quantum algebras as quantizations of dual poisson lie groups a systematic computational approach for the explicit construction of any quantum hopf algebra uz g z starting from the lie bialgebra g that gives the first order deformation of the coproduct map z is presented the procedure is based on the well known quantum duality principle namely the fact that any quantum algebra can be viewed as the quantization of the unique poisson lie structure g g on the dual group g which is obtained by exponentiating the lie algebra g defined by the dual map from this perspective the coproduct for uz g is just the pull back of the group law for g and the poisson analogues of the quantum commutation rules for uz g are given by the unique poisson lie structure g on g whose linearization is the poisson analogue of the initial lie algebra g this approach is shown to be a very useful technical tool in order to solve the lie bialgebra quantization problem explicitly since once a lie bialgebra g is given the full dual poisson lie group g can be obtained either by applying standard poisson lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs as a consequence the quantization of g will give rise to the full uz g quantum algebra provided that ordering problems are appropriately fixed through the choice of certain local coordinates on g whose coproduct fulfils a precise quantum symmetry property the applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant lie algebras such as the 2 1 anti de sitter algebra so 2 2 and the poincare algebra in 3 1 dimensions

hausdorff dimension of the set of nonergodic directions it is known that nonergodic directions in a rational billiard form a subset of the unit circle with hausdorff dimension at most 1 2 explicit examples realizing the dimension 1 2 are constructed using diophantine numbers and continued fractions a lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction

train algebras of rank 3 with finiteness conditions abstract in this paper we prove that a train algebra of rank 3 which is finitely generated n therian or artinian is finite dimensional

classification of real three dimensional lie bialgebras and their poisson lie groups classical r matrices of the three dimensional real lie bialgebras are obtained in this way all three dimensional real coboundary lie bialgebras and their types triangular quasitriangular or factorizable are classified then by using the sklyanin bracket the poisson structures on the related poisson lie groups are obtained

solving the two dimensional constant quantum yang baxter equation a detailed analysis of the constant quantum yang baxter equation rk1k2j1j2 rl1k3k1j3rl2l3k2k3 rk2k3j2j3 rk1l3j1k3rl1l2k1k2 in two dimensions is presented leading to an exhaustive list of its solutions the set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the grobner basis methods each solution was then transformed into a canonical form based on the various trace matrices of r for final elimination of duplicates and subcases if we use homogeneous parametrization the solutions can be combined into 23 distinct cases modulo the well known c p and t reflections and rotations and scalings r q q r q q 1

on a type of commutative algebras abstract we introduce some basic concepts for jacobi jordan algebras such as representations crossed products or frobenius metabelian co flag objects a new family of solutions for the quantum yang baxter equation is constructed arising from any 3 step nilpotent jacobi jordan algebra crossed products are used to construct the classifying object for the extension problem in its global form for a given jacobi jordan algebra a and a given vector space v of dimension c a global non abelian cohomological object g h 2 a v is constructed it classifies from the view point of the extension problem all jacobi jordan algebras that have a surjective algebra map on a with kernel of dimension c the object g h 2 a k responsible for the classification of co flag algebras is computed all 1 dim a dimensional jacobi jordan algebras that have an algebra surjective map on a are classified and the automorphism groups of these algebras is determined several examples involving special sets of matrices and symmetric bilinear forms as well as equivalence relations between them generalizing the isometry relation are provided

limit sets of teichm uller geodesics with minimal nonuniquely ergodic vertical foliation ii given a sequence of curves on a surface we provide con ditions which ensure that 1 the sequence is an innite quasi geodesic in the curve complex 2 the limit in the gromov boundary is repre sented by a nonuniquely ergodic ending lamination and 3 the sequence divides into a nite set of subsequences each of which projectively con verges to one of the ergodic measures on the ending lamination the conditions are suciently robust allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension 3g 3 for example we also study the limit sets in the thurston boundary of teichmuller geodesic rays dened by quadratic dierentials whose vertical foliations are obtained from the constructions mentioned above we prove that such examples exist for which the limit is a cycle the 1 skeleton of the simplex of projective classes of measures visiting every vertex

classification of three dimensional lie bialgebras the paper contains a classification of the three dimensional real and complex lie bialgebras with a description of the corresponding drinfel d doubles

the drinfeld double gl n tn we present a self dual drinfeld double structure underlying the an series of simple lie algebras such double is constructed through a central extension tn of gl n and is obtained by pairing two disjoint solvable subalgebras coming from positive and negative roots the cartan weyl basis of gl n is shown to be completely determined by the compatibility conditions in the double a natural lie bialgebra structure on gl n is obtained that offers a new perspective for the construction of its quantum deformations

the moduli space of 4 dimensional nilpotent complex associative algebras abstract in this paper we study 4 dimensional nilpotent complex associative algebras this is a continuation of the study of the moduli space of 4 dimensional algebras the nonnilpotent algebras were analyzed in an earlier paper even though there are only 15 families of nilpotent 4 dimensional algebras the complexity of their behavior warranted a separate study which we give here

limit sets of teichmuller geodesics with minimal non uniquely ergodic vertical foliation we describe a method for constructing teichmuller geodesics where the vertical foliation is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the teichmuller geodesic the construction depends on various parameters and we show that one can ad just the parameters to ensure that the set of accumulation points of such a geodesic in the thurston boundary is exactly the projective 1 simplex of all projective measured foliations that are topologically equivalent to with fur ther adjustment of the parameters one can further assume that the transverse measure is an ergodic measure on the non uniquely ergodic foliation

quantum poincare group related to the kappa poincare algebra the classical r matrix implied by the quantum kappa poincare algebra of lukierski nowicki and ruegg is used to generate a poisson structure on the poincare group a quantum deformation of the poincare group on the hopf algebra level is obtained by a trivial quantization

on nuclear bernstein algebras abstract in 1975 p holgate proved that the core of any orthogonal bernstein algebra is a special train algebra and consequently a genetic algebra let a be a bernstein algebra then a is a nuclear algebra if and only if the core of a is a the present paper proves that the core of a bernstein algebra is a special train algebra

classical r matrices for the osp 2 2 lie superalgebra the co lie structures compatible with the osp 2 2 lie superalgebra structure are investigated and found to be all of the coboundary type the corresponding classical r matrices are classified into several disjoint families the osp 1 2 u 1 lie superbialgebras are also classified

drinfel d superdoubles and poisson lie t plurality in low dimensions defining the real lie superalgebra as real z2 graded vector space we classify real manin supertriples and drinfel d superdoubles of superdimensions 2 2 4 2 and 2 4 the drinfel d doubles of the superdimension 2 2 are then used for construction of the simplest models related by poisson lie t plurality

quantum heisenberg weyl algebras all lie bialgebra structures on the heisenberg weyl algebra are classified and explicitly quantized the complete list of quantum heisenberg weyl algebras so obtained includes new multiparameter deformations most of them being of the non coboundary type

jacobi jordan algebras abstract we study finite dimensional commutative algebras which satisfy the jacobi identity such algebras are jordan algebras we describe some of their properties and give a classification in dimensions n 7 over algebraically closed fields of characteristic not 2 or 3

three dimensional quantum algebras a cartan like point of view a perturbative quantization procedure for lie bialgebras is introduced the relevance of the choice of a completely symmetrized basis of the quantum universal enveloping algebra is stressed sets of elements of the quantum algebra that play a role similar to generators in the case of lie algebras are considered and a cartan like procedure applied to find a representative for each class of quantum algebras the method is used to construct and classify all three dimensional complex quantum algebras that are compatible with a given type of coproduct the quantization of all lie algebras that in the classical limit belong to the most relevant sector in the classification for three dimensional lie bialgebras given in 1 is thus performed new quantizations of solvable algebras whose simplicity makes them suitable for possible physical applications are obtained and already known related quantum algebras recovered

on train algebras of rank 3 abstract we classify train algebras of rank 3 having type n 1 or 3 n 2 for an arbitrary value of n then we classify train algebras of rank 3 which have an anisotropic kernel

quantum e 2 groups and lie bialgebra structures lie bialgebra structures on e 2 are classified for two lie bialgebra structures which are not coboundaries i e those which are not determined by a classical r matrix we solve the cocycle condition find the lie poisson brackets and obtain quantum group relations there is one to one correspondence between lie bialgebra structures on e 2 and possible quantum deformations of u e 2 and e 2

classification of low dimensional lie super bialgebras a thorough analysis of lie super bialgebra structures on lie super algebras osp 1 2 and super e 2 is presented a combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied in all the cases poisson lie brackets on supergroups are found the possibility of quantizing them in order to obtain quantum groups is discussed it turns out to be straightforward for all but one structure for the super e 2 group

special and exceptional mock lie algebras abstract we observe several facts and make conjectures about commutative algebras satisfying the jacobi identity the central question is which of those algebras admit a faithful representation i e in lie parlance satisfy the ado theorem or in jordan parlance are special

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

lie bi algebra structures for centrally extended two dimensional galilei algebra and their lie poisson counterparts all bi algebra structures for centrally extended galilei algebra are classified the corresponding lie poisson structures on centrally extended galilei group are found

constructing r matrices on simple lie superalgebras abstract we construct r matrices for simple lie superalgebras with non degenerate killing forms using belavin drinfeld type triples this construction gives us the standard r matrices and some non standard ones

classification of the quantum deformations of the superalgebra gl 1 1 we present a classification of the possible quantum deformations of the supergroup and its lie superalgebra in each case the super commutation relations and the hopf structures are explicitly computed for each r matrix one finds two inequivalent co products whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras in the braided case one recovers the classical algebra for suitable limits of the deformation parameters but this is no longer true in the unbraided case

on the classification of quantum poincare groups using the general theory of 10 quantum poincare groups without dilatations are described and investigated the description contains a set of numerical parameters which satisfy certain polynomial equations for most cases we solve them and give the classification of quantum poincare groups each of them corresponds to exactly one quantum minkowski space the poincare series of these objects are the same as in the classical case we also classify possibler matrices for the fundamental representation of the group

quadratic symplectic lie superalgebras and lie bi superalgebras abstract we establish some relations between quadratic symplectic lie superalgebras and manin superalgebras next we introduce some concepts of double extensions of quadratic symplectic lie superalgebras and of manin superalgebras in order to give inductive descriptions of quadratic symplectic lie superalgebras

twist deformation of the rank one lie superalgebra the drinfeld twist is applied to deform the rank one orthosymplectic lie superalgebra the twist element is the same as for the sl 2 lie algebra due to the embedding of the sl 2 into the superalgebra the r matrix has the direct sum structure in the irreducible representations of the dual quantum group is defined using the frt formalism it includes the jordanian quantum group as subalgebra and grassmann generators as well

classification of four and six dimensional drinfel d superdoubles using adjoint representation we first classify two and three dimensional lie superbialgebras obtained from decomposable lie superalgebras in this way we complete the classification obtained by eghbali et al j math phys 51 073503 2010 e print arxiv 0901 4471 math ph then we classify all four and six dimensional drinfel d superdoubles

quantum deformation of the poincare supergroup and kappa deformed superspace the classical r matrix for the n 1 super poincare algebra given by lukierski et al 1993 is used to describe the graded poisson structure on the n 1 poincare supergroup the standard correspondence principle between the even odd poisson brackets and anti commutators leads to the consistent quantum deformation of the super poincare group with the deformation parameter q described by the fundamental mass parameter k k 1 in q the k deformation of n 1 superspace as dual to the k deformed supersymmetry algebra is discussed

noncommutative geometry of super jordanian osph 2 1 covariant quantum space extending a recently proposed procedure of construction of various elements of differential geometry on noncommutative algebras we obtain these structures on noncommutative superalgebras as an example a quantum superspace covariant under the action of super jordanian osph 2 1 is studied it is shown that there exist a two parameter family of torsionless connections and the curvature computed from this family of connections is bilinear it is also shown that the connections are not compatible with the metric

super jordanian algebra as a contraction limit we develop a generic representation independent contraction procedure for obtaining for instance matrices and l operators of arbitrary dimensions for the quantized super jordanian algebra from the pertinent quantities of the standard q deformed algebra

lie bialgebra quantizations of the oscillator algebra and their universal r matrices all coboundary lie bialgebras and their corresponding poisson lie structures are constructed for the oscillator algebra generated by quantum oscillator algebras are derived from these bialgebras by using the lyakhovsky and mudrov formalism and for some cases quantizations at both algebra and group levels are obtained including their universal r matrices

n 2 structures on solvable lie algebras the c 9 classification let mathfrak g be a finite dimensional lie algebra not necessarily semisimple it is known that if mathfrak g is self dual that is if it possesses an invariant metric then it admits ann 1 affine sugawara construction under certain additional hypotheses thisn 1 structure admits ann 2 extension if this is the case mathfrak g is said to possess ann 2 structure it is also known that ann 2 structure on a self dual lie algebra mathfrak g is equivalent to a vector space decomposition mathfrak g mathfrak g oplus mathfrak g where mathfrak g pm are isotropic lie subalgebras in other words n 2 structures on mathfrak g in one to one correspondence with manin triples mathfrak g mathfrak g mathfrak g in this paper we exploit this correspondence to obtain a classification of thec 9n 2 structures on solvable lie algebras in the process we also give some simple proofs for a variety of lie algebras in the process we also give some simple proofs for a variety of lie algebraic results concerning self dual lie algebras admitting symplectic or kahler structures

quantization of drinfel d doubles hopf algebra quantizations of four dimensional and six dimensional real classical drinfel d doubles are studied by following a direct analytic approach and the full quantization is explicitly obtained for most of them several new four and six dimensional quantum algebras are presented and some general features of the method are emphasized

structure des algebres de bernstein abstract the theory of bernstein s algebras is revisited many results concerning the structure of bernstein algebras ideals nilpotency and so on are stated for bernstein jordan algebras we give a definitive formulation of the theorem which characterizes them finally we emphasize the importance of the invariant l this ideal doesn t depend on the peirce decomposition of the algebra in the study of the structure of bernstein s algebras

hopf superalgebra contractions and r matrix for fermions different contractions are applied to the quantum superalgebra ospq 1 mod 2 in the first of them the graded analogue s hq 1 of the one dimensional heisenberg quantum algebra is obtained and its r matrix explicitly calculated in the second contraction a z2 graded version of the q oscillator is proposed and finally the supersymmetric two dimensional euclidean quantum algebra s eq 2 is found

duals of quasitriangular hopf superalgebras and the classical limit the duality of quasitriangular hopf superalgebras and superbialgebras with so called main commutation relations mcr which is established by means of bilinear forms with values in a finite dimensional representation of the universal r matrix r is used to calculate relations of quantum supercoordinates for several examples the tensor product of representations of r with respect to the comultiplication then leads to the construction of an accordingly deformed tensor product of mcr type superbialgebras the correspondingly deformed tensor product of quantum vector superspaces is then established in the classical limit of lie superbialgebras one similarly obtains supercommutation relations which may be understood in case of quantum groups with deformation parameter q 1 as mcr in the first order of ln q again the deformed tensor product of such semiclassical mcr type lie superbialgebras is constructed

non coboundary poisson lie structures on the book group all possible poisson lie pl structures on the 3d real lie group generated by a dilation and two commuting translations are obtained their classification is fully performed by relating these pl groups to the corresponding lie bialgebra structures on the corresponding book lie algebra by construction all these poisson structures are quadratic poisson hopf algebras for which the group multiplication is a poisson map in contrast to the case of simple lie groups it turns out that most of the pl structures on the book group are non coboundary ones moreover from the viewpoint of poisson dynamics the most interesting pl book structures are just some of these non coboundaries which are explicitly analysed in particular we show that the two different q deformed poisson versions of the sl 2 r algebra appear as two distinguished cases in this classification as well as the quadratic poisson structure that underlies the integrability of a large class of 3d lotka volterra equations finally the quantization problem for these pl groups is sketched

poisson lie groups bi hamiltonian systems and integrable deformations given a lie poisson completely integrable bi hamiltonian system on we present a method which allows us to construct under certain conditions a completely integrable bi hamiltonian deformation of the initial lie poisson system on a non abelian poisson lie group of dimension n where is the deformation parameter moreover we show that from the two multiplicative poisson lie hamiltonian structures on that underly the dynamics of the deformed system and by making use of the group law on one may obtain two completely integrable hamiltonian systems on by construction both systems admit reduction via the multiplication in to the deformed bi hamiltonian system in the previous approach is applied to two relevant lie poisson completely integrable bi hamiltonian systems the lorenz and euler top systems

classical lie algebras and drinfeld doubles the drinfeld double structure underlying the cartan series an bn cn dn of simple lie algebras is discussed this structure is determined by two disjoint solvable subalgebras matched by a pairing for the two nilpotent positive and negative root subalgebras the pairing is natural and in the cartan subalgebra is defined with the help of a central extension of the algebra a new completely determined basis is found from the compatibility conditions in the double and a different perspective for quantization is presented other related drinfeld doubles on are also considered

poisson lie t duality for quasitriangular lie bialgebras we introduce a new 2 parameter family of sigma models exhibiting poisson lie t duality on a quasitriangular poisson lie group g the models contain previously known models as well as a new 1 parameter line of models having the novel feature that the lagrangian takes the simple form where the generalised metric e is constant not dependent on the field u as in previous models we characterise these models in terms of a global conserved g invariance the models on g su 2 and its dual g are computed explicitly the general theory of poisson lie t duality is also extended notably the reduction of the hamiltonian formulation to constant loops as integrable motion on the group manifold the approach also points in principle to the extension of t duality in the hamiltonian formulation to group factorisations d g m where the subgroups need not be dual or connected to the drinfeld double

classification of two and three dimensional lie superbialgebras using adjoint representation of lie superalgebras we obtain the matrix form of super jacobi and mixed super jacobi identities of lie superbialgebras by direct calculations of these identities and use of automorphism supergroups of two and three dimensional lie superalgebras we obtain and classify all two and three dimensional lie superbialgebras