convection diffusion lattice boltzmann scheme for irregular lattices in this paper a lattice boltzmann lb scheme for convection diffusion on irregular lattices is presented which is free of any interpolation or coarse graining step the scheme is derived using the axioma that the velocity moments of the equilibrium distribution equal those of the maxwell boltzmann distribution the axioma holds for both bravais and irregular lattices implying a single framework for lb schemes for all lattice types by solving benchmark problems we have shown that the scheme is indeed consistent with convection diffusion furthermore we have compared the performance of the lb schemes with that of finite difference and finite element schemes the comparison shows that the lb scheme has a similar performance as the one step second order lax wendroff scheme it has little numerical diffusion but has a slight dispersion error by changing the relaxation parameter the dispersion error can be balanced by a small increase of the numerical diffusion

flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

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

sur la convexite holomorphe des revetements lineaires reductifs d une variete projective algebrique complexe let x be a smooth connected compact projective variety over we study the shafarevich conjecture concerning holomorphic convexity along the lines of kollar s approach when the fundamental group of x admits large finite dimensionnal representations we prove that given n the topological covering space widetilde x n of a projective algebraic compact complex manifold x corresponding to the intersection of the kernels of all linear reductive representations of 1 x to gln is holomorphically convex in the surface case this is a corollary of a theorem due to katzarkov and ramachandran

algebraic varieties with quasi projective universal cover we prove that the universal cover of a normal projective variety x is quasi projective if and only if a finite etale cover of x is a fiber bundle over an abelian variety with simply connected fiber

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

liftings of nichols algebras of diagonal type iii cartan type g2 abstract we complete the classification of hopf algebras whose infinitesimal braiding is a principal yetter drinfeld realization of a braided vector space of cartan type g 2 over a cosemisimple hopf algebra we develop a general formula for a class of liftings in which the quantum serre relations hold we give a detailed explanation of the procedure for finding the relations based on the recent work of andruskiewitsch angiono and rossi bertone

grothendieck rings of basic classical lie superalgebras the grothendieck rings of nite dimensional representations of the basic classical lie superalgebras are explicitly described in terms of the corresponding generalized root systems we show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called super weyl groupoids

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

rigidity of graded regular algebras we prove a graded version of alev polo s rigidity theorem the homogenization of the universal enveloping algebra of a semisimple lie algebra and the rees ring of the weyl algebras a n k cannot be isomorphic to their fixed subring under any finite group action we also show the same result for other classes of graded regular algebras including the sklyanin algebras

logarithmic widehat s ell 2 cft models from nichols algebras i we construct chiral algebras that centralize rank 2 nichols algebras with at least one fermionic generator this gives logarithmic w algebra extensions of a fractional level algebra we discuss crucial aspects of the emerging general relation between nichols algebras and logarithmic conformal field theory cft models i the extra input beyond the nichols algebra proper needed to uniquely specify a conformal model ii a relation between the cft counterparts of nichols algebras connected by weyl groupoid maps and iii the common double bosonization u x of such nichols algebras for an extended chiral algebra candidates for its simple modules that are counterparts of the u x simple modules are proposed as a first step toward a functorial relation between u x and w algebra representation categories

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

linear shafarevich conjecture in this paper we settle armatively shafarevich s uniformization conjecture for varieties with linear fundamental groups we prove the strongest to date uniformization result the universal covering space of a complex projective manifold with a linear fundamental group is holomorphically convex the proof is based on both known and newly developed techniques in non abelian hodge theory

lattice boltzmann method for the convection diffusion equation in curvilinear coordinate systems a lattice boltzmann method for simulating convection and diffusion using a curvilinear grid system is presented the proposed method does not require an interpolation or coarse graining procedure and thus maintains the algorithmic simplicity of the original lattice boltzmann scheme the lattice boltzmann scheme is based on uniformly distributed lattice points in a transformed coordinate system and the apparent anisotropy of diffusion that arises due to the coordinate transformation is properly handled using the multiple relaxation time collision operator an asymptotic analysis of the lattice boltzmann equation shows that the proposed method appropriately reproduces the transformed convection diffusion equation several specific problems are numerically analyzed in order to validate the proposed method including an axially symmetric two dimensional problem in which the diffusion flux at an oblate hemispheroid is simulated using a body fitted orthogonal curvilinear grid system

class of cellular automata for reaction diffusion systems finite difference methods for solving the pde s finite difference methods then proceed to solve the resulting coupled system of n s ordinary differential equations n points in space s equations in the pde system by any of a number of numerical methods operating on floating point numbers the use of floating point numbers on computers implies a discretization of the continuous variables the errors introduced by this discretization and the ensuing roundoff errors are often not considered explicitly but assumed to be small because the precision is rather high 8 decimal digits for usual floating point numbers in contrast in the ca approach all variables are explicitly discretized into relatively small integers this discretization allows the use of lookup tables to replace the evaluation of the nonlinear rate functions it is this table lookup combined with the fact that all calculations are performed using integers instead of floating point variables that accounts for an improvement in speed of orders of magnitude on a conventional multi purpose computer the undesirable effects of discretization are overcome by using probabilistic rules for the updating of the ca the state of the ca is given by a regular array of concentration vectors y residing on a d dimensional lattice each y r is a s vector of integers s is the number of reactive species for reasons of efficiency and to fulfill the finiteness condition of the definition of cellular automata each component yi r can only take integer values between 0 and bi where the bi s can be different for each species i the position index r is a d dimensional vector in the ca lattice for cubic lattices r is a d vector of integers the central operation of the automaton consists of calculating the sum

galerkin model for turing patterns on a sphere we carry out a lorenz like truncation of reaction diffusion systems to investigate pattern formation on a spherical surface stability analysis suggests spots to be favored more than stripes in general numerical calculations on the truncated model are in agreement with the theoretical predictions

nichols algebras over groups with finite root system of rank two iii abstract we study nichols algebras of direct sums of two absolutely simple yetter drinfeld modules over non abelian groups and arbitrary fields we construct families of new examples where the nichols algebra is finite dimensional and not of diagonal type

embedding a quantum nonsingular quadric in a quantum p3 a definition of regularity has been given for non commutative graded algebras and results of artin schelter tate and van den bergh classify the regular algebras of global dimension three that are generated by degree one elements our purpose is to classify a certain class of quadratic regular algebras of global dimension four letsbe a twisted homogeneous coordinate ring of a nonsingular quadricq p3 our interest is in algebrasrsuch that there is an embedding proj s proj r in this paper we classify all the quadratic regular algebrasrof global dimension four which have the same hilbert series as that of the polynomial ring on four variables and which map ontosvia a graded degree zero homomorphism our approach makes use of the point modules ofrand their associated geometric data we classify the algebrasraccording to their point scheme pand corresponding automorphism aut p those algebrasrwhich are determined by p belong to at most a five parameter family but those which are not determined by p belong to at most a four parameter family in the first case pis either p3or consists ofqtogether with a linel while in the second casep q it is also proved that under certain sufficient conditions the zero locus of the defining relations of a quadratic regular algebra of global dimension four is the graph of an automorphism

multi brace cotensor hopf algebras and quantum groups we give a construction of the whole quantum group associated to a symmetrizable kac moody lie algebra as a quantum quasi symmetric algebra a particular case of multi brace cotensor hopf algebras in the spirit of the quantum shuffle construction of the plus part all highest weight irreducible representations are constructed using this machinery it also provides a systematic way to construct simple modules over the quantum double of a quantum group

primitive cohomology of hopf algebras abstract primitive cohomology of a hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra among many of its applications two classifications are presented firstly we classify all non locally pi pointed hopf algebra domains of gelfand kirillov dimension two and secondly we classify all pointed hopf algebras of rank one the first classification extends some results of brown goodearl and others in an ongoing project to understand all hopf algebras of low gelfand kirillov dimension the second generalizes results of krop radford and wang you chen which classified hopf algebras of rank one under extra hypothesis properties and algebraic structures of the primitive cohomology are discussed

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

rings with auslander dualizing complexes abstract a ring with an auslander dualizing complex is a generalization of an auslander gorenstein ring we show that many results which hold for auslander gorenstein rings also hold in the more general setting on the other hand we give criteria for existence of auslander dualizing complexes which show these occur quite frequently the most powerful tool we use is the local duality theorem for connected graded algebras over a field filtrations allow the transfer of results to nongraded algebras we also prove some results of a categorical nature most notably the functoriality of rigid dualizing complexes

on defining ideals and differential algebras of nichols algebras abstract this paper is devoted to understanding the defining ideal of a nichols algebra from the decomposition of specific elements in the group algebra of braid groups a family of primitive elements is found and algorithms are proposed to prove the main result the differential algebra of a nichols algebra is constructed moreover another point of view on serre relations is provided

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

quantum deformations of projective three space abstract we describe the possible noncommutative deformations of complex projective three space by exhibiting the calabi yau algebras that serve as their homogeneous coordinate rings we prove that the space parametrizing such deformations has exactly six irreducible components and we give explicit presentations for the generic members of each family in terms of generators and relations the proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic poisson structures in four dimensions which we extract from cerveau and lins neto s classification of degree two foliations on projective space corresponding to the exceptional component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical schwarzenberger bundles

symmetric differentials and the fundamental group helene esnault asked whether a smooth complex projective variety x with infinite fundamental group must have a nonzero symmetric differential meaning that h x s x 6 0 for some i 0 this was prompted by the second author s work 23 in fact severi had wondered in 1949 about possible relations between symmetric differentials and the fundamental group 34 p 36 we know from hodge theory that the cotangent bundle x has a nonzero section if and only if the abelianization of 1x is infinite the geometric meaning of other symmetric differentials is more mysterious and it is intriguing that they may have such a direct relation to the fundamental group in this paper we prove the following result on esnault s question in the slightly broader setting of compact kahler manifolds

dualizing complexes over noncommutative graded algebras the motivation for the work presented in this paper was two questions posed by m artin let a be a d dimensional regular graded algebra over the field k with augmentation ideal m see def 4 13 denote by hi a resp hi a the dth left resp right local cohomology of a at the ideal m i e hi a lim extd a m a and similarly for hi a as left modules and as right modules hi a and h a are both isomorphic to a hom a k up to a twist in degrees all operations are taking place in the graded category

a classification result on prime hopf algebras of gk dimension one abstract in this paper we classify all prime hopf algebras h of gk dimension one satisfying the following two conditions 1 h has a 1 dimensional representation of order pi deg h and 2 the invariant components of h with respect to this 1 dimensional representation are domains see section 2 for related definitions as consequences 1 a number of new hopf algebras of gk dimension one are found and some of them are not pointed 2 we give a partial answer to a question posed in 12 and 3 the question 1 1 is solved

quadratic algebras associated with the union of a quadric and a line in p3 abstract we define a family of graded quadratic algebras a on 4 generators depending on a fixed nonsingular quadric q in p 3 a fixed line l in p 3 and an automorphism aut q l this family contains o q m2 c the coordinate ring of quantum 2 2 matrices many of the algebraic properties of a are shown to be determined by the geometric properties of q l for instance when a o q m2 c then the quantum determinant is the unique up to a scalar multiple homogeneous element of degree 2 in o q m2 c that vanishes on the graph in p 3 p 3 of q but not on the graph of l following results of m artin j tate and m van den bergh the grothendieck festschrift birkhauser basel 1990 and invent math 106 1991 335 388 we study point and line modules over the algebras a and find that their algebraic properties are consequences of the geometric data in particular the point modules are in one to one correspondence with the points of q or l and the line modules are in bijection with the lines in p 3 that either lie on q or meet l in the case of o q m2 c when q is not a root of unity the quantum determinant annihilates all the line modules m l corresponding to lines l q the determinant generates the whole annihilator for such l q if and only if l and l

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

representations of u qsl 2 1 at even roots of unity we construct all projective modules of the restricted quantum group u qsl 2 1 at an even 2p th root of unity this 64p4 dimensional hopf algebra is a common double bosonization of two rank 2 nichols algebras x1d505 x with fermionic generator s we show that the category of u qsl 2 1 modules is equivalent to the category of yetter drinfeld x1d505 x modules in c hhy kern 1ptd for h 2p 2p where the coaction is defined by a universal r matrix h h as an application of the projective module construction we study the basic algebra of u qsl 2 1 and find the associative algebra structure and the dimension 5p2 p 4 of its center

on the classification of finite dimensional pointed hopf algebras we classify finite dimensional complex hopf algebras a which are pointed that is all of whose irreducible comodules are one dimensional and whose group of group like elementsg a is abelian such that all prime divisors of the order of g a are 7 since these hopf algebras turn out to be deformations of a natural class of generalized small quantum groups our result can be read as an axiomatic description of generalized small quantum groups

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

verma and simple modules for quantum groups at non abelian groups abstract the drinfeld double d of the bosonization of a finite dimensional nichols algebra b v over a finite non abelian group g is called a quantum group at a non abelian group we introduce verma modules over such a quantum group d and prove that a verma module has simple head and simple socle this provides two bijective correspondences between the set of simple modules over d and the set of simple modules over the drinfeld double d g as an example we describe the lattice of submodules of the verma modules over the quantum group at the symmetric group s 3 attached to the 12 dimensional fomin kirillov algebra computing all the simple modules and calculating their dimensions

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

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

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

on subsequences of quiddity cycles and nichols algebras abstract we provide a tool to obtain local descriptions of quiddity cycles as an application we classify rank two affine nichols algebras of diagonal type

lattice boltzmann equation on a two dimensional rectangular grid abstract we construct a multirelaxation lattice boltzmann model 1 on a two dimensional rectangular grid the model is partly inspired by a previous work of koelman 2 to construct a lattice bgk model on a two dimensional rectangular grid the linearized dispersion equation is analyzed to obtain the constraints on the isotropy of the transport coefficients and galilean invariance for various wave propagations in the model the linear stability of the model is also studied the model is numerically tested for three cases a a vortex moving with a constant velocity on a mesh with periodic boundary conditions b poiseuille flow with an arbitrary inclined angle with respect to the lattice orientation and c a cylinder asymmetrically placed in a channel the numerical results of these tests are compared with either analytic solutions or the results obtained by other methods satisfactory results are obtained for the numerical simulations

on pointed hopf algebras associated with unmixed conjugacy classes in sm let sm be a product of disjoint cycles of the same length c the conjugacy class of and an irreducible representation of the centralizer of we prove that either the nichols algebra b c is infinite dimensional or the braiding of the yetter drinfeld module is negative

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

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

harmonic maps to buildings and singular perturbation theory the notion of a uni versal building associated with a point in the hitchin base is introduced the universal building is a building equipped with a harmonic map from the universal cover of the given riemann surface that is initial among harmonic maps which induce the given cameral cover of the riemann surface the notion of a versal building is obtained by relaxing the uniqueness condition in the definition of an initial object in the rank one case the universal building is the leaf space of the quadratic differential defining the point in the hitchin base the main conjectures of this paper are 1 a versal building always exists 2 the asymptotics of the riemann hilbert correspondence and the non abelian hodge correspondence are controlled by the harmonic map associated to any versal building 3 spectral networks arise as inverse images of singularities of the versal building and 4 versal buildings encode the data of a 3d calabi yau category whose space of stability conditions has a connected component that contains the hitchin base the main theorem establishes the existence of the universal building conjecture 3 as well as the riemann hilbert part of conjecture 2 in the case of the rank two example introduced in the seminal work of berk nevins roberts on higher order stokes phenomena it is also shown that the asymptotics of the riemann hilbert correspondence are always controlled by a harmonic map to a certain building which is constructed as the asymptotic cone of a symmetric space

nichols algebras of group type with many quadratic relations abstract we classify nichols algebras of irreducible yetter drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two all such nichols algebras are finite dimensional and all known finite dimensional nichols algebras of nonabelian group type appear in the result of our classification we find a new finite dimensional nichols algebra over fields of characteristic two

on a factorization of graded hopf algebras using lyndon words abstract we find a generalization of the restricted pbw basis for pointed hopf algebras over abelian groups constructed by kharchenko we obtain a factorization of the hilbert series for a wide class of graded hopf algebras these factors are parametrized by lyndon words and they are the hilbert series of certain graded hopf algebras

yetter drinfeld modules over bosonizations of dually paired hopf algebras abstract let r r be a dual pair of hopf algebras in the category of yetter drinfeld modules over a hopf algebra h with bijective antipode we show that there is a braided monoidal isomorphism between rational left yetter drinfeld modules over the bosonizations of r and of r respectively as an application of this very general category isomorphism we obtain a natural proof of the existence of reflections of nichols algebras of semi simple yetter drinfeld modules over h

a simplicial complex of nichols algebras we translate the concept of restriction of an arrangement in terms of hopf algebras in consequence every nichols algebra gives rise to a simplicial complex decorated by nichols algebras with restricted root systems as applications some of these nichols algebras provide weyl groupoids which do not arise for diagonal nichols algebras and in fact we realize all root systems of finite weyl groupoids of rank greater than three further our result explains the root systems of the folded nichols algebras over nonabelian groups and of generalized satake diagrams

lattice boltzmann model for complex ginzburg landau equation in curvilinear coordinates a lattice boltzmann model is proposed for solving the complex ginzburg landau equation cgle with curvilinear coordinates the method maintains the algorithmic simplicity of the original lattice boltzmann scheme and does not require an interpolation or coarse graining procedure this lattice boltzmann scheme is based on uniformly distributed lattice points in these curvilinear coordinate systems the algorithm provides advantages similar to the previous lattice boltzmann method in that it is easily adapted to cgle the numerical simulations show spiral wave on a disc the surface of a sphere and the inside of a sphere examples show that the model accurately reproduces the phenomena in the cgle