terms of the group scheme osp v over mathbb c where v is a finite dimensional super space or as a statement about the orthosymplectic lie supergroup over the infinite dimensional grassmann algebra lambda we take the latter point of view here and also state a corresponding theorem for the orthosymplectic lie superalgebra which involves an extra invariant generator the super pfaffian

classification of finite dimensional unitary irreps for uq gl m n all finite dimensional irreducible unitary representations of the quantum supergroup uq gl m n are classified in terms of their highest weights for generic q 0

characters dimensions and branching rules for covariant irreps of u n m the theory of symmetric functions is used to obtain simple expressions for the characters and dimensions of the covariant irreps of the supergroup u n m schur function expressions are given for a number of important branching rules

representations of the lie superalgebra gl 1 n in a gel fand zetlin basis and wigner quantum oscillators an explicit construction of all finite dimensional irreducible representations of the lie superalgebra in a gel fand zetlin basis is given particular attention is paid to the so called star type i representations unitary representations and to a simple class of representations v p with p any positive integer then the notion of wigner quantum oscillators wqos is recalled in these quantum oscillator models the unitary representations of are physical state spaces of the n particle d dimensional oscillator so far physical properties of wqos were described only in the so called fock spaces w p leading to interesting concepts such as non commutative coordinates and a discrete spatial structure here we describe physical properties of wqos for other unitary representations including certain representations v p of gl 1 dn these new solutions again have remarkable properties following from the spectrum of the hamiltonian and of the position momentum and angular momentum operators formulae are obtained that give the angular momentum content of all the representations v p of associated with the n particle three dimensional wqo for these representations v p we also consider in more detail the spectrum of the position operators and their squares leading to interesting consequences in particular a classical limit of these solutions is obtained that is in agreement with the correspondence principle

character formulae for ortho symplectic lie superalgebras osp n 2 abstract the characters of finite dimensional irreducible modules of the lie superalgebra osp n 2 are completely determined in terms of those of generalized verma modules as a by product the irreducible decomposition of the tensor of any finite dimensional irreducible module with the natural module is obtained

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

classification of unitary and grade star irreps for uq osp 2 2n all the unitary and grade star irreps of the quantum supergroup uq osp 2 2n are classified in terms of their highest weight labels

harmonic oscillators coupled by springs discrete solutions as a wigner quantum system we consider a quantum system consisting of a one dimensional chain of m identical harmonic oscillators with natural frequency coupled by means of springs such systems have been studied before and appear in various models in this paper we approach the system as a wigner quantum system not imposing the canonical commutation relations but using instead weaker relations following from the compatibility of hamilton s equations and the heisenberg equations in such a setting the quantum system allows solutions in a finite dimensional hilbert space with a discrete spectrum for all physical operators we show that a class of solutions can be obtained using generators of the lie superalgebra gl 1 m then we study from a mathematical point of view the properties and spectra of the physical operators in a class of unitary representations of gl 1 m these properties are both interesting and intriguing in particular we can give a complete analysis of the eigenvalues of the hamiltonian and of the pos

matrix elements and duality for type 2 unitary representations of the lie superalgebra gl m n the characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible gl m n modules in particular we give matrix element formulae for all gl m n generators including the non elementary generators together with their phases on finite dimensional type 2 unitary irreducible representations which include the contravariant tensor representations and an additional class of essentially typical representations remarkably we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non vanishing matrix elements up to a phase

finite and infinite dimensional representations of the orthosymplectic superalgebra osp 3 2 the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the orthosymplectic superalgebra osp 3 2 for all cases the star or grade star conditions for the algebra are investigated only two finite dimensional representations are grade star representations if the representation space is required to be a graded hilbert space when the even part is so 3 sp 2 r su 2 su 1 1 an infinite class of infinite dimensional star representations is found one of them can be realized in terms of two valued functions of a complex variable this representation reduces to the sum of two metaplectic representations of sp 2 we show that it is precisely this metaplectic representation for osp 3 2 which gives the spin energy eigenstates for the one dimensional harmonic oscillator with spin 1 2 states