extended kepler coulomb quantum superintegrable systems in three dimensions the quantum kepler coulomb system in three dimensions is well known to be second order superintegrable with a symmetry algebra that closes polynomially under commutators this polynomial closure is also typical for second order superintegrable systems in 2d and for second order systems in 3d with nondegenerate four parameter potentials however the degenerate three parameter potential for the 3d kepler coulomb system also second order superintegrable is an exception as its symmetry algebra does not close polynomially the 3d four parameter potential for the extended kepler coulomb system is not even second order superintegrable but verrier and evans 2008 j math phys 49 022902 showed it was fourth order superintegrable and tanoudis and daskaloyannis 2011 arxiv 11020397v1 showed that if a second fourth order symmetry is added to the generators the symmetry algebra closes polynomially here based on the tremblay turbiner and winternitz construction we consider an infinite class of quantum extended kepler coulomb three and four parameter systems indexed by a pair of rational numbers k1 k2 and reducing to the usual systems when k1 k2 1 we show these systems to be superintegrable of arbitrarily high order and determine the structure of their symmetry algebras we demonstrate that the symmetry algebras close algebraically only for systems admitting extra discrete symmetries is polynomial closure achieved underlying the structure theory is the existence of raising and lowering operators not themselves symmetry operators or even defined independent of basis that can be employed to construct the symmetry operators and their structure relations

recursive procedures for krall sheffer operators we consider the krall sheffer class of admissible partial differential operators in the plane we concentrate on algebraic structures such as the role of commuting operators and symmetries for the polynomial eigenfunctions we give explicit forms of the 3 level recurrence relations and differential raising operators which are shown to satisfy unusual commutation relations we present new generating functions for two of the cases

zariski theorems and diagrams for braid groups empirical properties of generating systems for complex reflection groups and their braid groups have been observed by orlik solomon and broue malle rouquier using shephard todd classification we give a general existence result for presentations of braid groups which partially explains and generalizes the known empirical properties our approach is invariant theoretic and does not use the classification the two ingredients are springer theory of regular elements and a zariski like theorem

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

matrix factorisations arising from well generated complex reflection groups abstract we discuss an interesting duality known to occur for certain complex reflection groups namely the duality groups our main construction yields a concrete representation theoretic realisation of this duality this allows us to naturally identify invariant vector fields with vector fields on the orbit space for the action of a duality group as another application we construct matrix factorisations of the highest degree basic invariant which give free resolutions of the module of kahler differentials of the coinvariant algebra a associated to such a reflection group from this one can explicitly calculate the dimension of each graded piece of a c and of der c a a adding a new formula to the numerology of reflection groups this applies for instance when a is the cohomology of any complete flag manifold and hence has geometric consequences

separable coordinates integrability and the niven equations finite polynomial solutions of laplace s equation are investigated the unifying features of this study are the so called niven equations which yield the dimension of the space of such solutions in carrying out this study complete sets of solutions are obtained on the n dimensional sphere in terms of ellipsoidal coordinates this corresponds to an integrable system having all the integrals of motion given by quadratic orbits of the universal enveloping algebra of o n 1 they call this system the n dimensional euler top the spectrum of the integrals of motion has been recently computed for n 3 by komarov and kuznetsov 1991 using results originally due to niven 1891 these calculations are extended to arbitrary dimension

homology computations for complex braid groups ii abstract we complete the computation of the integral homology of the generalized braid group b associated to an arbitrary irreducible complex reflection group w of exceptional type in order to do this we explicitly computed the recursively defined differential of a resolution of z as a z b module using parallel computing we also deduce from this general computation the rational homology of the milnor fiber of the singularity attached to most of these reflection groups

second cohomology groups and finite covers of infinite symmetric groups abstract for an infinite set k 2 and w the set of k sets from there is a natural closed permutation group k which is a non split extension 0 z 2 w k sym 1 we classify the closed subgroups of k which project onto sym the question arises in model theory as a problem about finite covers but here we formulate and solve it in algebraic terms

garside families and garside germs garside families have recently emerged as a relevant context for extending results involving garside monoids and groups which themselves extend the classical theory of generalized braid groups here we establish various characterizations of garside families that is equivalently various criteria for establishing the existence of normal decompositions of a certain type

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

path integration on hermitian hyperbolic space in this paper the path integral technique is applied to the quantum motion on the hermitian hyperbolic space hh 2 the schrodinger equation on this space separates in 12 coordinate systems which are closely related to the coordinate systems on the two dimensional hyperboloid for six coordinate systems out of the twelve it is possible to find a path integral solution

superintegrable three body systems on the line we consider classical three body interactions on a euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momentum first integrals these systems are multiseparable superintegrable and equivalent up to rescalings to a one particle system in the three dimensional euclidean space common features of the dynamics are discussed we show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics the conformal multiseparability is discussed and examples of conformal first integrals are given the systems considered here in generality include the calogero wolfes and other three body interactions widely studied in mathematical physics

completeness of multiseparable superintegrability on the complex 2 sphere the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability here we demonstrate for nondegenerate potentials how to establish a complete list of such potentials that are superintegrable on the complex 2 sphere using essentially algebraic means we classify all such potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters the method of proof generalizes to other spaces and dimensions we show for the 2 sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

finite groups generated in low real codimension abstract we study the intersection lattice of the arrangement a g of subspaces fixed by subgroups of a finite linear group g when g is a reflection group this arrangement is precisely the hyperplane reflection arrangement of g we generalize the notion of finite reflection groups we say that a group g is generated resp strictly generated in codimension k if it is generated by its elements that fix point wise a subspace of codimension at most k resp precisely k we prove that the alternating subgroup alt w of a reflection group w is strictly generated in codimension two further we compute the intersection lattice of all finite subgroups of g l 3 r and moreover we emphasize the groups that are minimally generated in real codimension two i e groups that are strictly generated in codimension two but have no real reflection representations we also provide several examples of groups generated in higher codimension

divisorial inertia and central elements in braid groups abstract given a complex reflection group w we will show how the generators of the centers of the parabolic subgroups of the pure braid group p w can be represented by loops around irreducible divisors of the corresponding minimal de concini procesi model x w we will also show that a more subtle construction gives representations of the generators of the centers of the parabolic subgroups of the braid group b w as loops in the not smooth quotient variety x w w

general nth order integrals of motion in the euclidean plane the general form of an integral of motion that is a polynomial of order n in the momenta is presented for a hamiltonian system in two dimensional euclidean space the classical and the quantum cases are treated separately emphasizing both the similarities and the differences between the two the main application will be to study nth order superintegrable systems that allow separation of variables in the hamilton jacobi and schrodinger equations respectively

cohomology of deligne lusztig varieties for short length regular elements in exceptional groups abstract we determine the cohomology of deligne lusztig varieties associated with some short length regular elements for split groups of type f 4 and e n as a byproduct we obtain conjectural brauer trees for the principal 14 block of e 7 and the principal 24 block of e 8

superintegrability in three dimensional euclidean space potentials for which the corresponding schrodinger equation is maximally superintegrable in three dimensional euclidean space are studied the quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

partial differential equations having orthogonal polynomial solutions abstract we show that if a second order partial differential equation l u auxx 2buxy cuyy dux euy nu has orthogonal polynomial solutions then the differential operator l must be symmetrizable and can not be parabolic in any nonempty open subset of the plane we also find rodrigues type formula for orthogonal polynomial solutions of such differential equations

master symmetries non hamiltonian symmetries and superintegrability of the generalized smorodinsky winternitz system the first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach it is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries in the second part the properties of dynamical but non hamiltonian symmetries are studied it is proved that the higher order superintegrability of the generalized smorodinsky winternitz system is a consequence of the existence of symplectic symmetries not preserving the hamiltonian function

quasi exactly solvable lie algebras of differential operators in two complex variables the authors completely classify all quasi exactly solvable lie algebras of first order differential operators in two complex variables applications to quasi exactly solvable quantum problems are indicated

decomposition of garside groups and self similar l algebras abstract picantin s iterated crossed product representation of garside monoids is extended and reproved for a wide class of not necessarily noetherian partially ordered groups with a right invariant lattice structure it is shown that the tree like structure of such an iterated crossed product is equivalent to a partial cycle set closely related to a class of set theoretic solutions of the quantum yang baxter equation the decomposition of finite square free solutions is related to the crossed product representation of the corresponding structure group

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

conformal laplace superintegrable systems in 2d polynomial invariant subspaces 2nd order conformal superintegrable systems in n dimensions are laplace equations on a manifold with an added scalar potential and independent 2nd order conformal symmetry operators they encode all the information about helmholtz eigenvalue superintegrable systems in an efficient manner there is a 1 1 correspondence between laplace superintegrable systems and stackel equivalence classes of helmholtz superintegrable systems in this paper we focus on superintegrable systems in two dimensions n 2 where there are 44 helmholtz systems corresponding to 12 laplace systems for each laplace equation we determine the possible two variate polynomial subspaces that are invariant under the action of the laplace operator thus leading to families of polynomial eigenfunctions we also study the behavior of the polynomial invariant subspaces under a stackel transform the principal new results are the details of the polynomial variables and the conditions on parameters of the potential corresponding to polynomial solutions the hidden gl 3 algebraic structure is exhibited for the exact and quasi exact systems for physically meaningful solutions the orthogonality properties and normalizability of the polynomials are presented as well finally for all helmholtz superintegrable solvable systems we give a unified construction of one dimensional 1d and two dimensional 2d quasi exactly solvable potentials possessing polynomial solutions and a construction of new 2d pt symmetric potentials is established

uniqueness of roots up to conjugacy for some affine and finite type artin groups let g be one of the artin groups of finite type b n c n and affine type tilde mathbf a n 1 tilde mathbf c n 1 in this paper we show that if and are elements of g such that k k for some nonzero integer k then and are conjugate in g for the artin group of type a n this was recently proved by gonzalez meneses in fact we prove a stronger theorem from which the above result follows easily by using descriptions of those artin groups as subgroups of the braid group on n 1 strands let p be a subset of 1 n an n braid is said to be p pure if its induced permutation fixes each i in p and p straight if it is p pure and it becomes trivial when we delete all the ith strands for i not in p exploiting the nielsen thurston classification of braids we show that if and are p pure n braids such that k k for some nonzero integer k then there exists a p straight n braid with 1 moreover if 1 in p the conjugating element can be chosen to have the first strand algebraically unlinked with the other strands especially in case of p 1 n our result implies the uniqueness of roots of pure braids which was known by bardakov and by kim and rolfsen

the super separability of the three body inverse square calogero system the geometrical theory of the variable separation for the hamilton jacobi equation is applied to the classical three body inverse square calogero system it is proved that this system is separable in infinitely many inequivalent ways related to five different kinds of separable webs in the euclidean three space and the corresponding systems of independent first integrals in involution are computed

quadratic poisson algebras of two dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems the integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic poisson algebra whose the general form is investigated each classical superintegrable problem has a quantum counterpart a quantum superintegrable system the quadratic poisson algebra is deformed into a quantum associative algebra the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique it is shown that the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system the calculation of energy eigenvalues is reduced to the solution of algebraic equations which are universal that is for all two dimensional superintegrable systems with quadratic integrals of motion

a new approach to the higher order superintegrability of the tremblay turbiner winternitz system the higher order superintegrability of systems separable in polar coordinates is studied using an approach that was previously applied for the study of the superintegrability of a generalized smorodinsky winternitz system the idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions here denoted by m and n this technique leads to a proof of the superintegrability of the tremblay turbiner winternitz system and to the explicit expression of the constants of motion a second family related with the first one of superintegrable systems is also studied

path integral solutions for deformed poschl teller like and conditionally solvable potentials i discuss in this paper the behaviour of the solutions of the so called q hyperbolic potentials i e poschl teller like and conditionally solvable potentials in terms of the path integral formalism the differences in comparison to the usual poschl teller like potentials are investigated including the discrete energy spectra and the bound state wavefunctions we also point out the relation of the q deformation with curvature on hyperboloids

necessary conditions for classical super integrability of a certain family of potentials in constant curvature spaces we formulate the necessary conditions for the maximal super integrability of a certain family of classical potentials defined in the constant curvature two dimensional spaces we give examples of homogeneous potentials of degree 2 on as well as their equivalents on and for which these necessary conditions are also sufficient we show explicit forms of the additional first integrals which can always be chosen as a polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta

proof of the k pi 1 conjecture for affine artin groups we prove the k pi 1 conjecture for affine artin groups the complexified complement of an affine reflection arrangement is a classifying space this is a long standing problem due to arnol d pham and thom our proof is based on recent advancements in the theory of dual coxeter and artin groups as well as on several new results and constructions in particular we show that all affine noncrossing partition posets are el shellable we use these posets to construct finite classifying spaces for dual affine artin groups we introduce new cw models for the orbit configuration spaces associated with arbitrary coxeter groups we construct finite classifying spaces for the braided crystallographic groups introduced by mccammond and sulway

reduction of superintegrable systems the anisotropic harmonic oscillator we introduce a 2n parametric family of maximally superintegrable systems in n dimensions obtained as a reduction of an anisotropic harmonic oscillator in a 2n dimensional configuration space these systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta they generalize known examples of superintegrable models in the euclidean plane

non crossing partitions of type e e r abstract we investigate a new lattice of generalised non crossing partitions constructed using the geometry of the complex reflection group g e e r for the particular case e 2 resp r 2 our lattice coincides with the lattice of simple elements for the type d n resp i 2 e dual braid monoid using this lattice we construct a garside structure for the braid group b e e r as a corollary one may solve the word and conjugacy problems in this group

eigenspace arrangements of reflection groups the lattice of intersections of reflecting hyperplanes of a complex reflection group w may be considered as the poset of 1 eigenspaces of the elements of w in this paper we replace 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset after posing the main question of whether this poset is shellable we show that all its upper intervals are geometric lattices and then answer the question in the affirmative for the infinite family g m p n of complex reflection groups and the first 31 of the 34 exceptional groups by constructing cl shellings in addition we completely determine when these eigenspaces of w form a k 1 resp free arrangement for the symmetric group we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets presented as particular upper order ideals of dowling lattices study the representation afforded by the top co homology group and give a simple map to the posets of pointed d divisible partitions 1 main question and results let v denote an n dimensional c vector space a reflection in v is any nonidentity element r in gl v of finite order that fixes some hyperplane hr and a finite subgroup w of gl v is called a reflection group if it is generated by reflections example the action of the symmetric group sn on n 1 2 n gives rise to a faithful action on c via ei e i where e1 e2 en denote the standard basis vectors since the transpositions act as reflections and generate the group this representation realizes sn as a reflection group in gl c we shall refer to it as the defining representation for an element g w and root of unity let v g denote the eigenspace of g in v define e w to be the w poset partially ordered set of all such eigenspaces v g g w ordered by reverse inclusion with w action given by h v g v hgh the eigenvalue 1 gives the lattice lw of intersections of reflecting hyperplanes for w see 29 lemma 4 4 the minimal elements of e w i e the inclusion maximal eigenspaces forw are the focus of springer s theory of regular elements 35 and each has dimension equal to the number a d of degrees d1 d2 dn of w that are divisible by d the order of see proposition 3 7 below when w is crystallographic the poset e w itself appears in broue malle and michel s sylow theory 8 2010 mathematics subject classification primary 20f55 05e10 05e18 55p20 14n20

wild ramification and k pi 1 spaces we prove that every connected affine scheme of positive characteristic is a k pi 1 space for the etale topology the main ingredient is the special case of the affine space mathbf a k n over a field k this is dealt with by induction on n using a key bertini type statement regarding the wild ramification of ell adic local systems on affine spaces which might be of independent interest its proof uses in an essential way recent advances in higher ramification theory due to t saito we also give rigid analytic and mixed characteristic versions of the main result

on realizations of polynomial algebras with three generators via deformed oscillator algebras we present the most general polynomial lie algebra generated by a second order integral of motion and one of order m construct the casimir operator and show how the jacobi identity provides the existence of a realization in terms of deformed oscillator algebra we also present the classical analogue of this construction for the most general polynomial poisson algebra two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2d superintegrable systems

completely integrable relativistic hamiltonian systems and separation of variables in hermitian hyperbolic spaces the hamilton jacobi and laplace beltrami equations on the hermitian hyperbolic space hh 2 are shown to allow the separation of variables in precisely 12 classes of coordinate systems the isometry group of this two complex dimensional riemannian space su 2 1 has four mutually nonconjugate maximal abelian subgroups these subgroups are used to construct the separable coordinates explicitly all of these subgroups are two dimensional and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones the symmetry reduction of the free hh 2 hamiltonian by a maximal abelian subgroup of su 2 1 reduces this hamiltonian to one defined on an o 2 1 hyperboloid and involving a nontrivial singular potential separation of variables on hh 2 and more generally on hh n thus provides a new method of generating nontrivial completely integrable relativistic hamiltonian systems

are all classical superintegrable systems in two dimensional space linearizable several examples of classical superintegrable systems in a two dimensional space are shown to possess hidden symmetries leading to their linearization they include those determined fifty years ago in the work of fris et al phys lett 13 354 356 1965 their generalizations and the more recent tremblay turbiner winternitz system f tremblay et al j phys a math theor 42 242001 2009 we conjecture that all classical superintegrable systems in the two dimensional space have hidden symmetries that make them linearizable

the bmm symmetrising trace conjecture for the exceptional 2 reflection groups of rank 2 abstract we prove the symmetrising trace conjecture of broue malle and michel for the generic hecke algebra associated to the exceptional irreducible complex reflection group g 13 our result completes the proof of the conjecture for the exceptional 2 reflection groups of rank 2

freeness of multi reflection arrangements via primitive vector fields abstract in 2002 terao showed that every reflection multi arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations we first generalize terao s result to multi arrangements stemming from well generated unitary reflection groups where the multiplicity of a hyperplane depends on the order of its stabilizer here the exponents depend on the exponents of the dual reflection representation we then extend our results further to all imprimitive irreducible unitary reflection groups in this case the exponents turn out to depend on the exponents of a certain galois twist of the dual reflection representation that comes from a beynon lusztig type semi palindromicity of the fake degrees

non crossing partition lattices in finite real reflection groups for a finite real reflection group w with coxeter element we give a case free proof that the closed interval i forms a lattice in the partial order on w induced by reflection length key to this is the construction of an isomorphic lattice of spherical simplicial complexes we also prove that the greatest element in this latter lattice embeds in the type w simplicial generalised associahedron and we use this fact to give a new proof that the geometric realisation of this associahedron is a sphere

finite complex reflection arrangements are k pi 1 let v be a finite dimensional complex vector space and w gl v be a finite complex reflection group let v reg be the complement in v of the reflecting hyperplanes we prove that v reg is a k 1 space this was predicted by a classical conjecture originally stated by brieskorn for complexified real reflection groups the complexified real case follows from a theorem of deligne and after contributions by nakamura and orliksolomon only six exceptional cases remained open in addition to solving these six cases our approach is applicable to most previously known cases including complexified real groups for which we obtain a new proof based on new geometric objects we also address a number of questions about 1 w v the braid group of w this includes a description of periodic elements in terms of a braid analog of springer s theory of regular elements

the center of pure complex braid groups abstract broue malle and rouquier 1998 conjectured in 2 that the center of the pure braid group of an irreducible finite complex reflection group is cyclic we prove this conjecture for the remaining exceptional types using the analogous result for the full braid group due to bessis and we actually prove the stronger statement that any finite index subgroup of such braid group has cyclic center

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

symmetry algebras for superintegrable systems it is shown that the symmetry algebra of quantum maximally superintegrable systems can always be chosen to be u n n being the number of degrees of freedom

a class of garside groupoid structures on the pure braid group we construct a class of garside groupoid structures on the pure braid groups one for each function called labelling from the punctures to the integers greater than 1 the object set of the groupoid is the set of ball decompositions of the punctured disk the labels are the perimeters of the regions our construction generalises garside s original garside structure but not the one by birman ko lee as a consequence we generalise the tamari lattice ordering on the set of vertices of the associahedron

a new approach to the word and conjugacy problems in the braid groups abstract a new presentation of the n string braid group b n is studied using it a new solution to the word problem in b n is obtained which retains most of the desirable features of the garside thurston solution and at the same time makes possible certain computational improvements we also give a related solution to the conjugacy problem but the improvements in its complexity are not clear at this writing

a dual braid monoid for the free group abstract we construct a quasi garside monoid structure for the free group this monoid should be thought of as a dual braid monoid for the free group generalising the constructions by birman ko lee and by the author of new garside monoids for artin groups of spherical type

completions branched covers artin groups and singularity theory we study the curvature of metric spaces and branched covers of riemannian manifolds with applications in topology and algebraic geometry here curvature bounds are expressed in terms of the cat inequality we prove a general cat extension theorem giving sufficient conditions on and near the boundary of a locally cat metric space for the completion to be cat we use this to prove that a branched cover of a complete riemannian manifold is locally cat if and only if all tangent spaces are cat 0 and the base has sectional curvature bounded above by we also show that the branched cover is a geodesic space using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally cat and the complement of the branch locus to be contractible we conjecture that the universal branched cover of cn over the mirrors of a finite coxeter group is cat 0 conditionally on this conjecture we use our machinery to prove the arnoldpham thom conjecture on k 1 spaces for artin groups also conditionally we prove the asphericity of moduli spaces of amply lattice polarized k3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in arnold s hierar

orthogonal polynomials in two variables and second order partial differential equations abstract we study the second order partial differential equations l u au xx 22 bu xy cu yy du x eu y n u which have orthogonal polynomials in two variables as solutions by using formal functional calculus on moment functionals we first give new simpler proofs and improvements of the results by krall and sheffer and littlejohn we then give a two variable version of al salam and chihara s characterization of classical orthogonal polynomials in one variable we also study in detail the case when l belongs to the basic class that is a y c x 0 in particular we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions

the algebra of two dimensional generalized chebyshev koornwinder oscillator in the previous works of borzov and damaskinsky chebyshev koornwinder oscillator theor math phys 175 3 765 772 2013 and ladder operators for chebyshev koornwinder oscillator in proceedings of the days on diffraction 2013 the authors have defined the oscillator like system that is associated with the two variable chebyshev koornwinder polynomials we call this system the generalized chebyshev koornwinder oscillator in this paper we study the properties of infinite dimensional lie algebra that is analogous to the heisenberg algebra for the chebyshev koornwinder oscillator we construct the exact irreducible representation of this algebra in a hilbert space of functions that are defined on a region which is bounded by the steiner hypocycloid the functions are square integrable with respect to the orthogonality measure for the chebyshev koornwinder polynomials and these polynomials form an orthonormalized basis in the space the generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators h h

wilson polynomials and the generic superintegrable system on the 2 sphere for complex two dimensional riemannian spaces every classical or quantum second order superintegrable system can be obtained from a single generic 3 parameter potential on the complex 2 sphere by delicate limit operations and through stackel transforms between manifolds here we derive families of finite and infinite dimensional irreducible representations of the corresponding quadratic quantum algebra for the 2 sphere and point out their role in explaining the degeneracy of the energy eigenspaces corresponding to bound state and continuous spectra of quantum and wave equation analogs of this system the algebra is exactly the one that describes the wilson and racah polynomials in their full generality

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

comment on dynamical systems and poisson structures j math phys 50 112703 2009 we show that theorems 8 9 and 11 in the work cited in the title are incorrect in general the existence of globally well defined first integrals or flow invariant functions for dynamical systems in r n cannot be taken for granted

computation of first cohomology groups of finite covers abstract we give several applications of standard methods of group cohomology to some problems arising in model theory concerning finite covers we prove a conjecture of the author that for g finite 0 categorical structures the kernels of minimal superlinked finite covers have bounded rank we show that the cohomology groups associated to finite covers of certain structures amongst them the primitive countable totally categorical structures have to be finite from this we deduce that the finite covers of these structures are determined up to finitely many possibilities by their kernels

krammer representations for complex braid groups abstract let b be the generalized braid group associated to some finite complex reflection group w we define a representation of b of dimension the number of reflections of the corresponding reflection group which generalizes the krammer representation of the classical braid groups and is thus a good candidate in view of proving the linearity of b we decompose this representation in irreducible components and compute its zariski closure as well as its restriction to parabolic subgroups we prove that it is faithful when w is a coxeter group of type ade and odd dihedral types and conjecture its faithfulness when w has a single class of reflections if true this conjecture would imply various group theoretic properties for these groups that we prove separately to be true for the other groups

on coxeter diagrams of complex reflection groups we study complex coxeter diagrams of some unitary reflection groups using solely the combinatorics of diagrams we give a new proof of the classification of root lattices defined over e z e 2 i 3 there are only four such lattices namely the e lattices whose real forms are a2 d4 e6 and e8 next we address the issue of characterizing the diagrams for unitary reflection groups a question that was raised by broue malle and rouquier to this end we describe an algorithm which given a unitary reflection group g picks out a set of complex reflections if g is the reflection group of a root lattice defined over e the reflections selected by the algorithm form the known diagram for g for the reflection group of the complex coxeter todd lattice k e 12 we find a new diagram that extends to an affine diagram withz 7z symmetry if g is a weyl group the algorithm immediately yields a set of simple roots otherwise experimental evidences indicate that the algorithm selects a minimal generating set of reflections if g is primitive and g has a set of roots whose z span is a discrete subset of the ambient vector space

a new realisation of dynamical groups and factorisation method a new method of algebraisation of quantum mechanical eigenvalue equations is presented in this method the dynamical algebra is represented on the space of group matrix elements the ladder operators of the dynamical algebra are obtained from infeld hull miller factorisations the method is used to study the first poschl teller equation even in the non symmetric case the energy spectrum and the exact normalised solutions are obtained in agreement with the results of non algebraic methods

totally categorical structures a first order theory is totally categorical if it has exactly one model in each infinite power we prove here that every such theory admits a finite language and is finitely axiomatizable in that language modulo axioms stating that the structure is infinite this was conjectured by vaught we also show that every n0 stable n0 categorical structure is a reduct of one that has finitely many models in small uncountable powers in the case of structures of disintegrated type we nearly find an explicit structure theorem and show that the remaining obstacle resides in certain nilpotent automorphism groups

superintegrability and associated polynomial solutions euclidean space and the sphere in two dimensions in this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems we do this for the corresponding systems defined in euclidean space and on the two dimensional sphere we present all of these cases from a unified point of view in particular all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros the principal new results are the details of the polynomial bases for each of the nonsubgroup bases not just the subgroup cartesian and polar coordinate cases and the details of the structure of the quadratic algebras we also study the polynomial eigenfunctions in elliptic coordinates of the n dimensional isotropic quantum oscillator

bounds in the theory of finite covers we give an upper bound for the number of conjugacy classes of closed subgroups of the full wreath product fwrwsym which project onto sym here is infinite w is the set of n tuples of distinct elements from for some finite n f is a finite nilpotent group and the topology on the wreath product is that of pointwise convergence in its imprimitive permutation action the result addresses a problem which arises in a natural model theoretic context about classifying certain types of finite covers

an infinite family of solvable and integrable quantum systems on a plane an infinite family of exactly solvable and integrable potentials on a plane is introduced it is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family the underlying algebraic structure of the new potentials is revealed as well as its hidden algebra we conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases a quasi exactly solvable and integrable generalization of the family is found

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

monoids in the fundamental groups of the complement of logarithmic free divisors in c3 abstract we study monoids generated by certain zariski van kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in c 3 listed by sekiguchi these monoids admit positive homogeneous presentations theorem 1 five of them are artin monoids and eight of them are free abelian monoids the remaining four monoids are not gausian and hence are neither garside nor artin theorem 2 however we introduce the concept of fundamental elements for positive homogeneously presented monoids and we show that all 17 monoids possess fundamental elements theorem 3 as an application of the study of monoids we solve some decision problems for the fundamental groups except in three cases

two dimensional krall sheffer polynomials and integrable systems two dimensional krall sheffer polynomials are analogues of the classical orthogonal polynomial they are eigenfunctions of second order linear partial differential operators and moreover satisfy orthogonality conditions we show that all krall sheffer polynomials are connected with two dimensional superintegrable systems on spaces with constant curvature

gaussian groups and garside groups two generalisations of artin groups it is known that a number of algebraic properties of the braid groups extend to arbitrary finite coxeter type artin groups here we show how to extend the results to more general groups that we call garside groups define a gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths and where left and right lowest common multiples exist a garside monoid is a gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition a gaussian group is the group of fractions of a gaussian monoid and a garside group is the group of fractions of a garside monoid braid groups and more generally finite coxeter type artin groups are garside groups we determine algorithmic criteria in terms of presentations for recognizing gaussian and garside monoids and groups and exhibit infinite families of such groups we describe simple algorithms that solve the word problem in a gaussian group show that these algorithms have a quadratic complexity if the group is a garside group and prove that garside groups have quadratic isoperimetric inequalities we construct normal forms for gaussian groups and prove that in the case of a garside group the language of normal forms is regular symmetric and geodesic has the 5 fellow traveller property and has the uniqueness property this shows in particular that garside groups are geodesically fully biautomatic finally we consider an automorphism of a finite coxeter type artin group derived from an automorphism of its defining coxeter graph and prove that the subgroup of elements fixed by this automorphism is also a finite coxeter type artin group that can be explicitly determined 1991 mathematics subject classification primary 20f05 20f36 secondary 20b40 20m05

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

superintegrability on the two dimensional hyperboloid ii this work is devoted to the investigation of the quantum mechanical systems on the two dimensional hyperboloid which admits separation of variables in at least two coordinate systems here we consider two potentials introduced in a paper of c p boyer e g kalnins and p winternitz j math phys 24 2022 1983 which have not yet been studied we give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion

on non conjugate coxeter elements in well generated reflection groups given an irreducible well generated complex reflection group w with coxeter number h we call a coxeter element any regular element in the sense of springer of order h in w this is a slight extension of the most common notion of coxeter element we show that the class of these coxeter elements forms a single orbit in w under the action of reflection automorphisms for coxeter and shephard groups this implies that an element c is a coxeter element if and only if there exists a simple system s of reflections such that c is the product of the generators in s we moreover deduce multiple further implications of this property in particular we obtain that all noncrossing partition lattices of w associated to different coxeter elements are isomorphic we also prove that there is a simply transitive action of the galois group of the field of definition of w on the set of conjugacy classes of coxeter elements finally we extend several of these properties to springer s regular elements of arbitrary order

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

parabolic deligne lusztig varieties abstract motivated by the broue conjecture on blocks with abelian defect groups for finite reductive groups we study parabolic deligne lusztig varieties and construct on those which occur in the broue conjecture an action of a braid monoid whose action on their l adic cohomology will conjecturally factor through a cyclotomic hecke algebra in order to construct this action we need to enlarge the set of varieties we consider to varieties attached to a ribbon category this category has a garside family which plays an important role in our constructions so we devote the first part of our paper to the necessary background on categories with garside families

braid groups of imprimitive complex reflection groups abstract we obtain new presentations for the imprimitive complex reflection groups of type d e e r and their braid groups b d e e r for d r 2 diagrams for these presentations are proposed the presentations have much in common with coxeter presentations of real reflection groups they are positive and homogeneous and give rise to quasi garside structures diagram automorphisms correspond to group automorphisms the new presentation shows how the braid group b d e e r is a semidirect product of the braid group of affine type a r 1 and an infinite cyclic group elements of b d e e r are visualised as geometric braids on r 1 strings whose first string is pure and whose winding number is a multiple of e we classify periodic elements and show that the roots are unique up to conjugacy and that the braid group b d e e r is strongly translation discrete

families of classical subgroup separable superintegrable systems we describe a method for determining a complete set of integrals for a classical hamiltonian that separates in orthogonal subgroup coordinates as examples we use it to determine complete sets of integrals polynomial in the momenta for some families of generalized oscillator and kepler coulomb systems hence demonstrating their superintegrability the latter generalizes recent results of verrier and evans and rodriguez tempesta and winternitz another example is given of a superintegrable system on a non conformally flat space

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators