quasi exact solvability and entropies of the one dimensional regularised calogero model the calogero model can be regularised through the introduction of a cutoff parameter which removes the divergence in the interaction term in this work we show that the one dimensional two particle regularised calogero model is quasi exactly solvable and that for certain values of the hamiltonian parameters the eigenfunctions can be written in terms of heun s confluent polynomials these eigenfunctions are such that the reduced density matrix of the two particle density operator can be obtained exactly as well as its entanglement spectrum we found that the number of non zero eigenvalues of the reduced density matrix is finite in these cases the limits for the cutoff distance going to zero calogero and infinity are analysed and all the previously obtained results for the calogero model are reproduced once the exact eigenfunctions are obtained the exact von neumann and renyi entanglement entropies are studied to characterise the physical traits of the model the quasi exactly solvable character of the model is assessed studying the numerically calculated renyi entropy and entanglement spectrum for the whole parameter space

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

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

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

novel relations and new properties of confluent heun s functions and their derivatives of arbitrary order the present paper reveals important properties of the confluent heun s functions we derive a set of novel relations for confluent heun s functions and their derivatives of arbitrary order specific new subclasses of confluent heun s functions are introduced and studied a new alternative derivation of confluent heun s polynomials is presented

new integrable generalizations of calogero moser quantum problem a one parameter deformation of calogero moser quantum problem is introduced it is shown that corresponding schrodinger operator is integrable for any value of the parameter and algebraically integrable in case of integer value

riemann roch theory for weighted graphs and tropical curves abstract we define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices we prove that the riemann roch theorem holds in both cases we extend baker s specialization lemma to weighted graphs

sklyanin algebras and hilbert schemes of points we construct projective moduli spaces for torsion free sheaves on noncommutative projective planes these moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual commutative projective plane p2 the generic noncommutative plane corresponds to the sklyanin algebra s skl e constructed from an automorphism of infinite order on an elliptic curve e p2 in this case the fine moduli space of line bundles over s with first chern class zero and euler characteristic 1 n provides a symplectic variety that is a deformation of the hilbert scheme of n points on p2 e

classification of arithmetic root systems arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property they can also be considered as generalizations of ordinary root systems with rich structure and many new examples on the other hand nichols algebras are fundamental objects in the construction of quantized enveloping algebras in the noncommutative differential geometry of quantum groups and in the classification of pointed hopf algebras by the lifting method of andruskiewitsch and schneider in the present paper arithmetic root systems are classified in full generality as a byproduct many new finite dimensional pointed hopf algebras are obtained

a freeness theorem for nichols algebras abstract we prove a freeness result for nichols algebras over certain nichols subalgebras this result allows us in particular to classify pointed hopf algebras of prime and prime squared index

application of the heun functions to rotor vibratory gyroscope their numerical computation and accuracy estimation abstract for an asymmetric rotor vibratory gyroscope that is oscillating in an elastic suspension means the equation of motion is derived from the eurler lagrange equation and the exact solution is obtained as heun functions hfs a fast and effective method for calculating heun functions by direct calculation of solutions of the heun differential equation hde using standard numerical integration methods is developed three methods of accuracy check are employed in this case the accuracy of the numerical solutions deteriorated in the vicinity of the singularity to overcome this difficulty an optimised method for calculating the hfs is developed which give a uniform accuracy of the calculated values on the interval the optimised method for calculating the numerical hfs by means of the solution of the governing initial value problem gave acceptable accuracy in modelling the behaviour of an asymmetric rotor gyroscope

complex calogero model with real energies recently the symmetry of many single particle non hermitian hamiltonians has been conjectured to be sufficient to keep their spectrum real we demonstrate this and how the similar concept of a weakened hermiticity can be extended to some exactly solvable two and three particle models

double extension regular algebras of type 14641 abstract we construct several families of artin schelter regular algebras of global dimension four using double ore extension and then prove that all these algebras are strongly noetherian auslander regular koszul and cohen macaulay domains many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an ore extension of an artin schelter regular algebra of global dimension three

exact finite reduced density matrix and von neumann entropy for the calogero model the information content of continuous quantum variables systems is usually studied using a number of well known approximation methods the approximations are made to obtain the spectrum eigenfunctions or the reduced density matrices that are essential to calculate the entropy like quantities that quantify the information even in the sparse cases where the spectrum and eigenfunctions are exactly known the entanglement spectrum the spectrum of the reduced density matrices that characterize the problem must be obtained in an approximate fashion in this work we obtain analytically a finite representation of the reduced density matrices of the fundamental state of the n particle calogero model for a discrete set of values of the interaction parameter as a consequence the exact entanglement spectrum and von neumann entropy is worked out

finite quantum groups and cartan matrices abstract we consider a class of hopf algebras having as an invariant a generalized cartan matrix for a hopf algebra of this kind we prove that it is finite dimensional if and only if its generalized cartan matrix is actually a finite cartan matrix under some mild hypothesis these results allow us to classify all the finite dimensional coradically graded pointed hopf algebras whose coradical has odd prime dimension p we also characterize coradically graded pointed hopf algebras of order p 4

on the simple representations of generalized quantum groups and quantum doubles abstract a highest weight theory is developed for a general class of algebras which includes generalizations of the quantum groups u q g and their finite dimensional versions g a semisimple lie algebra a basic finiteness result on irreducible representations is proved

on nichols algebras over basic hopf algebras this is a contribution to the classification of finite dimensional hopf algebras over an algebraically closed field mathbb k of characteristic 0 concretely we show that a finite dimensional hopf algebra whose hopf coradical is basic is a lifting of a nichols algebra of a semisimple yetter drinfeld module and we explain how to classify nichols algebras of this kind we provide along the way new examples of nichols algebras and hopf algebras with finite gelfand kirillov dimension

a class of as regular algebras of dimension five abstract we classify 5 dimensional artin schelter regular algebras generated by two generators of degree 1 with three generating relations of degree 4 under a generic condition all the algebras obtained are proved to be strongly noetherian auslander regular and cohen macaulay with respect to the gelfand kirillov dimension