smooth embeddings of sl 2 and pgl 2 an embedding of an algebraic homogeneous space g h is an algebraic normal variety with an action of g having an open dense orbit isomorphic to g h there is a beautiful theory for the embeddings of tori see for example kkms danl or oda in this theory the embeddings are classified by fans of rational convex cones by studying the combinatorics of the fans one can determine certain geometric properties of the associated embeddings for example one can check which embeddings are smooth complete or projective recently luna and vust developed a method to classify embeddings of homogeneous spaces g h where g is a connected algebraic reductive group lv in the present paper we analyse more closely the cases of embeddings of x 2 and pgl 2 each embedding is represented by a diagram containing combinatorical information about the local rings of orbits as in the case of torus embeddings one would like to be able to determine geometric properties from these diagrams in this article we find the conditions that the diagrams must satisfy to represent smooth embeddings the idea of the proof is as follows an embedding is smooth if and only if the local rings of all the orbits are regular the embedding is a three dimensional normal variety so all orbits of dimension two are smooth it can be shown that the fixed points are never smooth so one is left to treat the one dimensional orbits which are isomorphic to the projective line a result from blv states that a point of a one dimensional orbit has an affme neighborhood which is the product of the affrne line and a certain two dimensional variety the orbit projects down to a point in the two dimensional variety thus the problem is reduced to checking the regularity of certain local rings of points in twodimensional varieties in each case either one can find explicitly two

the quantum deformation of su 1 1 as the dynamical symmetry of the anharmonic oscillator as an application of q deformed algebras to standard quantum mechanics the author shows that the su 1 1 dynamical symmetry of the quantum harmonic oscillator is deformed in the first order of approximation to the dynamical symmetry defined by the quantized universal enveloping algebra of su 1 1 when the harmonic potential is perturbed with a potential in x4 the resolution of the anharmonic oscillator is carried out algebraically in terms of generalized lowering and rising operators

generic singularities of nilpotent orbit closures according to a theorem of brieskorn and slodowy the intersection of the nilpotent cone of a simple lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity at the opposite extremity of the poset of nilpotent orbits the closure of the minimal nilpotent orbit is also an isolated symplectic singularity called a minimal singularity for classical lie algebras kraft and procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures specifically any such singularity is either a simple surface singularity a minimal singularity or a union of two simple surface singularities of type a 2k 1 in the present paper we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional lie algebras up to normalization in a few cases we summarize the results in some graphs at the end of the paper in most cases we also obtain simple surface singularities or minimal singularities though often with more complicated branching than occurs in the classical types there are however six singularities that do not occur in the classical types three of these are unibranch non normal singularities an sl 2 c variety whose normalization is a 2 an sp 4 c variety whose normalization is a 4 and a two dimensional variety whose normalization is the simple surface singularity a 3 in addition there are three 4 dimensional isolated singularities each appearing once we also study an intrinsic symmetry action on the singularities extending slodowy s work for the singularity of the nilpotent cone at a point in the subregular orbit

spherical conjugacy classes and involutions in the weyl group let g be a simple algebraic group over an algebraically closed field of characteristic zero or positive odd good characteristic let b be a borel subgroup of g we show that the spherical conjugacy classes of g intersect only the double cosets of b in g corresponding to involutions in the weyl group of g this result is used in order to prove that for a spherical conjugacy class mathcal o with dense b orbit v0 bwb there holds ell w rm rk 1 w dim mathcal o extending to the case of groups over fields of odd good characteristic a characterization of spherical conjugacy classes obtained by cantarini costantini and the author it is also shown that the weights occurring in the g module decomposition of the ring of regular functions on mathcal o are self adjoint and they lie in the 1 eigenspace of the element w

quadratic algebra as a hidden symmetry of the hartmann potential it is shown that operators commuting with the hamiltonian of the hartmann potential form the quadratic hahn algebra qh 3 the structure of this algebra and its finite dimensional representations are described an analysis of these representations is applied to obtain all the relevant physical results energy spectrum degree of degeneration and overlap functions

scattering and bound state green s functions on a plane via so 2 1 lie algebra we calculate the green s functions for the particle vortex system for two anyons on a plane with and without a harmonic regulator and in a uniform magnetic field these green s functions which describe scattering or bound states depending on the specific potential in each case are obtained exactly using an algebraic method related to the so 2 1 lie group from these green s functions we obtain the corresponding wave functions and for the bound states we also find the energy spectra

a connectedness property of algebraic moment maps let k be a connected compact lie group and m a hamiltonian k manifold i e a symplectic k manifold equipped with a moment map m k liek a theorem of kirwan implicitly in ki asserts if m is connected and compact then the level sets of are connected the purpose of this note is to prove such a statement in the category of algebraic varieties first we reformulate kirwan s theorem consider the map m k k which is the composition of with the quotient map for a point x k let h kx be its isotropy group and y kx k k its orbit then the fiber y is isomorphic to the fiber product k x thus since both k h and h are connected the connectedness of the fibers of is equivalent to the connectedness of the fibers of this formulation is more suitable for the algebraic category let g be a connected reductive group everything over c and z a hamiltonian gvariety with moment map z g lieg let g g specc g be the categorical quotient and let z g g be the composition of with the quotient map since the latter is not an orbit map the connection between fibers of and is much more loose than in the differential category and we concentrate on from now on the morphism is still not the right map since sometimes not even its generic fibers are connected an example is the action of g sl2 c on z c 2 c this is the cotangent bundle of c then g g a and u u in particular the generic fiber of has two connected components this is remedied by looking at the map u u instead then is the composition of with the finite map a a z 7 z the latter being responsible for the disconnected fibers this construction can be generalized as follows the morphism induces an homomorphism of algebras

normality of nilpotent varieties in e6 abstract we determine which nilpotent orbits in e6 have closures which are normal varieties and which do not at the same time we are able to verify a conjecture in e sommers comm math univ sancti pauli 49 1 2000 101 104 concerning functions on non special nilpotent orbits for e6

scattering and bound states for a class of non central potentials we obtain l2 series solutions of the three dimensional schrodinger wave equation for a large class of non central potentials that includes as special cases the aharonov bohm hartmann and magnetic monopole potentials it also includes contributions of the potential term cos r2 in spherical coordinates the solutions obtained are for all energies the discrete for bound states as well as the continuous for scattering states the l2 bases of the solution space are chosen such that the matrix representation of the wave operator is tridiagonal the expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three term recursion relations resulting from the matrix wave equation

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

on the coordinate ring of spherical conjugacy classes let g be a simple algebraic group over an algebraically closed field k of characteristic zero and mathcal o be a spherical conjugacy class of g we determine the decomposition of the coordinate ring k mathcal o of mathcal o into simple g modules

periodic reduction of the factorization chain and the hahn polynomials the n 4 periodic closure of the factorization chain is considered it is shown that the nonlinear operator algebra corresponding to this closure can be transformed into the quadratic hahn algebra as a result the three term recurrence coefficients for the hahn polynomials provide a special realization of the n 4 periodic factorization chain

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

auxiliary fields for treating two time lagrangian quantum mechanics functional integrals over complex and real auxiliary fields for propagators relating to two time memory actions are constructed the integrals over fields result from partial replacement of the path variable via functional integration in the path integral form of the propagator by a field variable in a way that allows the remaining path integration to be performed thus leaving a functional integral two new cases of actions involving memory a quadratic and a quartic are treated explicitly a functional integral identity is also developed furthermore the method of coupling an auxiliary dynamical field for generating memory actions is also employed and formulae are developed in the case of certain quadratic actions giving the propagator exactly the method is flexible enough embracing all explicitly known cases of two time quadratic actions and in addition enabling further cases to be treated an example of this nature is given explicitly

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

dynamical invariance algebra of the hartmann potential the accidental degeneracy occurring in the quantum mechanical treatment of the ring shaped potential v eta sigma 2r 1 1 2q eta 2 sigma 2 r sin theta 2 is explained by constructing an su 2 dynamical invariance algebra the schrodinger equation is solved in parabolic coordinates written in the framework of the kustaanheimo stiefel transformation and the hamilton jacobi equations are solved in ordinary parabolic coordinates all finite trajectories are found to be periodic

nilpotent orbits normality and hamiltonian group actions the theory of coadjoint orbits of lie groups is central to a number of areas in mathematics a list of such areas would include 1 group representation theory 2 symmetry related hamiltonian mechanics and attendant physical theories 3 symplectic geometry 4 moment maps and 5 geometric quantization from many points of view the most interesting cases arise when the group g in question is semisimple for semisimple g the most familiar of the orbits are orbits of semisimple elements in that case the associated representation theory is pretty much understood the borel weil bott theorem and noncompact analogs e g zuckerman functors the isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations a totally different experience is encountered with nilpotent orbits of semisimple groups here the associated representation theory unipotent representations is poorly understood and there is a loss of reductivity of isotropy subgroups to make matters worse or really more interesting orbits are no longer closed and there can be a failure of normality for orbit closures in addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations the interest in nilpotent orbits of semisimple lie groups has increased sharply over the last two decades this perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits e g the springer correspondence with representations of the weyl group this paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry the starting point is the recognition made also by others that the ring r of regular functions on any g homogeneous cover m of 0 is not only a poisson algebra the case for any coadjoint orbit but that r is also naturally graded the key theme is that the same nilpotent orbit may be shared by more than one simple group the key result is the determination of all pairs of simple lie groups having a shared nilpotent orbit furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

exact solution for a noncentral electric dipole ring shaped potential in the tridiagonal representation the schrodinger equation with noncentral electric dipole ring shaped potential is investigated by working in a complete square integrable basis that supports an infinite tridiagonal matrix representation of the wave operator the three term recursion relations for the expansion coefficients of both the angular and radial wavefunctions are presented the discrete spectrum for the bound states is obtained by the diagonalization of the radial recursion relation some potential applications of this system in different fields are discussed

an introduction to wonderful varieties with many examples of type f4 we give an introduction to the theory of wonderful g varieties with many examples when g is simple of type f4 we present results and open problems about these varieties on their classification on their isotropy groups on morphisms between them and on their relations with the representation theory of g

some amazing properties of spherical nilpotent orbits let g be a simple algebraic group defined over an algebraically closed field k of characteristic zero write g for its lie algebra let x g be a nilpotent element and g x g the corresponding nilpotent orbit the maximal number m such that adx 6 0 is called the height of x or of g x denoted ht x recall that an irreducible g variety x is called g spherical if a borel subgroup of g has an open orbit in x it was shown in pa1 that g x is g spherical if and only if adx 0 this means that the spherical nilpotent orbits are precisely the orbits of height 2 and 3 unfortunately whereas my proof for the orbits of height 2 and height 4 was completely general the argument for the orbits of height 3 explicitly used their classification in this paper we give a proof of sphericity that does not rely on the classification of nilpotent orbits see theorem 3 3 we begin with some properties of invariants of symplectic representations for instance we prove that 1 if h sp v is an irreducible representation without nonconstant invariants then h sp v and 2 if h has non constant invariants then it has an invariant of degree 4 applying these results to nilpotent orbits we prove that the centraliser zg x has a rather specific structure whenever ht x is odd from this description we deduce a conceptual proof of sphericity in case ht x 3 as another application we compute the index of zg x it will be shown that ind zg x rk g if ht x 3 this confirms elashvili s conjecture for such x see pa5 sect 3 about this conjecture in section 4 we prove that if the highest root of g is fundamental then g always has a specific orbit of height 3 which is denoted by o this orbit satisfies several arithmetical relations namely if g 3 i 3 g i is the z grading associated with an sl2 triple containing e o then dim g 3 2 and dim g 1 2 dim g 2 furthermore the weighted dynkin diagram of o can explicitly be described let be the unique simple root that is not orthogonal to and let i be all simple roots adjacent to on the dynkin diagram of g then one has to put 1 at all i s and 0 at all other simple roots theorem 4 5 it is curious that these properties of o enables us to give an intrinsic construction of g2 grading in each simple g whose highest root is fundamental

hidden symmetry algebra and overlap coefficients for two ring shaped potentials a new approach is proposed to general systems possessing su 1 1 su 1 1 dynamical symmetry on the base of this approach the quadratic hahn algebra qh 3 is shown to serve as a hidden symmetry in both quantum and classical pictures for two potentials generalizing the hartmann and the oscillator ring shaped potentials the overlap coefficients between wavefunctions in spherical and parabolic cylindrical coordinates are shown to coincide with clebsch gordan coefficients for su 1 1 algebra

general potentials described by so 2 1 dynamical algebra in parabolic coordinate systems the authors propose general three dimensional potentials in rotational and cylindrical parabolic coordinates which are generated by direct products of the so 2 1 dynamical group then they construct their green functions algebraically and find their spectra particular cases of these potentials which appear in the literature are also briefly discussed

exact solutions of the schrodinger equation with non central potential by the nikiforov uvarov method the general solutions of the schrodinger equation for a non central potential are obtained by using the nikiforov uvarov method the schrodinger equation with general non central potential is separated into radial and angular parts and energy eigenvalues and eigenfunctions are derived analytically by making special selections the non central potential is reduced to coulomb and hartmann ring shaped potentials and the obtained results are compared with the solutions of coulomb and hartmann potentials given in the literature

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

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

automorphisms root systems and compactifications of homogeneous varieties let g be a complex semisimple group and let h c g be the group of fixed points of an involutive automorphism of g then x g h is called a symmetric variety in cp de concini and procesi have constructed an equivariant compactification x which has a number of remarkable properties some of them being i the boundary is the union of divisors d1 dr ii there are exactly 2r orbits their closures are the intersections di1 n ndi even schematically in particular there is only one closed orbit iii in case g is of adjoint type all orbit closures are smooth it is called the wonderful embedding of x or a complete symmetric variety and is the foundation for most deeper results about x independently luna and vust developed in lv a general theory of equivariant compactifications of homogeneous varieties under a connected reductive group g in particular they realized the reason which makes symmetric varieties behave so nicely a borel subgroup b has an open dense orbit in g h varieties with this property are called spherical luna and vust were able to describe all equivariant compactifications of them in terms of combinatorial data very similar to torus embeddings which are actually a special case they obtained in particular that every spherical embedding has only finitely many orbits nevertheless the reason for the existence of a compactification with properties i iii remained mysterious then brion and pauer established a relation with the automorphism group they proved in bp a spherical variety x g h possesses an equivariant compactification with exactly one closed orbit if and only if autg x ng h h is finite in this case there is a unique one which dominates all others the wonderful compactification x they also showed that the orbits of x correspond to the faces of a strictly convex polyhedral cone z then properties i and ii above are equivalent to z being simplicial this fact is much deeper and was proved by brion in brl in fact he showed much more let f be the set of characters of b which are the characters of a rational b eigenfunction on x this is a finitely generated free abelian group then the cone z is a subset of the real vector space hom f r brion showed that there is a finite reflection group wx acting on f such that z is one of its weyl chambers in case of a symmetric variety wx is its little weyl group

on a dimension formula for spherical twisted conjugacy classes in semisimple algebraic groups let g be a connected semisimple algebraic group over an algebraically closed field of characteristic zero and let be an automorphism of g we give a characterization of spherical twisted conjugacy classes in g by a formula for their dimensions in terms of certain elements in the weyl group of g generalizing a result of n cantarini g carnovale and m costantini when is the identity automorphism for g simple and an outer automorphism of g we also classify the weyl group elements that appear in the dimension formula

the generalized mic kepler system this paper deals with the dynamical system that generalizes the mic kepler system it is shown that the schrodinger equation for this generalized mic kepler system can be separated in spherical and parabolic coordinates the spectral problem in spherical and parabolic coordinates is solved

algebraic treatment of super integrable potentials the so 2 1 lie algebra is applied to three classes of two and three dimensional smorodinsky winternitz super integrable potentials for which the path integral discussion has been recently presented in the literature we have constructed the green s functions for two important super integrable potentials in r2 among the super integrable potentials in r3 we have considered two examples one is maximally super integrable and another one minimally super integrable the discussion is made in various coordinate systems the energy spectrum and the suitably normalized wave functions of bound and continuous states are then deduced

functional integral treatment of some quantum nondemolition systems in the scheme of a quantum nondemolition qnd measurement an observable is measured without perturbing its evolution in the context of studies of decoherence in quantum computing we examine the open quantum system of a two level atom or equivalently a spin 1 2 system in interaction with quantum reservoirs of either oscillators or spins under the qnd condition of the hamiltonian of the system commuting with the system reservoir interaction for completeness we also examine the well known non qnd spin bose problem for all these many body systems we use the methods of functional integration to work out the propagators the propagators for the qnd hamiltonians are shown to be analogous to the squeezing and rotation operators respectively for the two kinds of baths considered squeezing and rotation being both phase space area preserving canonical transformations this brings out an interesting connection between the energy preserving qnd hamiltonians and the homogeneous linear canonical transformations

l2 series solutions of the dirac equation for power law potentials at rest mass energy we obtain solutions of the three dimensional dirac equation for radial power law potentials at rest mass energy as an infinite series of square integrable functions these are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the dirac operator is tridiagonal the wave equation results in a three term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials these are modified versions of the meixner pollaczek polynomials and of the continuous dual hahn polynomials the choice depends on the values of the angular momentum and the power of the potential

solution of the nonrelativistic wave equation using the tridiagonal representation approach we choose a complete set of square integrable functions as a basis for the expansion of the wavefunction in configuration space such that the matrix representation of the nonrelativistic time independent linear wave operator is tridiagonal and symmetric consequently the matrix wave equation becomes a symmetric three term recursion relation for the expansion coefficients of the wavefunction the recursion relation is then solved exactly in terms of orthogonal polynomials in the energy some of these polynomials are not found in the mathematics literature the asymptotics of these polynomials give the phase shift for the continuous energy scattering states and the spectrum for the discrete energy bound states depending on the space and boundary conditions the basis functions are written in terms of either the laguerre or jacobi polynomials the tridiagonal requirement limits the number of potential functions that yield exact solutions of the wave equation nonetheless the class of exactly solvable prob

solution of the dirac equation with pseudospin symmetry for a new harmonic oscillatory ring shaped noncentral potential a new harmonic oscillatory ring shaped noncentral potential is proposed in which the noncentral electric dipole and a novel angle dependent component are included the pseudospin symmetry for this potential is investigated by working in a complete square integrable basis that supports a tridiagonal matrix representation of the wave operator the resulting three term recursion relation for the expansion coefficients of the wavefunctions both angular and radial is presented the angular radial wavefunction is written in terms of the jacobi laguerre polynomials the discrete spectrum of the bound states is obtained by diagonalization of the radial recursion relation

a new ring shaped potential and its dynamical invariance algebra a new ring shaped potential obtained by replacing the coulomb part of the hartmann potential by a harmonic oscillator term is investigated the schrodinger equation is solved in spherical circular cylindrical prolate and oblate spheroidal coordinates as in the case of the hartmann potential the accidental degeneracies occurring in the spectrum are shown to be due to an su 2 dynamical invariance algebra this establishes a close connection between both ring shaped potentials

non commutative differential calculus and q analysis starting from the formulation of covariant noncommutative differential calculus recently given by wess and zumino 1990 the author constructs a deformation of the virasoro algebra which allows him to identify the variables and differential operators on the quantum plane r2q to those on the classical plane r2 this correspondence indicates how noncommutative geometry can be understood in terms of q analysis on the commutative plane he generalizes this result to the general n dimensional case and discuss some of its consequences

the many body problem for q oscillators the correct hamiltonian for the n body problem of free q oscillators is found which promotes the symmetries of the standard q 1 oscillator systems to q symmetries the spectrum of the system is found to be rich exhibiting interactions between the levels of the individual oscillators

path integral quantization of certain noncentral systems with dynamical symmetries path integral quantization is done for the five classes of potentials appearing in the systematic search for nonrelativistic systems with dynamical symmetries done by makarov smorodinsky valiev and winternitz nuovo cimento a 52 1061 1967 by an iterated application of bateman s series formula to the polar coordinate path integral an expansion is obtained of the feynman kernel or the green s function whichever is possible in terms of hypergeometric functions of the polar and azimuthal parts and a radial path integral is obtained whose evaluation yields the energy eigenvalues and the normalized wave functions special cases include the hartmann potential and the ring shaped oscillator

pseudospin symmetry for a new oscillatory ring shaped noncentral potential a new oscillatory ring shaped noncentral potential is proposed and the pseudospin symmetry for this potential is investigated by solving the dirac equation with equally mixed scalar and vector potentials with opposite signs the spinor wave functions and the energy spectrum for the bound states are obtained by using the nikiforov uvarov method the effect of angle dependent part on the radial solutions and the algebraic property of the energy equation are discussed

on the irreducibility of associated varieties of w algebras abstract we investigate the irreducibility of the nilpotent slodowy slices that appear as the associated variety of w algebras furthermore we provide new examples of vertex algebras whose associated variety has finitely many symplectic leaves

uniqueness property for spherical homogeneous spaces let g be a connected reductive group recall that a homogeneous g space x is called spherical if a borel subgroup b g has an open orbit on x to x one assigns certain combinatorial invariants the weight lattice the valuation cone and the set of b stable prime divisors we prove that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic further we recover the group of g equivariant automorphisms of x from these invariants

quartic poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras we introduce the most general quartic poisson algebra generated by a second and a fourth order integral of motion of a 2d superintegrable classical system we obtain the corresponding quartic associative algebra for the quantum analog extend daskaloyannis construction obtained in context of quadratic algebras and also obtain the realizations as deformed oscillator algebras for this quartic algebra we obtain the casimir operator and discuss how these realizations allow to obtain the finite dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems we apply the construction and the formula obtained for the structure function on a superintegrable system related to type i laguerre exceptional orthogonal polynomials introduced recently

q deformation of radial problems the simple harmonic oscillator in two dimensions the possibility of q deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem the algebraic structure of the d dimensional simple harmonic oscillator is analysed from this point of view in the two dimensional case q deformed bosons are introduced for the radial degrees of freedom which together with the standard angular operators yield the so 2 soq 2 1 dynamical algebra the corresponding hilbert space is identified and possible q deformed hamiltonians described

the quantum algebra uq sl2 and its equitable presentation abstract we show that the quantum algebra u q sl 2 has a presentation with generators x 1 y z and relations x x 1 x 1 x 1 q x y q 1 y x q q 1 1 q y z q 1 z y q q 1 1 q z x q 1 x z q q 1 1 we call this the equitable presentation we show that y respectively z is not invertible in u q sl 2 by displaying an infinite dimensional u q sl 2 module that contains a nonzero null vector for y respectively z we consider finite dimensional u q sl 2 modules under the assumption that q is not a root of 1 and char k 2 where k is the underlying field we show that y and z are invertible on each finite dimensional u q sl 2 module we display a linear operator that acts on finite dimensional u q sl 2 modules and satisfies 1 x y 1 y z 1 z x on these modules we define using the q exponential function

rings of regular functions on nilpotent orbits and their covers letg be a complex semisimple algebraic group with lie algebra mathfrak g let mathcal o subset mathfrak g be a nilpotentg orbit r mathcal o its ring of regular functions we derive a formula for r mathcal o as ag module and prove some partial results on r tilde mathcal o tilde mathcal o a cover of mathcal o we then relate this formula to various existing multiplicity formulas fork types in harish chandra bimodules ofg

on the connection between the hydrogen atom and the harmonic oscillator the continuum case the connection between a three dimensional nonrelativistic hydrogen atom with positive energy and a four dimensional isotropic harmonic oscillator with repulsive potential is established by applying jordan schwinger boson calculus to the algebra of the laplace runge lenz pauli vector the spectrum generating group so 4 2 both for the bound and free states of the three dimensional hydrogen atom arises as a quotient of the group sp 8 r associated to four dimensional isotropic harmonic oscillator with constraint

quantum superintegrable systems with quadratic integrals on a two dimensional manifold there are two classes of quantum integrable systems on a manifold with quadratic integrals the liouville and the lie integrable systems as it happens in the classical case the quantum liouville quadratic integrable systems are defined on a liouville manifold and the schrodinger equation can be solved by separation of variables in one coordinate system the lie integrable systems are defined on a lie manifold and are not generally separable ones but can be solved therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems the quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion there are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones analytic formulas for the involved integrals are calculated in all the cases all the known quan

introduction to leonard pairs in this survey paper we give an elementary introduction to the theory of leonard pairs a leonard pair is defined as follows let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and b v v that satisfy conditions i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we give several examples of leonard pairs using these we illustrate how leonard pairs arise in representation theory combinatorics and the theory of orthogonal polynomials

algebraic treatment of a general noncentral potential it is shown that the problem of a particle moving in a noncentral potential generalizing the coulomb potential and the hartmann ring shaped potential accepts so 2 1 so 2 1 as a dynamical group within the framework of the kustaanheimo stiefel transformation the green s function relative to this compound potential is calculated through the algebraic approach so 2 1 and with the help of the baker campbell hausdorff formulas the energy spectrum and the normalized wave functions of the bound states can then be deduced eventually the coulomb potential the hartmann ring shaped potential and also due to its close link with the latter the compound coulomb plus aharonov bohm potential may all be considered as particular cases

extremal contractions stratified mukai flops and springer maps we prove that two springer maps of the same degree over a nilpotent orbit closure are connected by stratified mukai flops and the latter is obtained by contractions of extremal rays of a natural resolution of the nilpotent orbit closure

a characterization of nilpotent orbit closures among symplectic singularities we prove that a conical symplectic variety with maximal weight 1 is isomorphic to i an affine space with the standard symplectic form or ii a normal nilpotent orbit closure of a complex semisimple lie algebra

exact solution of non adiabatic model hamiltonians in solid state physics and optics simple non adiabatic model hamiltonians are treated using the realisation of the vibrational modes on bargmann s hilbert space of analytic functions in this formulation the schrodinger equation is a system of linear first order differential equations the energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions the solutions are given in terms of neumann series the recurrence relations for the expansion coefficients have a simple structure under particular conditions for the interaction constant they allow for terminating neumann series isolated exact solutions in the general case the conditions for the eigenvalues are transcendental equations involving a continued fraction the continued fraction can be approximated to any desired degree of accuracy in a rapidly convergent process based on worpitzky s theorem and its relation to conformal mapping the eigenvalues are calculated a physical interpretation of the solutions which makes use of intuitive arguments is also given

alternative hamiltonian quantization the authors take a standard hamiltonian quadratic in a set of creation and annihilation operators and demonstrate an alternative quantization by quommutators i e a set of rules for transposition of these operators different from the those of the usual canonical commutation relations which yields a new spectrum of a type reminiscent of logarithmic trajectories

perverse coherent sheaves and the geometry of special pieces in the unipotent variety let x be a scheme of finite type over a noetherian base scheme s admitting a dualizing complex and let ux be an open set whose complement has codimension at least 2 we extend the deligne bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension or intersection cohomology functor from perverse sheaves on u to perverse sheaves on x may be defined for a much broader class of perversities than has previously been known we also introduce a derived category version of the coherent intermediate extension functor under suitable hypotheses we introduce a construction called s2 extension in terms of perverse coherent sheaves of algebras on x that takes a finite morphism to u and extends it in a canonical way to a finite morphism to x in particular this construction gives a canonical s2 ification of appropriate x the construction also has applications to the macaulayfication problem and it is particularly well behaved when x is gorenstein our main goal however is to address a conjecture of lusztig on the geometry of special pieces certain subvarieties of the unipotent variety of a reductive algebraic group the conjecture asserts in part that each special piece is the quotient of some variety previously unknown for the exceptional groups and in positive characteristic by the action of a certain finite group we use s2 extension to give a uniform construction of the desired variety