relative hopf modules for dual quasi hopf algebras univ bucharest fac math ro 70109 bucharest 1 romania limburgs univ ctr dept math b 3590 diepenbeek belgium bulacu d univ bucharest fac math str acad 14 ro 70109 bucharest 1 romania

two sided two cosided hopf modules and doi hopf modules for quasi hopf algebras let h be a finite dimensional quasi hopf algebra over a field k and a a right h comodule algebra we introduce the category of two sided hopf modules and prove that it is isomorphic to a module category we also show that two sided hopf modules are coalgebra over a certain comonad we introduce doi hopf modules and show that they are comodules over a certain coring if the underlying h module coalgebra is finite dimensional then doi hopf modules are modules over a certain smash products a similar result holds for two sided two cosided hopf modules

gauge theory on nonassociative spaces we show how to do gauge theory on the octonions and other nonassociative algebras such as quasi r4 models proposed in string theory we use the theory of quasialgebras obtained by cochain twist introduced previously the gauge theory in this case is twisting equivalent to the usual gauge theory on the underlying classical space we give a general u 1 yang mills example for any quasialgebra and a full description of the moduli space of flat connections in this theory for the cube z23 and hence for the octonions we also obtain further results about the octonions themselves an explicit moyal product description of them as a nonassociative quantization of functions on the cube and a characterization of their cochain twist as invariant under fourier transform

quasialgebra structure of the octonions abstract we show that the octonions are a twisting of the group algebra of z 2 z 2 z 2 in the quasitensor category of representations of a quasi hopf algebra associated to a group 3 cocycle in particular we show that they are quasialgebras associative up to a 3 cocycle isomorphism we show that one may make general constructions for quasialgebras exactly along the lines of the associative theory including quasilinear algebra representation theory and an automorphism quasi hopf algebra we study the algebraic properties of quasialgebras of the type which includes the octonions further examples include the higher 2n onion cayley algebras and examples associated to hadamard matrices

action of truncated quantum groups on quasi quantum planes and a quasi associative differential geometry and calculus ifq is apth root of unity there exists a quasi coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofu q sl2 whose coproduct yields the truncated tensor product of physical representations ofu q sl2 and whoser matrix satisfies quasi yang baxter equations these truncated quantum group algebras are examples of weak quasitriangular quasi hopf algebras quasi quantum group algebras open image in new window we describe a space open image in new window of functions on the quasi quantum plane i e of polynomials in noncommuting complex coordinate functionsz a on which multiplication operatorsz a and the elements of open image in new window can act so thatz a will transform according to some representation f of open image in new window open image in new window can be made into a quasi associative graded algebra open image in new window on which elements of open image in new window act as generalized derivations in the special case of the truncatedu q sl2 algebra we show that the subspaces open image in new window of monomials inz a ofnth degree vanish forn p 1 and that open image in new window carries the 2j 1 dimensional irreducible representation of open image in new window ifn 2j j 0 1 2 1 2 p 2 assuming that the representation f of the quasi quantum group algebra gives rise to anr matrix with two eigenvalues we develop a quasi associative differential calculus on open image in new window this implies construction of an exterior differentiation a graded algebra open image in new window of forms and partial derivatives a quasi associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms it is covariant under open image in new window gauge transformations

hom l r smash products hom diagonal crossed products and the drinfeld double of a hom hopf algebra abstract we introduce the hom analogue of the l r smash product and use it to define the hom analogue of the diagonal crossed product when h is a finite dimensional hom hopf algebra with bijective antipode and bijective structure map we define the drinfeld double of h its algebra structure is a hom diagonal crossed product and it has all expected properties namely it is quasitriangular and modules over it coincide with left right yetter drinfeld modules over h

braided cyclic cocycles and nonassociative geometry we use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a drinfeld type cochain twist these are the so called quasialgebras and include the octonions as braided commutative but nonassociative coordinate rings as well as quasialgebra versions cq g of the standard q deformation quantum groups we introduce the notion of ribbon algebras in the category which are algebras equipped with a suitable generalized automorphism and obtain the required generalization of cyclic cohomology we show that this braided cyclic cohomology is invariant under a cochain twist we also extend to our generalization the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra the paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry as well as the algebraic noncommutative torus as an associative example

diagonal crossed products and l r smash products over weak hopf algebras the notions of the diagonal crossed product and the l r smash product over weak hopf algebras which can be seen as generalizations of smash products over weak hopf algebras are introduced and studied in particular the weak hopf algebraic structure of the drinfeld double of a finite dimensional weak hopf algebra which is a special case of diagonal crossed products is given moreover two canonical examples the kadison products and the connes moscovici products over weak hopf algebras are introduced and investigated

aeroservoelastic aspects of wing control surface planform shape optimization equivalent plate structural modeling and doublet point lifting surface unsteady aerodynamics are used to obtain analytic sensitivities of aeroelastic and aeroservoelastic response with respect to wing and control surface planform shape parameters rational function approximations for unsteady aerodynamic forces their shape sensitivities and the resulting linear time invariant state space models of aeroservoelastic systems and their shape sensitivities are examined the goal is to develop effective and numerically efficient approximation techniques for wing shape optimization for use with nonlinear programming and approximation concepts as a multidisciplinary optimization strategy effects of structural and unsteady aerodynamic modeling errors are studied examination of approximation accuracy using alternative approximation techniques and the resulting move limits provide insight and experience on the way to realistic wing control surface shape optimization with active controls and aeroservoelastic constraints

noncommutative finite dimensional manifolds i spherical manifolds and related examples we exhibit large classes of examples of noncommutative finite dimensional manifolds which are non formal deformations of classical manifolds the main result of this paper is a complete description of noncommutative three dimensional spherical manifolds a noncommutative version of the sphere s 3 defined by basic k theoretic equations we find a 3 parameter family of deformations of the standard 3 sphere s 3 and a corresponding 3 parameter deformation of the 4 dimensional euclidean space 4 for generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed 4 u only depend on two parameters and are isomorphic to the algebras introduced by sklyanin in connection with the yang baxter equation it follows that different can span the same this equivalence generates a foliation of the parameter space this foliation admits singular leaves reduced to a point these critical points are either isolated or fall in two 1 parameter families up to the simple operation of taking the fixed algebra by an involution these two families are identical and we concentrate here on c for the above isomorphism with the sklyanin algebra breaks down and the corresponding algebras are special cases of deformations a notion which we generalize in any dimension and various contexts and study in some detail here and this point is crucial the dimension is not an artifact i e the dimension of the classical model but is the hochschild dimension of the corresponding algebra which remains constant during the deformation besides the standard noncommutative tori examples of deformations include the recently defined noncommutative 4 sphere as well as m dimensional generalizations noncommutative versions of spaces and quantum groups which are deformations of various classical groups we develop general tools such as the twisting of the clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative dimensional spherical manifolds a key result is the differential self duality properties of these projections which generalize the self duality of the round instanton

hopf modules and the double of a quasi hopf algebra we give a different proof for a structure theorem of hausser and nill on hopf modules over quasi hopf algebras we extend the structure theorem to a classification of two sided two cosided hopf modules by yetter drinfeld modules which can be defined in two rather different manners for the quasi hopf case the category equivalence between hopf modules and yetter drinfeld modules leads to a new construction of the drinfeld double of a quasi hopf algebra as proposed by majid and constructed by hausser and nill

actions of monoidal categories and generalized hopf smash products let r be a k algebra and c a monoidal category assume given the structure of a c category on the category rm of left r modules that is the monoidal category c is assumed to act on the category rm by a coherently associative bifunctor c rm rm we assume that this bifunctor is right exact in its right argument in this setup we show that every algebra a respectively coalgebra c in c gives rise to an r ring a r respectively an r coring c r whose modules respectively comodules are the a modules respectively c comodules within the category rm we show that this very general scheme for constructing co associative co rings gives conceptual explanations for the double of a quasi hopf algebra as well as certain doubles of hopf algebras in braided categories each time avoiding ad hoc computations showing associativity

doubles of quasi quantum groups in dr1 drinfeld showed that any finite dimensional hopf algebra extends to a quasitriangular hopf algebra the quantum double of based on the construction of a so called diagonal crossed product developed by the authors in hn we generalize this result to the case of quasi hopf algebras as for ordinary hopf algebras as a vector space the quasi quantum double is isomorphic to where denotes the dual of we give explicit formulas for the product the coproduct the r matrix and the antipode on and prove that they fulfill drinfeld s axioms of a quasitriangular quasi hopf algebra in particular becomes an associative algebra containing as a quasi hopf subalgebra on the other hand is not a subalgebra of unless the coproduct on is strictly coassociative it is shown that the category of finite dimensional representations of coincides with what has been called the double category of modules by s majid m2 thus our construction gives a concrete realization of majid s abstract definition of quasi quantum doubles in terms of a tannaka krein like reconstruction procedure the whole construction is shown to generalize to weak quasi hopf algebras with now being linearly isomorphic to a subspace of

minimum state unsteady aerodynamics for aeroservoelastic configuration shape optimization of flight vehicles the minimum state technique for creating linear time invariant state space models of unsteady aerodynamic loads leads to major savings in terms of the dimension of the resulting aeroservoelastic systems compared to alternative techniques the upfront effort of the generation of minimum state approximants however is large and involves an iterative double least squares function fitting process when the overall shape of a configuration varies during the repetitive analysis required for configuration design optimization the cost of creating new minimum state approximations becomes prohibitive this paper presents an efficient method for obtaining shape sensitivity analysis of minimum state approximants the resulting sensitivities are then used based on a single detailed analysis and sensitivity analysis at some reference design to generate approximate aeroelastic poles divergence speed and flutter speed results for configurations that are subject to considerable planform shape changes

frobenius and separable functors for the category of entwined modules over cowreaths ii applications abstract let h be a quasi hopf algebra we apply results obtained in 8 to give necessary and sufficient conditions for the forgetful functor from doi hopf modules two sided hopf modules or yetter drinfeld modules over h to representations of the underlying algebra to be frobenius resp separable we show that in some situations these conditions reduce to the unimodularity and or co semisimplicity of the quasi hopf algebra h

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

generalized diagonal crossed products and smash products for quasi hopf algebras applications in this paper we introduce generalizations of diagonal crossed products two sided crossed products and two sided smash products for a quasi hopf algebra h the results we obtain may then be applied to h hopf bimodules and generalized yetter drinfeld modules the generality of our situation entails that the generating matrix formalism cannot be used forcing us to use a different approach this pays off because as an application we obtain an easy conceptual proof of an important but very technical result of hausser and nill concerning iterated two sided crossed products

minimum state unsteady aerodynamic shape sensitivities using sensitivity of optimal solutions to problem parameters parameters involved including the aerodynamic lag roots its solution is unique and convergence of the solution process based on standard mathematical programming techniques allows accurate determination of all unknown variables an improvement of the math programming minimum state fitting is presented here and it is developed further to include sensitivity analysis for the resulting minimum state series free of the nonuniqueness and convergence issues that plagued previous attempts to obtain sensitivities of minimum state fits a variable planform double swept forward wing case is used to demonstrate performance of the fitting technique itself shape sensitivity calculations and taylor series based approximations that can be used to replace detailed full order analysis in the course of multidisciplinary design optimization of flight vehicles nomenclature a0 a1 a2 minimum state approximation matrices corresponding to aerodynamic stiffness damping and inertia respectively a 0 a 1 a 2 elements of the matrices a0 a1 a2 b reference semichord length of wing c structural viscous damping matrix c reference chord length of wing d e minimum state approximation matrices