a generalization of coxeter groups root systems and matsumoto s theorem the root systems appearing in the theory of lie superalgebras and nichols algebras admit a large symmetry extending properly the one coming from the weyl group based on this observation we set up a general framework in which the symmetry object is a groupoid we prove that in our context the groupoid is generated by simple reflections and coxeter relations in a broad sense this answers a question of serganova our weak version of the exchange condition allows us to prove matsumoto s theorem therefore the word problem is solved for the groupoid

maps between non commutative spaces let j be a graded ideal in a not necessarily commutative graded k algebra a a 0 a 1 in which dim k a i for all i we show that the map a a j induces a closed immersion i proj nc a j proj nc a between the non commutative projective spaces with homogeneous coordinate rings a and a j we also examine two other kinds of maps between non commutative spaces first a homomorphism a b between not necessarily commutative n graded rings induces an affine map proj nc b u proj nc a from a non empty open subspace u c proj nc b second if a is a right noetherian connected graded algebra not necessarily generated in degree one and a n is a veronese subalgebra of a there is a map proj nc a proj nc a n we identify open subspaces on which this map is an isomorphism applying these general results when a is a quotient of a weighted polynomial ring produces a non commutative resolution of a closed subscheme of a weighted projective space

semiprime graded algebras of dimension two semiprime noetherian connected graded k algebras r of quadratic growth are described in terms of geometric data a typical example of such a ring is obtained as follows let y be a projective variety of dimension at most one over the base field k and let e be an oy order in a finite dimensional semisimple algebra a over k k y then for any automorphism of a that restricts to an automorphism of y and any ample invertible e bimodule b van den bergh constructs a noetherian twisted homogeneous coordinate ring b h0 y b b n 1 we show that r is noetherian if and only if some veronese ring r m of r has the form k i where i is a left ideal of such a ring b and where i b at each point p y at which has finite order this allows one to give detailed information about the structure of r and its modules

the band structure of the general periodic schrodinger operator with point interactions the spectrum of schrodinger operators h with periodic point potentials in dimensions d 2 3 is studied in the general case of n points in the wigner seitz cell it is proven that h has a band structure with at most a finite number of gaps bethe sommerfeld conjecture it is also proven that in the case of a generic local point perturbation no singular continuous components are present in the non local case a fractal component like the cantor set is exhibited this component can either consist of a singular continuous or a dense point spectrum

spectral theory of semibounded sturm liouville operators with local interactions on a discrete set we study the hamiltonians hx q with type point interactions at the centers xk on the positive half line in terms of energy forms we establish analogs of some classical results on operators hq d2 dx2 q with locally integrable potentials q lloc1 r in particular we prove that the hamiltonian hx q is self adjoint if it is lower semibounded this result completes the previous results of brasche perturbation of schrodinger hamiltonians by measures selfadjointness and semiboundedness j math phys 26 621 1985 on lower semiboundedness also we prove the analog of molchanov s discreteness criteria birman s result on stability of a continuous spectrum and investigate discreteness of a negative spectrum in the recent paper kostenko a and malamud m 1 d schrodinger operators with local point interactions on a discrete set j differ equations 249 253 2010 it was shown that the spectral properties of hx hx 0 correlate with the corresponding spectral properties of a certain clas

scattering matrices and weyl functions for a scattering system a a 0 consisting of self adjoint extensions a and a 0 of a symmetric operator a with finite deficiency indices the scattering matrix s and a spectral shift function are calculated in terms of the weyl function associated with a boundary triplet for a and a simple proof of the krein birman formula is given the results are applied to singular sturm liouville operators with scalar and matrix potentials to dirac operators and to schrodinger operators with point interactions

rank 4 finite dimensional nichols algebras of diagonal type in positive characteristic abstract nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed hopf algebras by lifting method of andruskiewitsch and schneider arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property in the present paper all rank 4 nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified our proof uses the classification of the finite arithmetic root systems of rank 4

spectral theory of schrodinger operators with infinitely many point interactions and radial positive definite functions abstract a number of results on radial positive definite functions on r n related to schoenberg s integral representation theorem are obtained they are applied to the study of spectral properties of self adjoint realizations of two and three dimensional schrodinger operators with countably many point interactions in particular we find conditions on the configuration of point interactions such that any self adjoint realization has purely absolutely continuous non negative spectrum we also apply some results on schrodinger operators to obtain new results on completely monotone functions

constructing clifford quantum p3s with finitely many points abstract we present an algebro geometric technique for constructing regular clifford algebras a of global dimension four with associated point scheme consisting of a prespecified finite number of points in particular if a has more than one point in its point scheme then the number of points in the point scheme can be obtained from the number of intersection points of two planar cubic divisors these cubic divisors correspond to regular clifford subalgebras of a of global dimension three if a has exactly a finite number n of distinct points in its point scheme then n 1 3 4 5 13 14 16 18 20 and all these possibilities occur we also prove that if a regular clifford algebra r of global dimension d 2 has exactly a finite number n of distinct isomorphism classes of point modules then n is odd if and only if r is an ore extension of a regular clifford subalgebra of r of global dimension d 1

on symmetries in the theory of finite rank singular perturbations for a nonnegative self adjoint operator a0 acting on a hilbert space h singular perturbations of the form a0 v v 1nbij j i are studied under some additional requirements of symmetry imposed on the initial operator a0 and the singular elements j a concept of symmetry is defined by means of a one parameter family of unitary operators u that is motivated by results due to r s phillips the abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials v and the corresponding self adjoint realizations of a0 v the results are applied for the investigation of singular perturbations of the schrodinger operator in l2 r3 and for the study of a fractional p adic schrodinger type operator with point interactions

a krein like formula for singular perturbations of self adjoint operators and applications abstract given a self adjoint operator a d a h h and a continuous linear operator d a x with range h 0 x a banach space we explicitly construct a family a of self adjoint operators such that any a coincides with the original a on the kernel of such a family is obtained by giving a krein like formula where the role of the deficiency spaces is played by the dual pair x x the parameter belongs to the space of symmetric operators from x to x when x c one recovers the h 2 construction of kiselev and simon and so to some extent our results can be regarded as an extension of it to the infinite rank case considering the situation in which h l 2 r n and is the trace restriction operator along some null subset we give various applications to singular perturbations of non necessarily elliptic pseudo differential operators thus unifying and extending previously known results

on the spectral theory of gesztesy seba realizations of 1 d dirac operators with point interactions on a discrete set abstract we investigate spectral properties of gesztesy seba realizations d x and d x of the 1 d dirac differential expression d with point interactions on a discrete set x x n n 1 r here n n 1 and n n 1 r the gesztesy seba realizations d x and d x are the relativistic counterparts of the corresponding schrodinger operators h x and h x with and interactions respectively we define the minimal operator d x as the direct sum of the minimal dirac operators on the intervals x n 1 x n then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator d x in the case d x inf x i x j i j 0 it turns out that the boundary operators b x and b x parameterizing the realizations d x and d x are jacobi matrices these matrices substantially differ from the ones appearing in spectral theory of schrodinger operators with point interactions we show that certain spectral properties of the operators d x and d x correlate with the corresponding spectral properties of the jacobi matrices b x and b x respectively using this connection we investigate spectral properties self adjointness discreteness absolutely continuous and singular spectra of gesztesy seba realizations moreover we investigate the non relativistic limit as the velocity of light c most of our results are new even in the case d x 0

some finite quantum p3s that are infinite modules over their centers abstract a result of m artin j tate and m van den bergh asserts that a regular algebra of global dimension three is a finite module over its center if and only if the automorphism encoded in the point scheme has finite order we prove that the analogous result for a regular algebra of global dimension four is false by presenting families of quadratic noetherian regular algebras a of global dimension four such that i a is an infinite module over its center ii a has a finite point scheme which is the graph of an automorphism of finite order and iii a has a one parameter family of line modules such algebras are candidates for generic regular algebras of global dimension four the methods used to construct the algebras provide new techniques for creating other potential candidates

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

spectral theory for perturbed krein laplacians in nonsmooth domains abstract we study spectral properties for h k the krein von neumann extension of the perturbed laplacian v defined on c 0 where v is measurable bounded and nonnegative in a bounded open set r n belonging to a class of nonsmooth domains which contains all convex domains along with all domains of class c 1 r r 1 2 in particular in the aforementioned context we establish the weyl asymptotic formula j n k j 2 n v n n 2 o n 1 2 2 as where v n n 2 n 2 1 denotes the volume of the unit ball in r n and k j j n are the non zero eigenvalues of h k listed in increasing order according to their multiplicities we prove this formula by showing that the perturbed krein laplacian i e the krein von neumann extension of v defined on c 0 is spectrally equivalent to the buckling of a clamped plate problem and using an abstract result of kozlov from the mid 1980s our work builds on that of grubb in the early 1980s who has considered similar issues for elliptic operators in smooth domains and shows that the question posed by alonso and simon in 1980 pertaining to the validity of the above weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting we also study certain exterior type domains r n k n 3 with k r n compact and vanishing bessel capacity b 2 2 k 0 to prove equality of friedrichs and krein laplacians in l 2 d n x that is c 0 has a unique nonnegative self adjoint extension in l 2 d n x

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

krein like extensions and the lower boundedness problem for elliptic operators abstract for selfadjoint extensions a of a symmetric densely defined positive operator a min the lower boundedness problem is the question of whether a is lower bounded if and only if an associated operator t in abstract boundary spaces is lower bounded it holds when the friedrichs extension a has compact inverse grubb 1974 also gorbachuk and mikhailets 1976 this applies to elliptic operators a on bounded domains for exterior domains a 1 is not compact and whereas the lower bounds satisfy m t m a the implication of lower boundedness from t to a has only been known when m t m a we now show it for general t the operator a a corresponding to t a i generalizing the krein von neumann extension a 0 appears here its possible lower boundedness for all real a is decisive we study this krein like extension showing for bounded domains that the discrete eigenvalues satisfy n t a a c a t n 2 m o t n 1 e 2 m for t

homological properties of sklyanin algebras to a pair consisting of an elliptic curve and a point on it odeskii and feigin associate certain quadratic algebras sklyanin algebras having the hilbert series of a polynomial algebra in this paper we show that sklyanin algebras have good homological properties and we obtain some information about their so called linear modules we also show how the construction by odeskii and feigin may be generalized so as to yield other sklyanin type algebras

the weyl groupoid of a nichols algebra of diagonal type the theory of nichols algebras of diagonal type is known to be closely related to that of semi simple lie algebras in this paper the connection between both theories is made closer for any nichols algebra of diagonal type invertible transformations are introduced which remind one of the action of the weyl group on the root system associated to a semi simple lie algebra they give rise to the definition of a groupoid as an application an alternative proof of classification results of rosso andruskiewitsch and schneider is obtained without using any technical assumptions on the braiding

irreducible representations of the 4 dimensional sklyanin algebra at points of infinite order abstract in 1982 sklyanin funct anal appl 16 1982 27 34 defined a family of graded algebras a e depending on an elliptic curve e and a point e which is not 4 torsion basic properties of these algebras were established in smith and stafford compositio math 83 1992 259 289 and a study of their representation theory was begun in levasseur and smith bull soc math france121 1993 35 90 the present paper classifies the finite dimensional simple a modules when is a point of infinite order sklyanin funct anal appl 17 1983 273 284 defines for each k n a representation of a in a certain k dimensional subspace of theta functions of order 2 k 1 we prove that these are irreducible representations and that any other simple module is obtained by twisting one of these by an automorphism of a the automorphism group of a is explicitly computed the method of proof relies on results in levasseur and smith in particular it is proved that every finite dimensional simple module is a quotient of a line module an important part of the analysis is a determination of the 1 critical a modules and the fact that such a module is equivalent to a quotient of a line module by a shifted line module

point interactions in two dimensions basic properties approximations and applications to solid state physics on etudie rigoureusement l operateur de schrodinger formellement donne par h y dans l 2 r 2 et on montre comment l approcher en termes d hamiltoniens a courte portee transformes d echelles on considere des questions semblables pour l operateur h j 1 n j y j n n on etudie les proprietes spectrales du modele a un electron d un cristal a 2 dimensions avec des interactions ponctuelles

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

1 d schrodinger operators with local point interactions on a discrete set abstract spectral properties of 1 d schrodinger operators h x d 2 d x 2 x n x n x x n with local point interactions on a discrete set x x n n 1 are well studied when d inf n k n x n x k 0 our paper is devoted to the case d 0 we consider h x in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding weyl functions we show that the spectral properties of h x like self adjointness discreteness and lower semiboundedness correlate with the corresponding spectral properties of certain classes of jacobi matrices based on this connection we obtain necessary and sufficient conditions for the operators h x to be self adjoint lower semibounded and discrete in the case d 0 the operators with type interactions are investigated too the obtained results demonstrate that in the case d 0 as distinguished from the case d 0 the spectral properties of the operators with and type interactions are substantially different

the r matrix of quantum doubles of nichols algebras of diagonal type let h be the quantum double of a nichols algebra of diagonal type we compute the r matrix of 3 tuples of modules for general finite dimensional highest weight modules over h we also calculate a multiplicative formula for the universal r matrix when h is finite dimensional we show the unicity of a pbw basis or a lusztig type poincare birkhoff witt basis with a given convex order

on the unitary equivalence of absolutely continuous parts of self adjoint extensions the classical weyl von neumann theorem states that for any self adjoint operator a0 in a separable hilbert space h there exists a non unique hilbert schmidt operator c c such that the perturbed operator a0 c has purely point spectrum we are interesting whether this result remains valid for non additive perturbations by considering the set exta of self adjoint extensions of a given densely defined symmetric operator a in h and some fixed a0 a0 exta we show that the ac parts a ac and a0ac of a a exta and a0 are unitarily equivalent provided that the resolvent difference ka a i 1 a0 i 1 is compact and the weyl function m of the pair a a0 admits weak boundary limits m t w limy 0m t iy for a e t r this result generalizes the classical kato rosenblum theorem moreover it demonstrates that for such pairs a a0 the weyl von neumann theorem is in general not true in the class exta

sturm liouville boundary value problems with operator potentials and unitary equivalence consider the minimal sturm liouville operator a amin generated by the differential expression a d2dt2 t in the hilbert space l2 r h where t t 0 in h we investigate the absolutely continuous parts of different self adjoint realizations of a in particular we show that dirichlet and neumann realizations ad and an are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the krein realization moreover if inf ess t inf t then the part a acea ad of any self adjoint realization a of a is unitarily equivalent to ad in addition we prove that the absolutely continuous part a ac of any realization a is unitarily equivalent to ad provided that the resolvent difference a i 1 ad i 1 is compact the abstract results are applied to elliptic differential expressions in the half space