segre product of artin schelter regular algebras of dimension 2 and embeddings in quantum p3 s the segre embedding of p1 p1as a smooth quadricqin p3corresponds to the surjection of the four dimensional polynomial ring onto the segre productsof two copies of the homogeneous coordinate ring of p1 we study segre products of noncommutative algebras if in particularaandbare two copies of a quantum p1thens i ai kbi is a twisted homogeneous coordinate ring of the quadricq the main result of this paper is the classification of all embeddings of the segre product of two quantum planes into so called quantum p3 s these are the proj of artin schelter regular algebrasrof global dimension four with the hilbert series of a commutative polynomial ring and which map ontos ifris not a twist of a polynomial ring then the point scheme ofreither is the union of the quadricqwith a line or is only the quadricq in the first case ris a central extension of a three dimensional artin schelter regular algebra and a twist of an algebra mapping onto the commutative homogeneous coordinate ring ofq in the second case such an algebraris the first known example of a four dimensional artin schelter regular algebra which is not determined by its point scheme

the fundamental group of a hopf linear category abstract we define the fundamental group of a hopf algebra over a field for this purpose we first consider gradings of hopf algebras and galois coverings the latter are given by linear categories with new additional structure which we call hopf linear categories over a finite group we compare this invariant to the fundamental group of the underlying linear category and we compute those groups for families of examples

flows and stochastic taylor series in ito calculus for general stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the ito flow map is given the computation relies on the lift to quasi shuffle algebras of formulas involving products of ito integrals of semimartingales whereas the chen strichartz formula computing the logarithm of the stratonovich flow map is classically expanded as a formal sum indexed by permutations the analogous formula in ito calculus is naturally indexed by surjections this reflects the change of algebraic background involved in the transition between the two integration theories lastly we extend our formula for the quasi shuffle chen strichartz series for the logarithm of the flow map to the non commutative case for linear matrix valued sdes driven by arbitrary semimartingales we obtain a similar formula

twisted descent algebras and the solomon tits algebra abstract the notion of descent algebra of a bialgebra is lifted to the barratt joyal setting of twisted bialgebras the new twisted descent algebras share many properties with their classical counterparts for example there are twisted analogs of classical lie idempotents and of the peak algebra moreover the universal twisted descent algebra is equipped with two products and a coproduct and there is a fundamental rule linking all three this algebra is shown to be naturally related to the geometry of the coxeter complex of type a

lie theory for hopf operads the present article takes advantage of the properties of algebras in the category of s modules twisted algebras to investigate further the fine algebraic structure of hopf operads we prove that any hopf operad p carries naturally the structure of a comonoid in the category of twisted p algebras many properties of hopf algebraic structures are then shown to be encapsulated in this remarquable structure in particular various classical theorems of the theory of free lie algebras are lifted to arbitrary hopf operads

agreement between two independent groups of raters we propose a coefficient of agreement to assess the degree of concordance between two independent groups of raters classifying items on a nominal scale this coefficient defined on a population based model extends the classical cohen s kappa coefficient for quantifying agreement between two raters weighted and intraclass versions of the coefficient are also given and their sampling variance is determined by the jackknife method the method is illustrated on medical education data which motivated the research

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

parametric tests for agreement amongst groups of judges summary series approximations to the logarithm of the probability of a permutation of the natural numbers 1 k using an extension of the thurstone model are investigated the asymptotic bias of the quadratic and cubic approximations using the normal density is shown to be small approximate likelihood ratio tests are considered for testing for agreement between and within groups of judges and x2 approximations seem adequate for significance levels

a class of nonparametric tests based on multiresponse permutation procedures summary a class of univariate rank tests based on multiresponse permutation procedures is introduced asymptotic equivalence is established between one member of this class and the kruskal wallis test simulated power comparisons for location shift detection indicate that another member of the class provides superior detection efficiency for location shifts of bimodal distributions and of heavy tailed unimodal distributions furthermore both tests possess substantial computational advantages over other members of the class

growth of graded twisted calabi yau algebras abstract we initiate a study of the growth and matrix valued hilbert series of n graded twisted calabi yau algebras that are homomorphic images of path algebras of weighted quivers generalizing techniques previously used to investigate artin schelter regular algebras and graded calabi yau algebras several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras we give particular attention to twisted calabi yau algebras of dimension d 3 giving precise descriptions of their matrix valued hilbert series and partial results describing which underlying quivers yield algebras of finite gk dimension for d 2 we show that these are algebras with mesh relations for d 3 we show that the resulting algebras are a kind of derivation quotient algebra arising from an element that is similar to a twisted superpotential

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

regularity of algebras related to the sklyanin algebra this paper continues the research of ss by finding further examples of artin schelter regular rings of dimension four unlike the threedimensional case studied in atv1 2 these examples show that the fourdimensional regular rings are not uniquely determined by their associated geometric data indeed we find a one parameter family of regular algebras associated to this data

lie idempotent algebras abstract besides the inner product inherited from each symmetric group algebra ksn on the direct sum ks nksn a second convolution product arises from the hopf algebra structure on ks for any subalgebra a of ks for both the inner and the convolution product it is shown that the lie idempotent algebra l a of a generated by all lie idempotents in a with respect to the convolution product is also closed under inner products its nth homogeneous component l n a is thus a subalgebra of ksn for all n provided that a contains a lie idempotent in ksk for all k n solomon s descent algebra d n is a subalgebra of l n a in this case a number of results on the structure of l n a is derived including a description of its jacobson radical its principal indecomposables and its cartan invariants furthermore solomon s epimorphism from d n onto the ring of class functions clk sn of sn extends to an epimorphism of algebras c n l n a cl k s n additional results are obtained concerning a bialgebra structure of l a in the particular case of a n d n this amounts to a new approach to several well known results on d n

hyperoctahedral chen calculus for effective hamiltonians abstract we tackle the problem of unraveling the algebraic structure of computations of effective hamiltonians this is an important subject in view of applications to chemistry solid state physics or quantum field theory we show among other things that the correct framework for these computations is provided by the hyperoctahedral group algebras we define several structures on these algebras and give various applications for example we show that the adiabatic evolution operator in the time dependent interaction representation of an effective hamiltonian can be written naturally as a picard type series and has a natural exponential expansion

embedding a quantum nonsingular quadric in a quantum p3 a definition of regularity has been given for non commutative graded algebras and results of artin schelter tate and van den bergh classify the regular algebras of global dimension three that are generated by degree one elements our purpose is to classify a certain class of quadratic regular algebras of global dimension four letsbe a twisted homogeneous coordinate ring of a nonsingular quadricq p3 our interest is in algebrasrsuch that there is an embedding proj s proj r in this paper we classify all the quadratic regular algebrasrof global dimension four which have the same hilbert series as that of the polynomial ring on four variables and which map ontosvia a graded degree zero homomorphism our approach makes use of the point modules ofrand their associated geometric data we classify the algebrasraccording to their point scheme pand corresponding automorphism aut p those algebrasrwhich are determined by p belong to at most a five parameter family but those which are not determined by p belong to at most a four parameter family in the first case pis either p3or consists ofqtogether with a linel while in the second casep q it is also proved that under certain sufficient conditions the zero locus of the defining relations of a quadratic regular algebra of global dimension four is the graph of an automorphism

homologically homogeneous rings in this paper we study the structure of a right noetherian ring r of finite right global dimesion integral over a central subring c and satisfying the following condition if v w are irreducible right r modules with rc v rc w then pr dim v

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

on koszul algebras and a new construction of artin schelter regular algebras abstract we prove the simple fact that the factor ring of a koszul algebra by a regular normal quadratic element is a koszul algebra this fact leads to a new construction of quadratic artin schelter regular algebras this construction generalizes the construction of artin schelter regular clifford algebras

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

dualizing complexes over noncommutative graded algebras the motivation for the work presented in this paper was two questions posed by m artin let a be a d dimensional regular graded algebra over the field k with augmentation ideal m see def 4 13 denote by hi a resp hi a the dth left resp right local cohomology of a at the ideal m i e hi a lim extd a m a and similarly for hi a as left modules and as right modules hi a and h a are both isomorphic to a hom a k up to a twist in degrees all operations are taking place in the graded category

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

non commutative cohen macaulay rings abstract we introduce a concept of cohen macaulayness for left noetherian semilocal rings and their modules which generalizes the corresponding notion of commutative algebra and naturally applies to orders

some noncommutative minimal surfaces abstract in the ongoing programme to classify noncommutative projective surfaces connected graded noetherian domains of gelfand kirillov dimension three a natural question is to determine the minimal models within any birational class in this paper we show that the generic noncommutative projective plane corresponding to the three dimensional sklyanin algebra as well as noncommutative analogues of p 1 p 1 and the more general van den bergh quadrics satisfy very strong minimality conditions translated into an algebraic question where one is interested in a maximality condition we prove the following result theorem a let r be a sklyanin algebra or a van den bergh quadric that is infinite dimensional over its centre and let a r be any connected graded noetherian maximal order with the same graded quotient ring as r then up to taking veronese rings a is isomorphic to r let t be an elliptic algebra that is the coordinate ring of a noncommutative surface containing an elliptic curve then under an appropriate homological condition we prove that every connected graded noetherian overring of t is obtained by blowing down finitely many lines line modules of self intersection 1

down up algebras and their representation theory abstract a class of algebras called down up algebras was introduced by g benkart and t roby 1998 j algebra 209 305 344 we classify the finite dimensional simple modules over noetherian down up algebras and show that in some cases every finite dimensional module is semisimple we also study the question of when two down up algebras are isomorphic

non commutative castelnuovo mumford regularity and as regular algebras abstract let a be a connected graded k algebra with a balanced dualizing complex we prove that a is a koszul as regular algebra if and only if that the castelnuovo mumford regularity and the ext regularity coincide for all finitely generated a modules this can be viewed as a non commutative version of t romer on the regularity over positively graded algebras j algebra 319 2008 1 15 theorem 1 3 by using castelnuovo mumford regularity we prove that any koszul standard as gorenstein algebra is as regular as a preparation to prove the main result we also prove that the following statements are equivalent 1 a is as gorenstein 2 a has finite left injective dimension 3 the dualizing complex has finite left projective dimension this generalizes i mori homological properties of balanced cohen macaulay algebras trans amer math soc 355 2002 1025 1042 corollary 5 9

diversity as a concept and its measurement abstract this paper puts forth the view that diversity is an average property of a community and identifies that property as species rarity an intrinsic diversity ordering of communities is defined and is shown to be equivalent to stochastic ordering also the sensitivity of an index to rare species is developed culminating in a crossing point theorem and a response theory to perturbations diversity decompositions analogous to the analysis of variance are discussed for two way classifications and mixtures the paper concludes with a brief survey of genetic diversity linguistic diversity industrial concentration and income inequality

skew group algebras of calabi yau algebras abstract the calabi yau property of skew group algebras is discussed it is shown that the skew group algebra a g of a koszul calabi yau algebra a with a finite subgroup g of automorphisms of a is calabi yau if and only if g is a finite subgroup of the special linear group sl a which is defined by means of the homological determinant using the a algebra structure on the yoneda algebra some results in bocklandt et al 2010 bsw are generalized say every connected graded p koszul calabi yau algebra is derived from a superpotential the superpotential for the skew group algebra a g is also constructed

a class of as regular algebras of dimension five abstract we classify 5 dimensional artin schelter regular algebras generated by two generators of degree 1 with three generating relations of degree 4 under a generic condition all the algebras obtained are proved to be strongly noetherian auslander regular and cohen macaulay with respect to the gelfand kirillov dimension

tree expansion in time dependent perturbation theory the computational complexity of time dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters combinatorial sequences goldstone and other feynman type diagrams etc we show that a very efficient perturbative expansion both for theoretical and numerical purposes can be obtained through an original parametrization by trees and generalized iterated integrals we emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics these tools are applied to the adiabatic approximation and the effective hamiltonian we prove perturbatively and nonperturbatively the convergence of morita s generalization of the gell mann and low wave function we show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms finally we recover the rayleigh schrodinger time independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree

a symmetry of the descent algebra of a finite coxeter group abstract the descent algebra d w of a finite coxeter group w discovered by solomon in 1976 is a subalgebra of the group algebra of w due to solomon it is intimately linked to the representation theory of w by means of a homomorphism of algebras mapping d w into the algebra of class functions of w for w of type a jollenbeck and reutenauer derived the identity x y y x for all x y d w where class functions of w have been extended to the group algebra of w linearly they conjectured that this symmetry property of d w holds for arbitrary finite coxeter groups w this conjecture actually a combinatorial refinement is proven here as a consequence several properties of the characters of w afforded by the primitive idempotents of d w may be derived at once including a symmetry of the corresponding character table and a combinatorial description of their intertwining numbers with the descent characters of w this recovers and extends results of gessel reutenauer and scharf thibon on the symmetric group and of poirier on the hyperoctahedral group

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

on the homology of associative algebras we present a new free resolution for k as an g module where g is an associative augmented algebra over a field k the resolution reflects the combinatorial properties of g introduction let k be a field and let g be an associative augmented k algebra for many purposes one wishes to have a projective resolution of k as a g module the bar resolution is always easy to define but it is often too large to use in practice at the other extreme minimal resolutions may exist but they are often hard to write down in a way that is amenable to calculations the main theorem of this paper presents a compromise resolution though rarely minimal it is small enough to offer some bounds but explicit enough to facilitate calculations as it relies heavily upon combinatorial constructions it is best suited for analyzing otherwise tricky algebras given via generators and relations since several results we get as consequences of the main theorem have been obtained before through other means this paper may be viewed as generalizing and unifying several seemingly unrelated ideas in particular we are generalizing priddy s results on koszul algebras 12 extending homology computations by govorov 9 and backelin 3 and complementing bergman s methods regarding the diamond lemma 6 three results may be of interest the homology of the mod p steenrod algebra is given in terms of the homology of a new chain complex smaller than the a algebra in theorem 3 5 formula 16 offers an efficient algorithm for the determination of hilbert series and theorem 4 2 asserts the existence of new bounds on the torsion groups of commutative graded rings 1 definitions and the main theorem throughout this paper k denotes any field and g is an associative k algebra with unity the field k embeds in g via 77 k g and we suppose that g has an augmentation i e a k algebra map s g k for which 77 is a right inverse s denotes a set of generators for g as a k algebra and k s is the free associative k algebra with unity on s there is a canonical surjection f k s g once s is chosen and the augmentation e is determined once we know s x for each x e s in particular this means that k s may be augmented such that f becomes a map of augmented algebras to s we associate a function e s z called a grading in the absence of a more compelling choice we often take e to be grading by length i e e x 1 for received by the editors may 23 1983 and in revised form february 22 1984 this paper was the subject of an invited one hour address in boulder colorado during the week long ams summer program on combinatorics and algebra june 1983 1980 mathematics subject classification 1985 revision primary 16a62 secondary 13d03 55s10 1986 american mathematical society 0002 9947 86 1 00 25 per page

double extension regular algebras of type 14641 abstract we construct several families of artin schelter regular algebras of global dimension four using double ore extension and then prove that all these algebras are strongly noetherian auslander regular koszul and cohen macaulay domains many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an ore extension of an artin schelter regular algebra of global dimension three

intergroup concordance definition and estimation summary a coefficient of intergroup concordance is proposed which is consistent with the concept of intragroup concordance as measured by kendall s coefficient of concordance kendall babington smith 1939 this approach reconciles the approaches of schucany frawley 1973 and of hollander sethuraman 1978 to the problem of two group concordance estimation and test procedures for the population are based on jackknife procedures extension to the problem of multiple intergroup concordance when groups have factorial structure is noted

homological properties of graded noetherian pi rings abstract let r be a connected graded noetherian pi ring if injdim r n r is auslander gorenstein and cohen macaulay with gelfand kirillov dimension equal to n if gldim r n r is a domain finitely generated as a module over its centre and a maximal order in its quotient division ring similar results hold if r is assumed to be local rather than connected graded alternatively suppose that r is a noetherian pi ring with gldim r r m 1 hd r m 2 for any two maximal ideals m i in the same clique then r is a direct sum of prime rings is integral over its centre and is auslander gorenstein if r is a prime ring then the centre z r of r is a krull domain and r equals its trace ring tr moreover hd r m height m for every maximal ideal m of r