q epsilon tensor for quantum and braided spaces the machinery of braided geometry introduced previously is used now to construct the e totally antisymmetric tensor on a general braided vector space determined by r matrices this includes natural q euclidean and q minkowski spaces the formalism is completely covariant under the corresponding quantum group such as soq 4 or soq 1 3 the hodge operator and differentials are also constructed in this approach

quantum spin coverings and statistics slq 2 at primitive odd roots of unity ql 1 is studied as a quantum cover of the complex rotation group so 3 in terms of the associated hopf algebras of quantum polynomial functions we work out the irreducible corepresentations the decomposition of their tensor products and a coquasitriangular structure with the associated braiding or statistics as an example the case l 3 is discussed in detail

wave equations on q minkowski space we give a systematic account of a component approach to the algebra of forms onq minkowski space introducing the corresponding exterior derivative hodge star operator coderivative laplace beltrami operator and lie derivative using this braided differential geometry we then give a detailed exposition of theq d alembert andq maxwell equation and discuss some of their non trivial properties such as for instance plane wave solutions for theq maxwell field we also give aq spinor analysis of theq field strength tensor

quantum instantons with classical moduli spaces we introduce a quantum minkowski space time based on the quantum group su 2 q extended by a degree operator and formulate a quantum version of the anti self dual yang mills equation we construct solutions of the quantum equations using the classical adhm linear data and conjecture that up to gauge transformations our construction yields all the solutions we also find a deformation of penrose s twistor diagram giving a correspondence between the quantum minkowski space time and the classical projective space p 3

free braided differential calculus braided binomial theorem and the braided exponential map braided differential operators i are obtained by differentiating the addition law on the braided covector spaces introduced previously such as the braided addition law on the quantum plane these are affiliated to a yang baxter matrix r the quantum eigenfunctions expr x v of the i braided plane waves are introduced in the free case where the position components xi are totally noncommuting a braided r binomial theorem and a braided taylor theorem expr a f x f a x are proven these various results precisely generalize to a generic r matrix and hence to n dimensions the well known properties of the usual one dimensional q differential and q exponential as a related application it is shown that the q heisenberg algebra px qxp 1 is a braided semidirect product c x c p of the braided line acting on itself a braided weyl algebra and similarly for its generalization to an arbitrary r matrix

q lorentz group and braided coaddition on q minkowski space we present a new version ofq minkowski space which has both a coaddition law and anslq 2 decomposition the additive structure forms a braided group rather than a quantum one in the process we obtain aq lorentz group which coacts covariantly on thisq minkowski space

q oscillator representations of hermitian braided matrices representations of the braided algebra of in a hilbert space are constructed the generators of this algebra are expressed in terms of two independent q oscillators it is shown that apart from a factor of u 1 these generators reduce to the ordinary angular momentum algebra generators in the limit

the quantum general linear supergroup and braid statistics we generalize to braid statistics some of manin s work 1989 on the quantum deformation of the general linear supergroup the key ingredient in this construction is the introduction of a non standard transposition map psi psi 2 not 1 which is defined in terms of a generalized permutation operator pmu pmu 2 not 1 the dual space enveloping algebra with pmu statistics is defined consideration of coactions on quantum spaces clarifies the resulting structure

quantum line bundles via cayley hamilton identity as shown by pyatov and saponov 1995 j phys a math gen 28 4415 21 and gurevich et al 1997 lett math phys 41 255 64 the matrix l lij whose entries lij are generators of the so called reflection equation algebra rea is subject to some polynomial identity resembling the cayley hamilton identity for a numerical matrix here a similar statement is presented for a matrix whose entries are generators of a filtered algebra that is a non commutative analogue of the rea in an appropriate limit we obtain a similar statement for the matrix formed by the generators of the algebra u gl n this property is used to introduce the notion of line bundles over quantum orbits in the spirit of the serre swan approach the quantum orbits in question are presented explicitly as some quotients of one of the algebras mentioned above both in the quasiclassical case i e that related to the quantum group uq sl n and a non quasiclassical one i e that arising from a hecke symmetry with non standard poincare series of the corresponding symmetric and skew symmetric algebras

noncommutative geometry of angular momentum space u su 2 we study the standard angular momentum algebra xi xj i eijkxk as a noncommutative manifold r 3 we show that there is a natural 4d differential calculus and obtain its cohomology and hodge operator we solve the spin 0 wave equation and some aspects of the maxwell or electromagnetic theory including solutions for a uniform electric current density and we find a natural dirac operator we embed r 3 inside a 4d noncommutative space time which is the limit q 1 of q minkowski space and show that r 3 has a natural quantum isometry group given by the quantum double c su 2 u su 2 which is a singular limit of the q lorentz group we view r 3 as a collection of all fuzzy spheres taken together we also analyze the semiclassical limit via minimum uncertainty states j approximating classical positions in polar coordinates

differential calculuses on the quantized braided groups braided differential calculuses on the quantized braided groups are constructed and their braided bialgebra hopf algebra structures are demonstrated these are a kind of generalization and unification of the differential calculuses on quantum groups braided groups quantum supergroups etc and contain the latter ones as special cases moreover it is shown that some quantum differential co vector algebras are covariant under the braided local coactions of the obtained braided differential bialgebras hopf algebras some examples are also given

neural data fusion algorithms based on a linearly constrained least square method two novel neural data fusion algorithms based on a linearly constrained least square lcls method are proposed while the lcls method is used to minimize the energy of the linearly fused information two neural network algorithms are developed to overcome the ill conditioned and singular problems of the sample covariance matrix arisen in the lcls method the proposed neural fusion algorithms are samples for implementation using both software and hardware compared with the existing fusion methods the proposed neural data fusion method has an unbiased statistical property and does not require any a priori knowledge about the noise covariance it is shown that the proposed neural fusion algorithms converge globally to the optimal fusion solution when the sample covariance matrix is singular and converge globally with exponential rate when the sample covariance matrix is nonsingular we apply the proposed neural fusion method to image and signal fusion and it is shown that the quality of the solution can be greatly enhanced by the proposed technique

algebraic framework for quantization of nonultralocal models extension of the braid relations to the multiple braided tensor product of algebras that can be used for quantization of nonultralocal models is presented the yang baxter type consistency conditions as well as conditions for the existence of the multiple coproduct monodromy matrix which can be used for construction of the conserved quantities are given homogeneous and local algebras are introduced and simplification of the yang baxter type conditions for them is shown the yang baxter type equations and multiple coproduct conditions for homogeneous and local of order 2 algebras are solved

braided geometry of the conformal algebra we show that the action of the special conformal transformations of the usual undeformed conformal group is the q 1 scaling limit of the braided adjoint action or r commutator of q minkowski space on itself we also describe the q deformed conformal algebra in r matrix form and its quasi structure

relative yetter drinfeld modules and comodules over braided groups let h1 be a quantum group and f h1 h2 a hopf algebra homomorphism assume that b is some braided group obtained by majid s transmutation process we first show that there is a tensor equivalence between the category of comodules over the braided group b and that of relative yetter drinfeld modules next we prove that the drinfeld centers of the two categories mentioned above are equivalent to the category of modules over some quantum double namely the category of ordinary yetter drinfeld modules over some radford s biproduct hopf algebra importantly the above results not only hold for a finite dimensional quantum group but also for an infinite dimensional one

braided momentum in the q poincare group the q poincare group of m schlieker et al z phys c 53 79 1992 is shown to have the structure of a semidirect product and coproduct b soq 1 3 where b is a braided quantum group structure on the q minkowski space of four momentum with braided coproduct p p 1 1 p here the necessary b is not a usual kind of quantum group but one with braid statistics similar braided vectors and covectors v r v r exist for a general r matrix the abstract structure of the q lorentz group is also studied

deformed minkowski spaces classification and properties using general but simple covariance arguments we classify the quantum minkowski spaces for dimensionless deformation parameters this requires a previous analysis of the associated lorentz groups which reproduces a previous classification by woronowicz and zakrzewski as a consequence of the unified analysis presented we give the commutation properties the deformed and central length element and the metric tensor for the different spacetime algebras

non levi closed conjugacy classes of sp q 2 n we construct an explicit quantization of semi simple conjugacy classes of the complex symplectic group sp 2n with non levi isotropy subgroups through an operator realization on highest weight modules over the quantum group u q bigl mathfrak sp 2n bigr

podleś spheres for the braided quantum su 2 abstract starting with the braided quantum group su q 2 for a complex deformation parameter q we perform the construction of the quotient su q 2 t which serves as a model of a quantum sphere then we follow the reasoning of podle who for real q classified quantum spaces with the action of su q 2 with appropriate spectral properties these properties can also be expressed in the context of the braided quantum su q 2 with complex q and we find that they lead to precisely the same family of quantum spaces as found by podle for the real parameter q

algebraic q integration and fourier theory on quantum and braided spaces an algebraic theory of integration on quantum planes and other braided spaces is introduced in the one dimensional case a novel picture of the jackson q integral as indefinite integration on the braided group of functions in one variable x is obtained here x is treated with braid statistics q rather than the usual bosonic or grassmann ones it is shown that the definite integral x x can also be evaluated algebraically as multiples of the integral of a q gaussian with x remaining as a bosonic scaling variable associated with the q deformation further composing the algebraic integration with a representation then leads to ordinary numbers for the integral integration is also used to develop a full theory of q fourier transformation f the braided addition x x 1 1 x and braided antipode s is used to define a convolution product and prove a convolution theorem it is also proven that f2 s the analogous results are proven on any braided group including integration and fourier transformation on quan

braided coaddition on the quantized braided matrices an additive braided coproduct also be called braided coaddition is introduced on the quantized braided matrices qbms it gives qbms another braided hopf algebra structure the coaddition is also shown to be compatible with the existing coproduct such that they together form a so called quantized braided ring and some quantized braided differential operator bialgebras hopf algebras relating to this braided coaddition are constructed these give a unification and generalization of the known results about braided and quantum matrices moreover the coaddition construction and the related differential calculi on the qbms are further extended to a kind of more general quantum matrix algebraic systems some examples are also given

extended reflection equation algebras the braid group on a handlebody and associated link polynomials the correspondence of the braid group on a handlebody of arbitrary genus to the algebra of yang baxter and extended reflection equation operators is shown representations of the infinite dimensional extended reflection equation algebra in terms of direct products of quantum algebra generators are derived they lead to a representation of this braid group in terms of r matrices restriction to the reflection equation operators only gives the colored braid group the reflection equation operators describing the effect of handles attached to a three ball satisfy characteristic equations that give rise to additional skein relations and thereby invariants of links on handlebodies the origin of the skein relations is explained and they are derived from an adequately adapted handlebody version of the jones polynomial relevance of these results to the construction of link polynomials on closed three manifolds via heegard splitting and surgery is indicated

generalized q exponentials related to orthogonal quantum groups and fourier transformations of noncommutative spaces essential prerequisites for the study of q deformed physics are particle states in position and momentum representation in order to relate x and p space by fourier transformations the appropriate q exponential series related to orthogonal quantum symmetries is constructed it turns out to be a new q special function giving rise to q plane wave solutions that transform with a noncommutative phase under translations

q euclidean space and quantum wick rotation by twisting the quantum matrix algebra r21x1x2 x2x1r is studied and proposed for the standard 2 2 case as the coordinates of q deformed euclidean space the algebra in this simplest case is isomorphic to the usual quantum matrices mq 2 but in a form which is naturally covariant under the euclidean rotations suq 2 suq 2 a quantum wick rotation is introduced that twists this system precisely into the approach to q minkowski space based on braided matrices and their associated spinorial q lorentz group

braided bialgebras of hecke type abstract the paper is devoted to prove a version of milnor moore theorem for connected braided bialgebras that are infinitesimally cocommutative namely in characteristic different from 2 we prove that for a given connected braided bialgebra a c a which is infinitesimally cocommutative for some element 0 that is not a root of one in the base field then the infinitesimal braiding of a is of hecke type of mark and a is isomorphic as a braided bialgebra to the symmetric algebra of the braided subspace of its primitive elements

on the addition of quantum matrices an addition law is introduced for the usual quantum matrices a r by means of a coaddition t t 1 1 t it supplements the usual comultiplication t t t and together they obey a codistributivity condition the coaddition does not form a usual hopf algebra but a braided one the same remarks apply for rectangular m n quantum matrices as an application left invariant vector fields are constructed on a r and other quantum spaces they close in the form of a braided lie algebra as another application the wave functions in the lattice approximation of kac moody algebras and other lattice fields can be added and functionally differentiated

structures on braided spaces structures on quantum and braided spaces of the type defined via an r matrix are studied these include q minkowski and q euclidean spaces as additive braided groups the duality between the braided groups of vectors and covectors is proved and some first applications to braided geometry are made

quantized braided groups algebras that represent a generalization of both quantum groups quantum supergroups and braided groups are defined they are given by a pair of solutions of the yang baxter equation that satisfy some additional conditions several examples are presented

reflection equation and twisted yangians with any involutive antialgebra and coalgebra automorphism of a quasitriangular bialgebra we associate a reflection equation algebra a hopf algebraic treatment of the reflection equation of this type and its universal solution is given applications to the twisted yangians are considered

examples of braided groups and braided matrices matrix braided groups are developed as an analog of the coordinate functions on a group or supergroup the 1 in the super case is replaced by braid statistics there are braided group analogs of all the classical simple lie groups as well as braided matrix groups and braided matrices b r for every regular solution r of the quantum yang baxter equations a direct verification of b r is provided and some of the simplest examples are computed in detail

the center of the reflection equation algebra via quantum minors abstract we give simple formulas for the elements c k appearing in a quantum cayley hamilton formula for the reflection equation algebra re algebra associated to the quantum group u q gl n answering a question of kolb and stokman the c k s are certain canonical generators of the center of the re algebra and hence of u q gl n itself they have been described by reshetikhin using graphical calculus by nazarov tarasov using quantum yangians and by gurevich pyatov and saponov using quantum schur functions however no explicit formulas for these elements were previously known as byproducts we prove a quantum girard newton identity relating the c k s to the so called quantum power traces and we give a new presentation for the quantum group u q gl n as a localization of the re algebra along certain principal minors

quantized braided linear algebras relating to quantized braided groups an r matrix pair r z solving a system of yang baxter type equations is needed to define a quantized braided matrix group l hlavaty j math phys 35 2560 1994 it is found that a series of such kind of r matrix pairs r n z n 0 1 2 can be constructed from a known pair r z and a series of realizations of the quantized braided matrix bialgebras a r n z in v r n 1 v r n can be obtained some covariant quantized braided linear algebras and their transformation properties under the braided coactions of the quantized braided group a r z are considered some examples are presented

bicrossproduct hopf superalgebras and d 4 κ deformed poincare supergroup the general framework of bicrossproduct hopf algebras given by majid is extended to z2 graded bicrossproduct hopf superalgebras as examples of bicrossproduct hopf superalgebras we provide the graded algebras of functions on undeformed as well as deformed d 4 poincare supergroups

the braided heisenberg group the braided groups and braided matrices b r for the solution r of the yang baxter equation associated to the quantum heisenberg group are computed it is also shown that a particular extension of the quantum heisenberg group is dual to the heisenberg universal enveloping algebra uq h and this result is used to derive an action of uq h on the braided groups the various covariance properties are then demonstrated using the braided heisenberg group as an explicit example in addition the braided heisenberg group is found to be self dual finally a physical application to a system of n braided harmonic oscillators is discussed an isomorphism is found between the n fold braided and unbraided tensor products and the usual free time evolution is shown to be equivalent to an action of a primitive generator of uq h on the braided tensor product

soul expansion of g superfunctions supersmooth or g superfunctions with coefficients in the banach grassmann rogers algebra b are described in terms of the soul expansion and the notion of pringsheim regular real function

c statistical quantum groups and weyl algebras c statistical quantum groups and algebras are obtained by a process of transmutation they are like super quantum groups and algebras but with statistics given now by a complex number among the examples it is shown that the weyl algebra of canonical quantization may be viewed equivalently as a c statistical plane related examples are the noncommutative torus and a double loop variable quantization of photons these results lead toward a reformulation of ordinary quantum mechanics as a classical theory with c statistics in this reformulation the role of the 1 factors of super statistics is played by the free particle time evolution

stability of linear nonautonomous multivariable systems with differentiable matrices abstract we consider linear nonautonomous multivariable systems with differentiable matrices explicit stability conditions are derived in the appropriate situations our results improve the traditional freezing method

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

coadditive differential complexes on quantum groups and quantum spaces a regular way to define an additive coproduct or coaddition on q deformed differential complexes is proposed for quantum groups and quantum spaces related to the hecke type r matrices several examples of braided coadditive differential bialgebras hopf algebras are presented

quantum and braided linear algebra quantum matrices a r are known for every r matrix obeying the quantum yang baxter equations it is also known that these act on vectors given by the corresponding zamalodchikov algebra this interpretation is developed in detail distinguishing between two forms of this algebra v r vectors and v r covectors a r v r21 v r is an algebra homomorphism i e quantum matrices are realized by the tensor product of a quantum vector with a quantum covector while the inner product of a quantum covector with a quantum vector transforms as a scaler it is shown that if v r and v r are endowed with the necessary braid statistics then their braided tensor product v r v r is a realization of the braided matrices b r introduced previously while their inner product leads to an invariant quantum trace introducing braid statistics in this way leads to a fully covariant quantum braided linear algebra the braided groups obtained from b r act on themselves by conjugation in a way impossible

a braided interpretation of fractional supersymmetry in higher dimensions a many variable q calculus is introduced using the formalism of braided covector algebras its properties are discussed in detail and related to fractional supersymmetry when certain of its deformation parameters are roots of unity the special cases of two dimensional supersymmetry and fractional supersymmetry are developed in detail

commuting quantum traces for quadratic algebras consistent tensor products on auxiliary spaces hereafter denoted fusion procedures and commuting transfer matrices are defined for general quadratic algebras nondynamical and dynamical inspired by results on reflection algebras applications of these procedures then yield integer indexed families of commuting hamiltonians

nichols woronowicz algebra model for schubert calculus on coxeter groups abstract we realise the cohomology ring of a flag manifold more generally the coinvariant algebra of an arbitrary finite coxeter group w as a commutative subalgebra of a certain nichols woronowicz algebra in the yetter drinfeld category over w this gives a braided hopf algebra version of the corresponding schubert calculus the nilcoxeter algebra and its action on the coinvariant algebra by divided difference operators are also realised in the nichols woronowicz algebra we discuss the relationship between fomin kirillov quadratic algebras kirillov maeno bracket algebras and our construction