matching rota baxter algebras matching dendriform algebras and matching pre lie algebras abstract we introduce the notion of a matching rota baxter algebra motivated by the recent work on multiple pre lie algebras arising from the study of algebraic renormalization of regularity structures 10 19 this notion is also related to iterated integrals with multiple kernels and solutions of the associative polarized yang baxter equation generalizing the natural connection of rota baxter algebras with dendriform algebras to matching rota baxter algebras we obtain the notion of matching dendriform algebras as in the classical case of one operation matching rota baxter algebras and matching dendriform algebras are related to matching pre lie algebras which coincide with the aforementioned multiple pre lie algebras more general notions and results on matching tridendriform algebras and matching postlie algebras are also obtained

cumulants free cumulants and half shuffles free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability there is a mix of similarities and differences when one considers the two families of cumulants whereas the combinatorics of classical cumulants is well expressed in terms of set partitions that of free cumulants is described and often introduced in terms of non crossing set partitions the formal series approach to classical and free cumulants also largely differs the purpose of this study is to put forward a different approach to these phenomena namely we show that cumulants whether classical or free can be understood in terms of the algebra and combinatorics underlying commutative as well as non commutative half shuffles and half unshuffles as a corollary cumulants and free cumulants can be characterized through linear fixed point equations we study the exponential solutions of these linear fixed point equations which display well the commutative respectively non commutative character of classical and free cumulants

on motives associated to graph polynomials the appearance of multiple zeta values in anomalous dimensions and functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions in this paper we start to hunt the motive restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions

comparability preserving maps on hilbert space effect algebras we describe the general form of bijective comparability preserving transformations of the hilbert space effect algebra thus improving several known characterizations of ortho order automorphisms

conditionally multiplicative maps on the set of all bounded observables preserving compatibility in this note we describe the form of all those transformations no linearity is assumed of the set of all self adjoint bounded linear operators acting on a hilbert space which preserve commutativity in both directions and which are multiplicative on commuting pairs of operators

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

representations of the q deformed algebra uq so4 we study the nonstandard q deformation uq so4 of the universal enveloping algebra u so4 obtained by deforming the defining relations for skew symmetric generators of u so4 this algebra is used in quantum gravity and algebraic topology we construct a homomorphism of uq so4 to the certain nontrivial extension of the drinfeld jimbo quantum algebra uq sl2 2 and show that this homomorphism is an isomorphism by using this homomorphism we construct irreducible finite dimensional representations of the classical type and of the nonclassical type for the algebra uq so4 it is proved that for q not a root of unity each irreducible finite dimensional representation of uq so4 is equivalent to one of these representations we prove that every finite dimensional representation of uq so4 for q not a root of unity is completely reducible it is shown how to construct by using the homomorphism tensor products of irreducible representations of uq so4 note that no hopf algebra structure is known fo

linear maps preserving the spectral radius abstract let x be a complex banach space if b x b x is a surjective linear map such that a and a have the same spectral radius for every a b x then c where is either an algebra automorphism or an antiautomorphism of b x and c is a complex constant such that c 1

linear maps preserving maximal deviation and the jordan structure of quantum systems in the algebraic approach to quantum theory a quantum observable is given by an element of a jordan algebra and a state of the system is modelled by a normalized positive functional on the underlying algebra maximal deviation of a quantum observable is the largest statistical deviation one can obtain in a particular state of the system the main result of the paper shows that each linear bijective transformation between jbw algebras preserving maximal deviations is formed by a jordan isomorphism or a minus jordan isomorphism perturbed by a linear functional multiple of an identity it shows that only one numerical statistical characteristic has the power to determine the jordan algebraic structure completely as a consequence we obtain that only very special maps can preserve the diameter of the spectra of elements nonlinear maps preserving the pseudometric given by maximal deviation are also described the results generalize hitherto known theorems on preservers of maximal deviation in the case of self adjoint parts of von neumann algebras proved by molnar

a lie theoretic approach to renormalization motivated by recent work of connes and marcolli based on the connes kreimer approach to renormalization we augment the latter by a combinatorial lie algebraic point of view our results rely both on the properties of the dynkin idempotent one of the fundamental lie idempotents in the theory of free lie algebras and on properties of hopf algebras encapsulated in the notion of associated descent algebras besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures the new techniques give rise to an algebraic approach to the galois theory of renormalization in particular they do not depend on the geometry underlying the case of dimensional regularization and the riemann hilbert correspondence this is illustrated with a discussion of the bphz renormalization scheme

preserving the measure of compatibility between quantum states in this article after defining the abstract concept of compatibility like functions on quantum states we prove that every bijective transformation on the set of all states which preserves such a function is implemented by an either unitary or antiunitary operator

coexistency on hilbert space effect algebras and a characterisation of its symmetry transformations the hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in ludwig s formulation of quantum mechanics each effect represents a quantum fuzzy event the relation of coexistence plays an important role in this theory as it expresses when two quantum events can be measured together by applying a suitable apparatus this paper s first goal is to answer a very natural question about this relation namely when two effects are coexistent with exactly the same effects the other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence in particular we will see that they can differ quite a lot from usual standard automorphisms which appear for instance in ludwig s theorem as a byproduct of our methods we also strengthen a theorem of moln ar

renormalization in quantum field theory and the riemann hilbert problem i the hopf algebra structure of graphs and the main theorem this paper gives a complete selfcontained proof of our result announced in 6 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the riemann hilbert problem we shall first show that for any quantum field theory the combinatorics of feynman graphs gives rise to a hopf algebra which is commutative as an algebra it is the dual hopf algebra of the enveloping algebra of a lie algebra whose basis is labelled by the one particle irreducible feynman graphs the lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa the corresponding lie group g is the group of characters of we shall then show that using dimensional regularization the bare unrenormalized theory gives rise to a loop where c is a small circle of complex dimensions around the integer dimension d of space time our main result is that the renormalized theory is just the evaluation at z d of the holomorphic part of the birkhoff decomposition of we begin to analyse the group g and show that it is a semi direct product of an easily understood abelian group by a highly non trivial group closely tied up with groups of diffeomorphisms the analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title

hilbert isometries and maximal deviation preserving maps on jb algebras abstract in this paper we characterize the surjective linear variation norm isometries on jb algebras variation norm isometries are precisely the maps that preserve the maximal deviation the quantum analogue of the standard deviation which plays an important role in quantum statistics consequently we characterize the hilbert s metric isometries on cones in jb algebras

spectral order of operators and range projections we study the effect algebra i e the positive part of the unit ball of an operator algebra and its relation to the projection lattice from the perspective of the spectral order a spectral orthomorphism is a map between effect algebras which preserves the spectral order and orthogonality of elements we show that if the spectral orthomorphism preserves the multiples of the unit then it is a restriction of a jordan homomorphism between the corresponding algebras this is an optimal extension of the dye s theorem on orthomorphisms of the projection lattices to larger structures containing the projections moreover results on automatic countable additivity of spectral homomorphisms are proved further we study the order properties of the range projection map assigning to each positive contraction in a jbw algebra its range projection it is proved that this map preserves infima of positive contractions in the spectral respectively standard order if and only if the projection lattice of the algebra in question is a modular lattice

new identities in dendriform algebras abstract dendriform structures arise naturally in algebraic combinatorics where they allow for example the splitting of the shuffle product into two pieces and through rota baxter algebra structures the latter appear among others in differential systems and in the renormalization process of pqft we prove new combinatorial identities in dendriform algebras that appear to be strongly related to classical phenomena such as the combinatorics of lyndon words rewriting rules in lie algebras or the fine structure of the malvenuto reutenauer algebra one of these identities is an abstract noncommutative dendriform generalization of the bohnenblust spitzer identity and of an identity involving iterated chen integrals due to c s lam

exponential renormalisation ii bogoliubov s r operation and momentum subtraction schemes this article aims at advancing the recently introduced exponential method for renormalisation in perturbative quantum field theory it is shown that this new procedure provides a meaningful recursive scheme in the context of the algebraic and group theoretical approach to renormalisation in particular we describe in detail a hopf algebraic formulation of bogoliubov s classical r operation and counterterm recursion in the context of momentum subtraction schemes this approach allows us to propose an algebraic classification of different subtraction schemes our results shed light on the peculiar algebraic role played by the degrees of taylor jet expansions especially the notion of minimal subtraction and oversubtractions

transformations on the sets of states and density operators in this paper we consider two physically relevant numerical quantities namely the concurrence and fidelity and an operation called product defined on the set of all quantum states or density operators our main aim is to present the complete descriptions of all bijective transformations which preserve any of these quantities or operation

representations of the q deformed algebras uq so2 1 and uq so3 1 representations of the algebra uq so2 1 are studied this algebra is a q deformation of the universal enveloping algebra u so2 1 of the lie algebra of the group so0 2 1 and differs from the quantum algebra uq su1 1 classifications of irreducible representations and of infinitesimally unitary irreducible representations of uq so2 1 with simple and discrete spectrum of one of generators are derived the set of infinitesimally unitary representations of uq so2 1 does not coincide with that for the algebra uq su1 1 the sets of irreducible representations and of infinitesimally unitary irreducible representations of the algebra uq so3 1 are given we also consider representations of uq son 1 which are of class 1 with respect to the subalgebra uq son

brst quantization of quasisymplectic manifolds and beyond we consider a class of factorizable poisson brackets which includes almost all reasonable poisson structures a particular case of the factorizable brackets are those associated with symplectic lie algebroids the brst theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case this can be viewed as an extension of the fedosov deformation quantization to a wide class of irregular poisson structures in a more general case the factorizable poisson brackets are shown to be closely connected with the notion of n algebroid a simple description is suggested for the geometry underlying the factorizable poisson brackets based on construction of an odd poisson algebra bundle equipped with an abelian connection it is shown that the zero curvature condition for this connection generates all the structure relations for the n algebroid as well as a generalization of the yang baxter equation for the symplectic structure

automorphisms of hilbert space effect algebras let h be a hilbert space and e h the effect algebra on h a bijective map is called an ortho order automorphism of e h if for every we have and the classical theorem of ludwig states that every such is of the form for some unitary or antiunitary operator u it is also known that each bijective map on e h preserving order and coexistency in both directions is of the same form can we improve these two theorems by relaxing the bijectivity assumption and or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion for both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results

the hopf algebra approach to feynman diagram calculations the hopf algebra structure underlying feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed recent progress is briefly summarized with an emphasis on further directions of research

renormalization in quantum field theory and the riemann hilbert problem ii the β function diffeomorphisms and the renormalization group we showed in part i that the hopf algebra of feynman graphs in a given qft is the algebra of coordinates on a complex infinite dimensional lie group g and that the renormalized theory is obtained from the unrenormalized one by evaluating at 0 the holomorphic part of the riemann hilbert decomposition 1 of the loop g provided by dimensional regularization we show in this paper that the group g acts naturally on the complex space x of dimensionless coupling constants of the theory more precisely the formula g 0 gz 1 z 3 3 2 for the effective coupling constant when viewed as a formal power series does define a hopf algebra homomorphism between the hopf algebra of coordinates on the group of formal diffeomorphisms to the hopf algebra this allows first of all to read off directly without using the group g the bare coupling constant and the renormalized one from the riemann hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms this shows that renormalization is intimately related with the theory of non linear complex bundles on the riemann sphere of the dimensional regularization parameter it also allows to lift both the renormalization group and the function as the asymptotic scaling in the group g this exploits the full power of the riemann hilbert decomposition together with the invariance of under a change of unit of mass this not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group g for the full higher pole structure of minimal subtracted counterterms in terms of the residue

from dynamical systems to renormalization in this paper we study logarithmic derivatives associated to derivations on completed graded lie algebra as well as the existence of inverses these logarithmic derivatives when invertible generalize the exp log correspondence between a lie algebra and its lie group such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms these concepts stemming from the theory of dynamical systems can be rephrased in the abstract setting of lie algebra and the same difficulties as in perturbative quantum field theory pqft arise here surprisingly one can adopt the same ideas as in pqft with fruitful results such as new constructions of normal forms with the help of the birkhoff decomposition the analogy goes even further locality of counter terms choice of a renormalization scheme and shall lead to more interactions between dynamical systems and quantum field theory

characterization of sum automorphisms and jordan triple automorphisms of quantum probabilistic maps in quantum mechanics it is often important for the representation of a quantum system to study the structure preserving bijective maps of the system such maps are also called isomorphisms or automorphisms in this paper using the uhlhorn type of wigner s theorem we show that both sum automorphisms and jordan triple automorphisms of the unit ball of density operators are implemented by either unitary or anti unitary operators of the underlying hilbert space

preservers on hilbert space effects in this paper we study a new class of transformations on the set of all hilbert space effects this consists of the bijective maps which preserve the order and zero product in both directions the main result of the paper gives a complete description of the structure of those transformations as applications we obtain additional new results and some former ones as easy corollaries in particular we obtain the form of the ortho order automorphisms as well as that of the sequential automorphisms in the last paragraph of the paper we show that the automorphisms of these two kinds belong to our class of transformations even when their domain is the set of all effects in a general von neumann algebra

towards cohomology of renormalization bigrading the combinatorial hopf algebra of rooted trees the renormalization of quantum field theory twists the antipode of a noncocommutative hopf algebra of rooted trees decorated by an infinite set of primitive divergences the hopf algebra of undecorated rooted trees r generated by a single primitive divergence solves a universal problem in hochschild cohomology it has two nontrivial closed hopf subalgebras the cocommutative subalgebra ladder of pure ladder diagrams and the connes moscovici noncocommutative subalgebra cm of noncommutative geometry these three hopf algebras admit a bigrading by n the number of nodes and an index k that specifies the degree of primitivity in each case we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of n and k and infer a simple generating procedure for the remainder the results for ladder are familiar from the theory of partitions while those for cm involve novel transforms of partitions most beautiful is the bigrading of r the largest of the three thanks to sloane s superseeker we discovered that it saturates all possible inequalities we prove this by using the universal hochschild closed one cocycle b which plugs one set of divergences into another and by generalizing the concept of natural growth beyond that entailed by the connes moscovici case we emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory

characterizations of the automorphisms of hilbert space effect algebras in this paper we characterize the automorphisms of hilbert space effect algebras by means of their preserving properties which concern certain relations and quantities appearing in quantum measurement theory

linear maps on the space of all bounded observables preserving maximal deviation abstract in this paper we determine all the bijective linear maps on the space of bounded observables which preserve a fixed moment or the variance nonlinear versions of the corresponding results are also presented

isometries of the spaces of self adjoint traceless operators abstract in this paper we describe the structure of all isometries of the space of self adjoint traceless operators on a finite dimensional hilbert space under the metrics coming from the operator norm or the schatten norms we prove that up to a translation and multiplication by 1 they are unitary or antiunitary similarity transformations

free tri dendriform family algebras abstract in this paper we first prove that a rota baxter family algebra indexed by a semigroup amounts to an ordinary rota baxter algebra structure on the tensor product with the semigroup algebra we show that the same phenomenon arises for dendriform and tridendriform family algebras then we construct free dendriform family algebras in terms of typed decorated planar binary trees finally we generalize typed decorated rooted trees to typed valently decorated schroder trees and use them to construct free tridendriform family algebras

order automorphisms of hilbert space effect algebras the two dimensional case it is well known that the order automorphisms of the effect algebra of a hilbert space of dimension not less than three are implemented by unitary or antiunitary operators the aim of this paper is to show that the same assertion also holds true in the two dimensional case

fedosov quantization of lagrange finsler and hamilton cartan spaces and einstein gravity lifts on co tangent bundles we provide a method of converting lagrange and finsler spaces and their legendre transforms to hamilton and cartan spaces into almost kahler structures on tangent and cotangent bundles in particular cases the hamilton spaces contain nonholonomic lifts of pseudo riemannian einstein metrics on effective phase spaces this allows us to define the corresponding fedosov operators and develop deformation quantization schemes for nonlinear mechanical and gravity models on lagrange and hamilton fedosov manifolds

spitzer s identity and the algebraic birkhoff decomposition in pqft in this paper we continue to explore the notion of rota baxter algebras in the context of the hopf algebraic approach to renormalization theory in perturbative quantum field theory we show in very simple algebraic terms that the solutions of the recursively defined formulae for the birkhoff factorization of regularized hopf algebra characters i e feynman rules naturally give a non commutative generalization of the well known spitzer s identity the underlying abstract algebraic structure is analysed in terms of complete filtered rota baxter algebras

on operators preserving commutativity abstract let l x be the algebra of all bounded operators on a non trivial complex banach space x and f l x l x a bijective linear operator such that f and f 1 both send commuting pairs of operators into commuting pairs then either f a uau 1 p a i or f a ua u 1 p a i where p is a linear functional on l x u is a bounded linear bijective operator between the appropriate two spaces is a complex constant and a is the adjoint of a the form of an operator f for which f and f 1 both send projections of rank one into projections of rank one is also determined

a geometric characterization of invertible quantum measurement maps a geometric characterization is given for invertible quantum measurement maps denote by s h the convex set of all states i e trace 1 positive operators on hilbert space h with dimh and 1 2 the line segment joining two elements 1 2 in s h it is shown that a bijective map s h s h satisfies 1 2 1 2 for any 1 2 s if and only if has one of the following forms m m tr m m or m tm tr m tm where m is an invertible bounded linear operator and t is the transpose of with respect to an arbitrarily fixed orthonormal basis

hilbert and thompson isometries on cones in jb algebras hilbert s and thompson s metric spaces on the interior of cones in jb algebras are important examples of symmetric banach finsler spaces in this paper we characterize the hilbert s metric isometries on the interiors of cones in jbw algebras and the thompson s metric isometries on the interiors of cones in jb algebras these characterizations generalize work by bosche on the hilbert s and thompson s metric isometries on symmetric cones and work by hatori and molnar on the thompson s metric isometries on the cone of positive self adjoint elements in a unital c algebra to obtain the results we develop a variety of new geometric and jordan algebraic techniques

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

on left hopf algebras within the framework of inhomogeneous quantum groups for particle algebras we deal with some matters needed to construct concrete left hopf algebras for inhomogeneous quantum groups produced as noncommutative symmetries of fermionic and bosonic creation annihilation operators we find a map for the bidimensional fermionic case produced as in manin s quantum groups and non commutative hopf geometry crm univ de montreal 1988 seminal work named preantipode that fulfills all the necessary requirements to be left but not right on the generators of the algebra due to the complexity and importance of the full task we consider our result as an important step that will be extended in the near future

order preserving bijections of c x i abstract let x be a compact hausdorff space which satisfies the first axiom of countability i 0 1 and c x i the set of all continuous functions from x to i in the paper we will give a description of bijective maps on c x i which preserve the order in both directions

representations of the quantum algebra uq su1 1 all irreducible representations of the quantum group uq su1 1 are given they are determined by two complex numbers infinitesimally unitary representations are separated from the set of irreducible representations it is shown that the symmetric operators of irreducible infinitesimally unitary representations corresponding to generators of the lie algebra su1 1 admit self adjoint extensions

lax pair equations and connes kreimer renormalization we find a lax pair equation corresponding to the connes kreimer birkhoff factorization of the character group of a hopf algebra this flow preserves the locality of counterterms in particular we obtain a flow for the character given by feynman rules and relate this flow to the renormalization group flow

birkhoff type decompositions and the baker campbell hausdorff recursion we describe a unification of several apparently unrelated factorizations arising from quantum field theory vertex operator algebras combinatorics and numerical methods in differential equations the unification is given by a birkhoff type decomposition that was obtained from the baker campbell hausdorff formula in our study of the hopf algebra approach of connes and kreimer to renormalization in perturbative quantum field theory there we showed that the birkhoff decomposition of connes and kreimer can be obtained from a certain baker campbell hausdorff recursion formula in the presence of a rota baxter operator we will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for lie algebras that arose in vertex operator algebra and conformal field theory and the even odd decomposition of combinatorial hopf algebra characters as well as the lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations

spectra eigenvectors and overlap functions for representation operators of q deformed algebras operators of representations corresponding to symmetric elements of theq deformed algebrasuq su1 1 uq so2 1 uq so3 1 uq son and representable by jacobi matrices are studied closures of unbounded symmetric operators of representations of the algebrasuq su1 1 anduq so2 1 are not selfadjoint operators for representations of the discrete series their deficiency indices are 1 1 bounded symmetric operators of these representations are trace class operators or have continuous simple spectra eigenvectors of some operators of representations are evaluated explicitly coefficients of transition to eigenvectors overlap coefficients are given in terms ofq orthogonal polynomials it is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq deformed algebras

order automorphisms of the set of bounded observables let h be a complex hilbert space and denote by bs h the set of all self adjoint bounded linear operators on h in this article we describe the form of all bijective maps no linearity or continuity is assumed on bs h which preserve the order in both directions