connes moscovici characteristic map is a lie algebra morphism let h be a hopf algebra with a modular pair in involution character 1 let a be a module algebra over h equipped with a non degenerated character invariant 1 trace tau we show that connes moscovici characteristic map varphi tau hc character 1 h rightarrow hc lambda a is a morphism of graded lie algebras we also have a morphism phi of batalin vilkovisky algebras from the cotorsion product of h text cotor h bbbk bbbk to the hochschild cohomology of a hh a a let k be both a hopf algebra and a symmetric frobenius algebra suppose that the square of its antipode is an inner automorphism by a group like element then this morphism of batalin vilkovisky algebras phi text cotor k vee mathbb f mathbb f cong text ext k mathbb f mathbb f hookrightarrow hh k k is injective

discovering multiple lyapunov functions for switched hybrid systems in this paper we analyze local asymptotic stability of switched hybrid systems whose subsystems have polynomial vector fields by discovering multiple lyapunov functions in quadratic forms we start with an algebraizable sufficient condition for the existence of quadratic multiple lyapunov functions then since different discrete modes are considered we apply real root classification together with a projection operator to underapproximate this sufficient condition step by step arriving at a set of semialgebraic sets which only involve the coefficients of the pre assumed multiple lyapunov function note that for each step we additionally use the information on the different discrete modes to optimize our intermediate computation results afterward we compute a sample point in the corresponding semialgebraic set for the coefficients resulting in a multiple lyapunov function finally we test our approach on some examples using a prototypical implementation and compare it with the generic quantifier e

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

homological stability for topological chiral homology of completions abstract by proving that several new complexes of embedded disks are highly connected we obtain several new homological stability results our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed e n algebras using this we prove a special case of a conjecture of vakil and wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

non commutative batalin vilkovisky algebras homotopy lie algebras and the courant bracket we consider two different constructions of higher brackets first based on a grassmann odd nilpotent operator we define a non commutative generalization of the higher koszul brackets which are used in a generalized batalin vilkovisky algebra and we show that they form a homotopy lie algebra secondly we investigate higher so called derived brackets built from symmetrized nested lie brackets with a fixed nilpotent lie algebra element q we find the most general jacobi like identity that such a hierarchy satisfies the numerical coefficients in front of each term in these generalized jacobi identities are related to the bernoulli numbers we suggest that the definition of a homotopy lie algebra should be enlarged to accommodate this important case finally we consider the courant bracket as an example of a derived bracket we extend it to the big bracket of exterior forms and multi vectors and give closed formulas for the higher courant brackets

reduction of jacobi manifolds a reduction procedure for jacobi manifolds is described in the algebraic setting of jacobi algebras as applications reduction by arbitrary submanifolds distributions and the reduction of jacobi manifolds with symmetry are discussed this generalized reduction procedure extends the well known reduction procedures for symplectic poisson contact and co symplectic structures

symplectic homology product via legendrian surgery this research announcement continues the study of the symplectic homology of weinstein manifolds undertaken by the authors bourgeois f ekholm t eliashberg y 2009 arxiv 0911 0026 where the symplectic homology as a vector space was expressed in terms of the legendrian homology algebra of the attaching spheres of critical handles here we express the product and batalin vilkovisky operator of symplectic homology in that context

on the deformation complex of homotopy affine actions abstract an affine action of an associative algebra a on a vector space v is an algebra morphism a v end v where v is a vector space and v end v is the algebra of affine transformations of v the one dimensional version of the swiss cheese operad denoted sc 1 is the operad whose algebras are affine actions of associative algebras this operad is koszul and admits a minimal model denoted by sc 1 algebras over this minimal model are called homotopy affine actions they consist of an a morphism a v end v where a is an a algebra in this paper we prove a relative version of deligne s conjecture in other words we show that the deformation complex of a homotopy affine action has the structure of an algebra over an sc 2 operad that structure is naturally compatible with the e 2 structure on the deformation complex of the a algebra

a homotopy theoretic realization of string topology let m be a closed oriented manifold of dimension d let lm be the space of smooth loops in m in 2 chas and sullivan defined a product on the homology h lm of degree d they then investigated other structure that this product induces including a batalin vilkovisky structure and a lie algebra structure on the s1 equivariant homology h s 1 lm these algebraic structures as well as others came under the general heading of the string topology of m in this paper we will describe a realization of the chas sullivan loop product in terms of a ring spectrum structure on the thom spectrum of a certain virtual bundle over the loop space we also show that an operad action on the homology of the loop space discovered by voronov has a homotopy theoretic realization on the level of thom spectra this is the cactus operad defined in 6 which is equivalent to operad of framed disks in this operad action realizes the chas sullivan bv structure on h lm we then describe a cosimplicial model of this ring spectrum and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the hochschild cohomology hh c m c m

n 2 structures in string theories the brst cohomology of any topological conformal field theory admits the structure of a batalin vilkovisky algebra and string theories are no exception loosely speaking we say that two topological conformal field theories are cohomologically equivalent if their brst cohomologies are isomorphic as batalin vilkovisky algebras in this paper we argue that any string theory regardless of the matter background is cohomologically equivalent to some twisted n 2 superconformal field theory we discuss three string theories in detail the bosonic string the nsr string and the w3 string in each case the way the cohomological equivalence is constructed can be understood as coupling the topological conformal field theory to topological gravity these results lend further supporting evidence to the conjecture that any topological conformal field theory is cohomologically equivalent to some topologically twisted n 2 superconformal field theory we end the paper with some speculative comments on massey products i

gerstenhaber algebras and bv algebras in poisson geometry the purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well known algebraic structures such as gerstenhaber algebras and bv algebras some applications are discussed in particular we find an explicit connection between the koszul brylinski operator and the modular class of a poisson manifold as a consequence we prove that poisson homology is isomorphic to poisson cohomology for unimodular poisson structures

bv structure on hochschild cohomology of the group ring of quaternion group of order eight in characteristic two abstract let k be an algebraically closed field of characteristic two and let q 8 be the quaternion group of order 8 we determine the gerstenhaber lie algebra structure and the batalin vilkovisky structure on the hochschild cohomology ring of the group algebra k q 8

jacobi structures revisited jacobi algebroids i e graded lie brackets on the grassmann algebra associated with a vector bundle which satisfy a property similar to that of the jacobi brackets are introduced they turn out to be equivalent to generalized lie algebroids in the sense of iglesias and marrero jacobi bialgebroids are defined in the same manner a lifting procedure of elements of this grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding lie brackets is developed this gives the possibility of associating canonically a lie algebroid with any local lie algebra in the sense of kirillov

homotopy classification of bosonic string field theory we prove the decomposition theorem for the loop homotopy lie algebra of quantum closed string field theory and use it to show that closed string field theory is unique up to gauge transformations on a given string background and given s matrix for the theory of open and closed strings we use results in open closed homotopy algebra to show that the space of inequivalent open string field theories is isomorphic to the space of classical closed string backgrounds as a further application of the open closed homotopy algebra we show that string field theory is background independent and locally unique in a very precise sense finally we discuss topological string theory in the framework of homotopy algebras and find a generalized correspondence between closed strings and open string field theories

open closed homotopy algebra in mathematical physics in this paper we discuss various aspects of open closed homotopy algebras ochas presented in our previous paper inspired by zwiebach s open closed string field theory but that first paper concentrated on the mathematical aspects here we show how an ocha is obtained by extracting the tree part of zwiebach s quantum open closed string field theory we clarify the explicit relation of an ocha with kontsevich s deformation quantization and with the b models of homological mirror symmetry an explicit form of the minimal model for an ocha is given as well as its relation to the perturbative expansion of open closed string field theory we show that our open closed homotopy algebra gives us a general scheme for deformation of open string structures a algebras by closed strings l algebras

new perspectives on the brst algebraic structure of string theory motivated by the descent equation in string theory we give a new interpretation for the action of the symmetry charges on the brst cohomology in terms of what we callthe gerstenhaber bracket this bracket is compatible with the graded commutative product in cohomology and hence gives rise to a new class of examples of what mathematicians call agerstenhaber algebra the latter structure was first discussed in the context of hochschild cohomology theory 11 off shell in the chiral brst complex all the identities of a gerstenhaber algebra hold up to homotopy applying our theory to thec 1 model we give a precise conceptual description of the brst gerstenhaber algebra of this model we are led to a direct connection between the bracket structure here and the anti bracket formalism in bv theory 29 we then discuss the bracket in string backgrounds with both the left and the right movers we suggest that the homotopy lie algebra arising from our gerstenhaber bracket is closely related to the hla recently constructed by witten zwiebach finally we show that our constructions generalize to any topological conformal field theory

from zwiebach invariants to getzler relation we introduce the notion of zwiebach invariants that generalize gromov witten invariants and homotopical algebra structures we outline the induction procedure that induces the structure of zwiebach invariants on the sub bicomplex that gives the structure of gromov witten invariants on sub bicomplex with zero differentials we propose to treat hodge dgbv with 1 12 axiom as the simplest set of zwiebach invariants and explicitly prove that it induces wdvv and getzler equations in genera 0 and 1 respectively

on operad structures of moduli spaces and string theory we construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy lie algebras gravity algebras and batalin vilkovisky algebras these algebras appeared recently in the context of string theory and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory

real solution isolation using interval arithmetic we propose a complete algorithm for real solution isolation for semialgebraic systems by using interval arithmetic the algorithm is implemented as a maple program nrealzero and its performance on several examples is reported

minimal models of quantum homotopy lie algebras via the bv formalism using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral

the frobenius properad is koszul we show koszulness of the prop governing involutive lie bialgebras and also of the props governing non unital and unital counital frobenius algebras solving a long standing problem this gives us minimal models for their deformation complexes and for deformation complexes of their algebras which are discussed in detail using an operad of graph complexes we prove with the help of an earlier result of one of the authors that there is a highly non trivial action of the grothendieck teichm uller group grt 1 on completed versions of the minimal models of the properads governing lie bialgebras and involutive lie bialgebras by automorphisms as a corollary one obtains a large class of universal deformations of any involutive lie bialgebra and any frobenius algebra parameterized by elements of the grothendieck teichm uller lie algebra we also prove that for any given homotopy involutive lie bialgebra structure in a vector space there is an associated homotopy batalin vilkovisky algebra structure on the associated chevalley eilenberg complex

a class of cubic differential systems with 6 tuple focus abstract this paper presents a class of cubic differential systems with the origin as a 6 tuple focus from which 6 limit cycles may be constructed for this class of differential systems the stability of the origin is given

weyl n algebras we introduce weyl n algebras and show how their factorization complex may be used to define invariants of manifolds in the appendix we heuristically explain why these invariants must be perturbative chern simons invariants

homotopy algebras inspired by classical open closed string field theory we define a homotopy algebra associated to classical open closed strings we call it an open closed homotopy algebra ocha it is inspired by zwiebach s open closed string field theory and also is related to the situation of kontsevich s deformation quantization we show that it is actually a homotopy invariant notion for instance the minimal model theorem holds also we show that our open closed homotopy algebra gives us a general scheme for deformation of open string structures a algebras by closed strings l algebras

background independent algebraic structures in closed string field theory we construct a batalin vilkovisky bv algebra on moduli spaces of riemann surfaces this algebra is background independent in that it makes no reference to a state space of a conformal field theory conformal theories define a homomorphism of this algebra to the bv algebra of string functionals the construction begins with a graded commutative free associative algebrac built from the vector space whose elements are orientable subspaces of moduli spaces of punctured riemann surfaces the typical element here is a surface with several connected components the operation of sewing two punctures with a full twist is shown to be an odd second order derivation that squares to zero it follows thatc is a batalin vilkovisky algebra we introduce the odd operator where is the boundary operator it is seen thatc and that consistent closed string vertices define a cohomology class of this cohomology class is used to construct a lie algebra on a quotient space ofc this lie algebra gives a manifestly background independent description of a subalgebra of the closed string gauge algebra

infinite loop spaces from operads with homological stability abstract motivated by the operad built from moduli spaces of riemann surfaces we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions we prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces the recent strong homological stability results of galatius and randal williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability as a consequence diffeomorphism groups and mapping class groups are shown to give rise to infinite loop spaces furthermore the map to k theory defined by the action of the diffeomorphisms on the middle dimensional homology is shown to be a map of infinite loop spaces

isolating the real roots of the piecewise algebraic variety the piecewise algebraic variety as a set of the common zeros of multivariate splines is a kind of generalization of the classical algebraic variety in this paper we present an algorithm for isolating the zeros of the zero dimensional piecewise algebraic variety which is primarily based on the interval zeros of univariate interval polynomials numerical example illustrates that the proposed algorithm is flexible

infinitesimal operations on complexes of graphs in two seminal papers kontsevich used a construction called graph homology as a bridge between certain infinite dimensional lie algebras and various topological objects includ ing moduli spaces of curves the group of outer automorphisms of a free group and invariants of odd dimensional manifolds in this paper we show that kontsevich s graph complexes which include graph complexes studied earlier by culler and vogtmann and by penner have a rich algebraic structure we define a lie bracket and cobracket on graph complexes and in fact show that they are batalin vilkovisky algebras and therefore gerstenhaber algebras we also find natural subcomplexes on which the bracket and cobracket are compatible as a lie bialgebra kontsevich s graph complex construction was generalized to the context of operads by ginzburg and kapranov with later generalizations by getzler kapranov and markl in cov we show that kontsevich s results in fact extend to general cyclic operads for some operads including the examples associated to moduli space and outer automorphism groups of free groups the subcomplex on which we have a lie bi algebra structure is quasi isomorphic to the entire con nected graph complex in the present paper we show that all of the new algebraic operations canonically vanish when the homology functor is applied and we expect that the resulting con straints will be useful in studying the homology of the mapping class group finite type manifold invariants and the homology of out f n