branching rules for the supergroup su n m from those of su n m the decomposition of representations of supergroups into representations of subgroups is needed in practical applications in this paper we set up and exploit a fruitful one to one correspondence between the lie group branching su n m su n su m u 1 and the supergroup branchings su n m su n su m u 1 and su n1 n2 m1 m2 su n1 m1 su n2 m2 u 1 a simple and useful prescription is discovered for obtaining the su n m branching rules from those of su n m for any representation a large class of examples sufficient for many physical applications we can foresee are explicitly worked out and tabulated

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

the quantum superalgebra uq osp 1 2n deformed para bose operators and root of unity representations we recall the relation between the lie superalgebra osp 1 2n and para bose operators the quantum superalgebra uq osp 1 2n defined as usual in terms of its chevalley generators is shown to be isomorphic to an associative algebra generated by so called pre oscillator operators satisfying a number of relations from these relations and the analogue with the non deformed case one can interpret these pre oscillator operators as deformed para bose operators some consequences for uq osp 1 2n cartan weyl basis poincare birkhoff win basis and its hopf subalgebra uq gl n are pointed out finally using a realization in terms of q commuting q bosons we construct an irreducible finite dimensional unitary fock representation of uq osp 1 2n and its decomposition in terms of uq gl n representations when q is a root of unity

generalized boson algebra and its entangled bipartite coherent states starting with a given generalized boson algebra known as the bosonized version of the quantum super hopf algebra we employ the hopf duality arguments to provide the dually conjugate function algebra with both the hopf algebras being finitely generated we produce a closed form expression of the universal matrix that caps the duality and generalizes the familiar exponential map relating a lie algebra with its corresponding group subsequently using an inverse mellin transform approach the coherent states of single node systems subject to the symmetry are found to be complete with a positive definite integration measure the nonclassical coalgebraic structure of the algebra is found to generate naturally entangled coherent states in bipartite composite systems

a conjectured character formula for atypical irreducible modules of the lie superalgebra sl m n a formula is used for the character of each finite dimensional irreducible representation of the lie superalgebra sl m n this new formula appears to cover not only typical but also both singly and multiply atypical representations a comparison is made with previous conjectures and the problem of generalising the formula to cover all of the basic classical lie superalgebras is discussed

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

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

character formulas for irreducible modules of the lie superalgebras sl m n kac distinguished between typical and atypical finite dimensional irreducible representations of the lie superalgebras sl m n and provided an explicit character formula appropriate to all the typical representations here the range of validity of some character formulas for atypical representations that have been proposed are discussed several of them are of the kac weyl type but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl 3 4 having ruled out therefore all such formulas a completely new extension of the kac weyl character formula is proposed the validity of this formula in the case of all covariant tensor irreducible representations is proved and some evidence in support of the conjecture that it covers all irreducible representations of sl m n is presented

finite dimensional representations of the special linear lie superalgebra sl 1 n i typical representations in a series of two papers all finite dimensional irreducible representations of the special linear lie superalgebra sl 1 n are written down in a matrix form this paper develops a background for constructing the representations expressions for the transformation of the basis under the action of the generators are given for all induced and hence for all typical sl 1 n modules

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

effective moduli of piezoelectric material with microcavities abstract the effective electroelastic moduli of microcavity weakened piezoelectric plates are investigated by the dilute self consistent mori tanaka and differential micromechanics theories the results of perturbed heat intensity strain and electric field sef due to the presence of voids are obtained for two dimensional 2 d piezoelectric plates with microcavities of various shapes and then the above four micromechanics models can be established with the results these models can be applicable to a wide range of microcavities such as ellipse circle crack triangle square and pentagon some numerical results are presented to illustrate the applicability of these models

branching rules for typical and atypical representations of gl n 1 the gl n 1 down arrow gl n gl 1 branching rules are determined for all finite dimensional irreducible typical and atypical representations of gl n mod 1 using a recently introduced induced module construction for atypical modules and confirm those found recently by palev 1988 by another method the lowest weights and characters of the irreducible representations are found and the validity for gl n mod 1 of the character formula conjectured by hughes and king 1987 is confirmed some generalisations and extensions of the method are discussed

mixed supertableaux of the superunitary groups ii su n m the young supertableaux of the superunitary groups are studied we give the conditions on the size of legal supertableaux the type of atypicity for tensor representations which are irreducible making the link between the kac dynkin parameters and those of the supertableau a discussion is given on nonfully reducible representations and generalized atypical supertableaux

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

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

general solutions for thermopiezoelectrics with various holes under thermal loading abstract the thermoelectroelastic problems for a hole of various shapes embedded in an infinite matrix are considered in this paper using stroh s formalism and the technique of conformal mapping a unified solution is obtained in closed form for an infinite thermopiezoelectric plate with various holes induced by thermal loads the loads may be uniform remote heat flow point heat source and temperature discontinuity further a detailed discussion on the critical points for the mapping function of a piezoelectric plate with polygonal holes is presented and the study shows that the transformation is nonsingle valued a simple approach is given to treat such a situation as an application of the proposed solutions a system of singular integral equations for the unknown temperature discontinuity and the dislocation of elastic displacement and electric potential defined on crack faces is developed and solved numerically numerical results are presented to illustrate the application of the proposed formulation

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

classification of star and grade star representations of c n 1 the two types of and grade representations of the lie superalgebra c n 1 are classified a type 1 irreducible representation is characterized by the single condition s 0 and s being the highest weight and the odd simple root respectively while the type 2 representations are duals of type 1 representations for n 1 only the identity and vector representations of c n 1 are shown to be grade the superalgebra c 2 proves to be an exception and admits several classes of nontrivial irreducible grade representations

equivalence of two character formulae for typical and singly atypical irreducible modules of sl m n the bernstein leites van der jeugt character formula known to be valid for all typical and singly atypical irreducible modules of sl m n is shown to be equivalent in all such cases to the formula conjectured by hughes and king 1987 which previously has only been proved valid in the case of sl m 1

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

on the composition factors of kac modules for the lie superalgebras sl m n in the classification of finite dimensional modules of lie superalgebras kac distinguished between typical and atypical modules kac introduced an induced module the so called kac module v with highest weight which was shown to be simple if is a typical highest weight if is an atypical highest weight the kac module is indecomposable and the simple module v can be identified with a quotient module of v in the present paper the problem of determining the composition factors of the kac modules for the lie superalgebra sl m n is considered an algorithm is given to determine all these composition factors and conversely an algorithm is given to determine all the kac modules containing a given simple module as a composition factor the two algorithms are presented in the form of conjectures and illustrated by means of detailed examples strong evidence in support of the conjectures is provided the combinatorial way in which the two algorithms are intertwined is both surprising and interesting and is a convincing argument in favor of the solution to the composition factor problem presented here

polynomial deformations of osp 1 2 and generalized parabosons we consider the algebra r generated by three elements a b h subject to three relations h a a h b b and a b f h when f h h this coincides with the lie superalgebra osp 1 2 when f is a polynomial we speak of polynomial deformations of osp 1 2 irreducible representations of r are described and in the case deg f 2 we obtain a complete classification showing some similarities but also some interesting differences with the usual osp 1 2 representations the relation with deformed oscillator algebras is discussed leading to the interpretation of r as a generalized paraboson algebra

a new kind of two parameter deformation of heisenberg and parabose algebras and related deformed derivative we propose a new kind of two parameter p q deformed heisenberg and parabose algebra which reduces to the heisenberg algebra for the p 1 case and to parabose algebra for q 1 case corresponding to the two parameter deformed oscillator we also introduce a new kind of p q deformed derivative which relates to the ordinary derivative and q deformed derivative in an explicit manner we study the structure of fock like space of the new p q deformed oscillators and derive a formal solution for the eigenvalue equation of the hamiltonian

hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the jordan schwinger map the aim of this paper is to show that there is a hopf structure of the parabosonic and parafermionic algebras and this hopf structure can generate the well known hopf algebraic structure of the lie algebras through a realization of lie algebras using the parabosonic and parafermionic extension of the jordan schwinger map the differences between the hopf algebraic and the graded hopf superalgebraic structure on the parabosonic algebra are discussed

on boson algebras as hopf algebras certain types of generalized undeformed and deformed boson algebras which admit a hopf algebra structure are introduced together with their fock type representations and their corresponding r matrices it is also shown that a class of generalized heisenberg algebras including those underlying physical models such as that of calogero sutherland is isomorphic with one of the types of boson algebra proposed and can be formulated as a hopf algebra

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

plane problems in piezoelectric media with defects abstract a two dimensional electroelastic analysis is performed on a transversely isotropic piezoelectric material containing defects a general solution is provided in terms of complex potentials with emphasis being placed on stress concentrations that arise in the vicinity of circular and elliptical holes it is shown that for this genre of problem both mechanical and electrical variables are responsible for the peak stresses

recursively minimally deformed oscillators a recursive deformation of the boson commutation relation is introduced each step consists of a minimal deformation of a commutator a a fk n into a a qk 1 fk n where stands for the set of deformation parameters that fk depends on followed by a transformation into the commutator a a fk 1 qk 1 n to which the deformed commutator is equivalent within the fock space starting from the harmonic oscillator commutation relation a a 1 we obtain the arik coon and macfarlane biedenharn oscillators at the first and second steps respectively followed by a sequence of multiparameter generalizations several other types of deformed commutation relations related to the treatment of integrable models and to parastatistics are also obtained the generic form consists of a linear combination of exponentials of the number operator and the various recursive families can be classified according to the number of free linear parameters involved that depends on the form of the initia

conformal mapping for the helmholtz equation acoustic wave scattering by a two dimensional inclusion with irregular shape in an ideal fluid the complex variables method with mapping function was extended to solve the linear acoustic wave scattering by an inclusion with sharp smooth corners in an infinite ideal fluid domain the improved solutions of helmholtz equation shown as bessel function with mapping function as the argument and fractional order bessel function were analytically obtained based on the mapping function the initial geometry as well as the original physical vector can be transformed into the corresponding expressions inside the mapping plane as all the physical vectors are calculated in the mapping plane this method can lead to potential vast savings of computational resources and memory in this work the results are validated against several published works in the literature the different geometries of the inclusion with sharp corners based on the proposed mapping functions for irregular polygons are studied and discussed the findings show that the variation of angles and frequencies of the incident waves ha

a class of quadratic deformations of lie superalgebras we study certain graded finite dimensional polynomial algebras of degree 2 which are a special class of deformations of lie superalgebras which we call quadratic lie superalgebras starting from the formal definition we discuss the generalized jacobi relations in the context of the koszul property and give a proof of the poincare birkhoff witt basis theorem we give several concrete examples of quadratic lie superalgebras for low dimensional cases and discuss aspects of their structure constants for the type i class we derive the equivalent of the kac module construction for typical and atypical modules and a related direct construction of irreducible modules due to gould we investigate in detail one specific case the quadratic generalization gl2 n 1 of the lie superalgebra sl n 1 we formulate the general atypicality conditions at level 1 and present an analysis of zero and one step atypical modules for a certain family of kac modules

irreducible representations of the superalgebras type ii the representation of the orthosymplectic algebras and the other members of their class are built explicitly with simple techniques

atypical representations for type i lie superalgebras a new modified induced module construction is presented for all finite dimensional irreducible typical and atypical modules of a type i basic classical lie superalgebra the method is illustrated with some low dimensional representations of gl m mod n and all representations of gl 2 mod 1

an exact and explicit treatment of an elliptic hole problem in thermopiezoelectric media abstract this paper presents an exact solution for the problem of an elliptic hole or a crack in a thermopiezoelectric solid first based on the extended version of eshelby stroh s formulation the generalized 2d problems of an elliptical hole in a thermopiezoelectric medium subject to uniform heat flow and mechanical electrical loads at infinity are studied according to exact boundary conditions at the rim of the hole the complex potentials in the medium and the electric field inside the hole are obtained in closed form respectively then when the hole degenerates into a crack the explicit solutions for the field intensity factors near the crack tip and the electric field inside the crack are presented it is shown that the singularities of all the field are dependent on the material constants the applied heat load and mechanical loads at infinity but not on the applied electric loads it is also found that the electric field inside the crack is linearly variable which is different from the result based on the impermeable crack model

algebraic structure of parabose fock space i the green s ansatz revisited a description of the fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed a new statement of the well known green s ansatz is introduced to help understand the significance of the bilinear commutation relations it is also applied to show how all parabose oscillators of even order respectively odd order can be obtained as irreducible constituents of the fock space associated with a green s ansatz of two three terms analogs of the form in which the parabose green s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order methods used in the paper allow various new results for q deformed parabose oscillators to be derived

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

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

some problems in plane thermopiezoelectric materials with holes piezoelectric materials have recently attracted considerable attention due to their potential use in intelligent structural systems in this paper we treat the plane problem of thermopiezoelasticity with various holes and subject to coupled mechanical electric and thermal loads an analytical solution is obtained by applying the technique of conformal mapping and some identities in the stroh formalism the solution has a simple unified form for various holes such as ellipse circle triangle and square by way of the solution the expressions for the energy release rate and stress intensity factors of cracks are presented numerical results for concentration coefficients of stress and electric displacement along the hole boundary are given to assess the acceptability of the proposed method

compressibility of two dimensional pores having n fold axes of symmetry the complex variable method and conformal mapping are used to derive a closed form expression for the compressibility of an isolated pore in an infinite two dimensional isotropic elastic body the pore is assumed to have an n fold axis of symmetry and be represented by at most four terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle the results are validated against some special cases available in the literature and against boundary element calculations by extrapolation of the results for pores obtained from three and four terms of the schwarz christoffel mapping function for regular polygons the compressibilities of a triangle square pentagon and hexagon are found to at least three digits specific results for some other pore shapes more general than the quasi polygons obtained from the schwarz christoffel mapping are also presented an approximate scaling law for the compressibility in terms of the ratio of perimeter squared to area is also tested this expression gives a reasonable approximation to the pore compressibility but may overestimate it by as much as 20

a braided look at green ansatz for parabosons bosons and parabosons are described as associative superalgebras with an infinite number of odd generators bosons are shown to be a quotient superalgebra of parabosons establishing thus an even algebra epimorphism which is an immediate link between their simple modules parabosons are a super hopf algebra the super hopf algebraic structure of parabosons combined with the projection epimorphism previously stated provides a braided interpretation of the green ansatz device and of the parabosonic fock like representations this braided interpretation combined with an old problem leads to the construction of straightforward generalizations of the green ansatz

unitarizable representations of the deformed para bose superalgebra uq osp 1 2 at roots of 1 the unitarizable irreps of the deformed para bose superalgebra pbq which is isomorphic to uq osp 1 2 are classified at q being root of 1 new finite dimensional irreps of uq osp 1 2 are found explicit expressions for the matrix elements are written down

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

normal ordering for the deformed heisenberg algebra involving the reflection operator we give the solution to the problem of normal ordering of monomials a a n in pairs of deformed annihilation and creation operators of the heisenberg algebra involving the reflection operator

coherent states for parabosons taking into consideration that the fock space of a parabose oscillator may be described by bilinear commutation relations involving creation and annihilation operators we construct the coherent states for odd and even order parabosons

representations of the orthosymplectic lie superalgebra mathfrak osp 1 4 and paraboson coherent states we introduce and obtain two mode paraboson coherent states in appropriate subspaces these coherent states provide a decomposition of unity where the measure when expressed using the cat type states is positive definite bicoherent states where the mutually commuting lowering operators are diagonalized are also obtained matrix elements in the coherent state basis are calculated

quantum algebra from generalized q hermite polynomials abstract in this paper we discuss new results related to the generalized discrete q hermite ii polynomials h n x q introduced by mezlini et al in 2014 our aim is to give a continuous orthogonality relation a q integral representation and an evaluation at unity of the poisson kernel for these polynomials furthermore we introduce q schrodinger operators and we give the raising and lowering operator algebra corresponding to these polynomials our results generate a new explicit realization of the quantum algebra su q 1 1 using the generators associated with a q deformed generalized para bose oscillator

algebraic structure of green s ansatz and its q deformed analogue the algebraic structure of green s ansatz 1953 is analysed in such a way that its generalization to the case of q deformed para bose and para fermi operators becomes evident to this end the underlying lie super algebraic properties of the parastatistics are essentially used

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