going from cohomology to hochschild cohomology abstract the relation between ordinary cohomology and hochschild cohomology is investigated for quotients of quiver algebras which are either graded or finite dimensional it is shown that a resolution of the direct sum of the simple modules of the algebra can often be made two sided and yield a resolution of the algebra itself over its enveloping algebra the multiplicative structures of the cohomologies are shown to be related by a spectral sequence of algebras converging to the hochschild cohomology whose first term is given by a tensor product of the algebra itself with its cohomology algebra

minimal resolutions of algebras abstract a method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself the method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations that is either homogeneous or admissible with some additional finiteness restrictions in the latter case in particular it applies to any finite dimensional algebra over an algebraically closed field the method is illustrated by a number of examples viz truncated algebras monomial algebras and koszul algebras with the aim of unifying existing treatments of these in the literature

the poset metrics that allow binary codes of codimension m to be m m 1 or m 2 perfect a binary poset code of codimension m of cardinality 2n m where n is the code length can correct maximum m errors all possible poset metrics that allow codes of codimension m to be m m 1 or m 2 perfect are described some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived as examples we prove the nonexistence of r perfect poset codes for some r in the case of the crown poset and in the case of the union of disjoint chains

mellin transforms and asymptotics finite differences and rice s integrals high order differences of simple number sequences may be analysed asymptotically by means of integral representations residue calculus and contour integration this technique akin to mellin transform asymptotics is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries digital search trees quadtrees and distributed leader election

the binomial transform and the analysis of skip lists to any sequence of real numbers an n 0 we can associate another sequence s s 0 which knuth calls its binomial transform this transform is defined through the rule s bsan n 1 n s n an we study the properties of this transform obtaining rules for its manipulation and a table of transforms that allow us to invert many transforms by inspection we use these methods to perform a detailed analysis of skip lists a probabilistic data structure introduced by pugh as an alternative to balanced trees in particular we obtain the mean and variance for the cost of searching for the first or the last element in the list confirming results obtained previously by other methods and also for the cost of searching for a random element whose variance was not known we obtain exact solutions although not always in closed form from them we are able to find the corresponding asymptotic expressions

automatic average case analysis of algorithms abstract many probabilistic properties of elementary discrete combinatorial structures of interest for the average case analysis of algorithms prove to be decidable this paper presents a general framework in which such decision procedures can be developed it is based on a combination of generating function techniques for counting and complex analysis techniques for asymptotic estimations we expose here the theory of exact analysis in terms of generating functions for four different domains the iterative recursive and unlabelled labelled data type domains we then present some major components of the associated asymptotic theory and exhibit a class of naturally arising functions that can be automatically analyzed a fair fragment of this theory is also incorporated into a system called lambda upsilon omega in this way using computer algebra one can produce automatically non trivial average case analyses of algorithms operating over a variety of decomposable combinatorial structures at a fundamental level this paper is part of a global attempt at understanding why so many elementary combinatorial problems tend to have elementary asymptotic solutions in several cases it proves possible to relate entire classes of elementary combinatorial problems whose structure is well defined with classes of elementary special functions and classes of asymptotic forms relative to counting probabilities or average case complexity

mds poset codes satisfying the asymptotic gilbert varshamov bound in hamming weights we prove that mds linear poset codes satisfy gilbert varshamov bound for their hamming weights asymptotically we also construct mds linear poset codes on arbitrary poset metric spaces by using the dilworth s chain decomposition theorem and results about the hermite interpolation problem over a finite field we prove that there exist linear poset codes with large weights for both poset metrics and hamming metrics as well

the alternating syzygy behavior of monomial algebras abstract in this paper we present a minimal projective resolution of a monomial algebra over its enveloping algebra the syzygies for this resolution exhibit an alternating behavior which is explained by the construction of a special sequence of paths from the quiver of

homological invariants relating the super jordan plane to the virasoro algebra abstract nichols algebras are an important tool for the classification of hopf algebras within those with finite gk dimension we study homological invariants of the super jordan plane that is the nichols algebra a b v 1 2 these invariants are hochschild homology the hochschild cohomology algebra the lie structure of the first cohomology space which is a lie subalgebra of the virasoro algebra and its representations h n a a and also the yoneda algebra we prove that the algebra a is k 2 moreover we prove that the yoneda algebra of the bosonization a k z of a is also finitely generated but not k 2

metrics based on finite directed graphs and coding invariants given a finite directed graph with n vertices we define a metric d g on mathbb f q n where mathbb f q is the finite field with q elements the weight of a word is defined as the number of vertices that can be reached by a directed path from a vertex within the support of the vector two canonical forms which do not affect the metric are given to each graph based on these forms we characterize each such metric we further use these forms to prove that two graphs with different canonical forms yield different metrics efficient algorithms to check if a set of metric weights define a metric based on a graph are given we provide tight bounds on the number of metric weights required to reconstruct the metric furthermore we give a complete description of the group of linear isometries of the graph metrics and a characterization of the graphs for which every linear code admits a g canonical decomposition considering those graphs we are able to derive an expression of the packing radius of linear codes in such metric spaces finally given a directed graph which determines a hierarchical poset we present sufficient and necessary conditions to ensure the validity of the macwilliams identity and the macwilliams extension property

hochschild homology and cohomology of down up algebras abstract we present a detailed computation of the cyclic and the hochschild homology and cohomology of generic and 3 calabi yau homogeneous down up algebras this family was defined by benkart and roby in 3 in their study of differential posets our calculations are completely explicit by making use of the koszul bimodule resolution and some arguments similar to those used in 13 to compute the hochschild cohomology of yang mills algebras

a classification of posets admitting the macwilliams identity in this paper all poset structures that admit the macwilliams identity are classified and the macwilliams identities for poset weight enumerators corresponding to such posets are derived it is proved that being a hierarchical poset is a necessary and sufficient condition for a poset to admit the macwilliams identity an explicit relation is also derived between the p weight distribution of a hierarchical poset code and the p weight distribution of the dual code

noncommutative mirror symmetry for punctured surfaces in 2013 abouzaid auroux efimov katzarkov and orlov showed that the wrapped fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities we generalize this result to other punctured riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models in particular given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is mathtt a infty isomorphic to a subcategory of the wrapped fukaya category of a punctured riemann surface the connection between the dimer model and the punctured riemann surface then has a nice interpretation in terms of a duality on dimer models

projective resolutions of associative algebras and ambiguities abstract the aim of this article is to give a method to construct bimodule resolutions of associative algebras generalizing bardzell s well known resolution of monomial algebras we stress that this method leads to concrete computations providing thus a useful tool for computing invariants associated to the considered algebras we illustrate how to use it by giving several examples in the last section of the article in particular we give necessary and sufficient conditions for noetherian down up algebras to be 3 calabi yau

derived equivalences of gentle algebras via fukaya categories following the approach of haiden katzarkov kontsevich publ math inst hautes etudes sci 126 247 318 2017 to any homologically smooth mathbb z graded gentle algebra a we associate a triple sigma a lambda a eta a where sigma a is an oriented smooth surface with non empty boundary lambda a is a set of stops on partial sigma a and eta a is a line field on sigma a such that the derived category of perfect dg modules of a is equivalent to the partially wrapped fukaya category of sigma a lambda a eta a modifying arguments of johnson and kawazumi we classify the orbit decomposition of the action of the symplectic mapping class group of sigma a on the homotopy classes of line fields as a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent our criterion uses numerical invariants generalizing those given by avella alaminos geiss in avella et al j pure appl algebra 212 1 228 243 2008 as well as some other numerical invariants as an application we find many new cases when the aag invariants determine the derived morita class as another application we establish some derived equivalences between the stacky nodal curves considered in lekili and polishchuk j topology 11 615 444 2018

finiteness conditions on the ext algebra of a cycle algebra abstract let a be a finite dimensional algebra given by quiver and monomial relations in e l green d zacharia manuscripta math 85 1994 11 23 we see that the ext algebra of a is finitely generated only if all the ext algebras of certain cycle algebras overlying a are finitely generated here a cycle algebra is a finite dimensional algebra given by quiver and monomial relations where the quiver is an oriented cycle the main result of this paper gives necessary and sufficient conditions for the ext algebra of such a to be finitely generated this is achieved by defining a computable invariant of the smo tube we also give necessary and sufficient conditions for the ext algebra of to be noetherian

comparison morphisms and the hochschild cohomology ring of truncated quiver algebras abstract a main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and bardzell s minimal resolution for truncated quiver algebras over arbitrary fields tqa s as a direct application we describe explicitly the yoneda product and derive several results on the structure of the cohomology ring of tqa s over a field of characteristic zero for instance we show that the product of odd degree cohomology classes is always zero we prove that tqa s associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings on the other side we exhibit a fundamental example of a tqa with nontrivial cohomology ring finally for truncated polynomial algebras in one variable we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring

rigidity of truncated quiver algebras a quiver q is a finite oriented graph which can contain more than one arrow between two vertices as well as loops and oriented cycles for n a positive integer q is the set of oriented paths of length n of q where the length is the number of arrows of the oriented path notice that q is the set of vertices and q the set of arrows let k be a field and kqo x st po ks be the commutative semi simple algebra with q0 as a k basis of idempotents for each arrow a e q with source vertex s a s and end vertex r u t the one dimensional vector space k has an evident kq bimodule structure for u e q we have au 6 sa and ua d a where 6 is the kronecker symbol in this way kq o ka is a kq bimodule and the quiver algebra kq is the tensor algebra over kqo of the kq bimodule kqt of course kq can also be described as the vector space kqo kql 0 kq 0 where the multiplication of e qj and ci e qi is 3 e qj if t a s p and 0 otherwise let now n be a finite dimensional k algebra we suppose that i is morita reduced and that the endomorphism ring of each simple n module is k this is equivalent to a r k x x k were r is the jacobson radical of i by definition the set of vertices of the gabriel s quiver q of i is the set of isomorphism classes of simple i modules if s and t are simple n modules the number of arrows from s to t is dim exti s t by an observation of gabriel 6 7 4 3 every k algebra n such that a r k x x k admits a presentation that is an algebra surjection cp kq a whose kernel z verifies f c zc f2 where f is the two sided ideal of kq generated by ql and m is some positive integer such ideals z of a quiver algebra are called admissible in general an algebra n has not a unique presentation that is two different admissible two sided ideals z and j of a quiver algebra kq can give isomorphic k algebras kq z and kq j by definition we say that a k algebra i is a truncated quiver algebra if

on solvability of the first hochschild cohomology of a finite dimensional algebra for an arbitrary finite dimensional algebra a we introduce a general approach to determining when its first hochschild cohomology rm hh 1 a considered as a lie algebra is solvable if a is moreover of tame or finite representation type we are able to describe rm hh 1 a as the direct sum of a solvable lie algebra and a sum of copies of mathfrak sl 2 we proceed to determine the exact number of such copies and give an explicit formula for this number in terms of certain chains of kronecker subquivers of the quiver of a as a corollary we obtain a precise answer to a question posed by chaparro schroll and solotar

quivers from matrix factorizations we discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity this method also yields explicit geometrical interpretations of d branes i e quiver representations on a resolution given in terms of grassmannians as an example we analyze some non toric singularities which are resolved by a single mathbb p 1 but have length greater than one these examples have a much richer structure than conifolds a picture is proposed that relates matrix factorizations in landau ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions

morse theory from an algebraic viewpoint forman s discrete morse theory is studied from an algebraic viewpoint and we show how this theory can be extended to chain complexes of modules over arbitrary rings as applications we compute the homologies of a certain family of nilpotent lie algebras and show how the algebraic morse theory can be used to derive the classical anick resolution as well as a new two sided anick resolution