modules and matrices abstract noncommutative ring theory was described in terms of matrices in its earliest days we give some examples showing how matrix theory can now in turn profit from the theory of rings and modules

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

an explicit description of the second cohomology group of a quandle we use the inflation restriction sequence and a result of etingof and grana on the rack cohomology to give a explicit description of 2 cocycles of finite indecomposable quandles with values in an abelian group several applications are given

knot theory for self indexed graphs we introduce and study so called self indexed graphs these are oriented finite graphs endowed with a map from the set of edges to the set of vertices such graphs naturally arise from classical knot and link diagrams in fact the graphs resulting from link diagrams have an additional structure an integral flow we call a self indexed graph with integral flow a comte the analogy with links allows us to define transformations of comtes generalizing the reidemeister moves on link diagrams we show that many invariants of links can be generalized to comtes most notably the linking number the alexander polynomials the link group etc we also discuss finite type invariants and quandle cocycle invariants of comtes

from racks to pointed hopf algebras abstract a fundamental step in the classification of finite dimensional complex pointed hopf algebras is the determination of all finite dimensional nichols algebras of braided vector spaces arising from groups the most important class of braided vector spaces arising from groups is the class of braided vector spaces c x c q where x is a rack and q is a 2 cocycle on x with values in c racks and cohomology of racks appeared also in the work of topologists this leads us to the study of the structure of racks their cohomology groups and the corresponding nichols algebras we will show advances in these three directions we classify simple racks in group theoretical terms we describe projections of racks in terms of general cocycles we introduce a general cohomology theory of racks containing properly the existing ones we introduce a fourier transform on racks of certain type finally we compute some new examples of finite dimensional nichols algebras

new large rank nichols algebras over nonabelian groups with commutator subgroup z2 abstract in this article we explicitly construct new finite dimensional indecomposable nichols algebras with dynkin diagrams of type a n c n d n e 6 7 8 f 4 over any group g of order 2 l with commutator subgroup isomorphic to z 2 the construction is generic in the sense that the type just depends on the rank and center of g and thus positively answers for all groups of this class a question raised by susan montgomery in 1995 22 4 our construction uses the new notion of a covering nichols algebra as a special case of a covering hopf algebra 18 and produces non faithful nichols algebras we give faithful examples of doi twists for type a 3 c 3 d 4 c 4 f 4 over several nonabelian groups of order 16 and 32 these are hence the first known examples of nondiagonal finite dimensional indecomposable nichols algebras of rank 2 over nonabelian groups

quadratic forms and iterated tilted algebras given a finite quiver a without oriented cycles an algebra a is called an iterated tilted algebra of type a 2 if there exists a sequence of algebras a a a 1 a where a is the path algebra of a and a sequence of tilting modules t 0 and every indecomposable a module satisfies either hom j t m 0 or exta t m 0 if m 1 a is called a tilted algebra of type a 25 the representation theory of iterated tilted algebras was proved to be closely related to that of a class of symmetric algebras namely the trivial extension algebras see 3 4 18 27 291 recently they were also shown to arise naturally in the study of the derived category of a finite dimensional algebra see 23 24 71 iterated tilted algebras of type a where the underlying graph of a is a dynkin diagram were studied in 1 2 8 223 and the iterated tilted algebras of euclidean type a m 3 1 were classified in s 55 0021 8693 90 3 00

resolutions of mesh algebras periodicity and calabi yau dimensions a triangulated category mathcal t sigma is said to be calabi yau of dimension d if d is a serre functor we determine which stable module categories of self injective algebras of finite type are calabi yau and compute their calabi yau dimensions correcting errors in previous work we first show that the calabi yau property of mod can be detected in the minimal projective resolution of the stable auslander algebra of over its enveloping algebra we then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the generalized preprojective algebras of dynkin graphs for representation finite self injective algebras of torsion order t 1 we obtain a complete description of their stable calabi yau properties but only partial results for those algebras of torsion order t 2 we also obtain some new information about the periods of the representation finite self injective algebras of torsion order t 1 finally we describe how these questions can also be approached by realizing the stable categories of representation finite self injective algebras as orbit categories of the bounded derived categories of hereditary algebras and illustrate this technique with several explicit computations that our previous methods left unsettled

subquandles of affine quandles abstract a quandle will be called quasi affine if it embeds into an affine quandle our main result is a characterization of quasi affine quandles by group theoretic properties of their displacement group by a universal algebraic condition coming from the commutator theory and by an explicit construction over abelian groups as a consequence we obtain efficient algorithms for recognizing affine and quasi affine quandles and we enumerate small quasi affine quandles we also prove that the abelian implies quasi affine problem of universal algebra has affirmative answer for the class of quandles

partial order in the self dual ashkin teller model corner transfer matrices are used to calculate the order parameters of the self dual ashkin teller model on the square lattice in the non critical regime it is found that the model is partially ordered i e the magnetisations s1 and t1 vanish while the polarisation s1t1 is non zero the polarisation which is given simply in terms of elliptic functions exhibits an essential singularity at criticality

finite dimensional pointed hopf algebras over finite simple groups of lie type i non semisimple classes in psln q abstract we show that nichols algebras of most simple yetter drinfeld modules over the projective special linear group over a finite field corresponding to non semisimple orbits have infinite dimension we spell out a new criterium to show that a rack collapses

quantum integrability in systems with finite number of levels we consider the problem of defining quantum integrability in systems with finite number of energy levels starting from commuting matrices and construct new general classes of such matrix models with a given number of commuting partners we argue that if the matrices depend on a real parameter one can define quantum integrability from this feature alone leading to specific results such as exact solvability poissonian energy level statistics and to level crossings

hard squares with diagonal attractions the authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next nearest neighbour diagonal interactions on the special surface corresponding to regimes iii and iv of the generalised hard hexagon model the interfacial tension correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row to row and corner transfer matrices the associated critical exponents are found to be mu v 5 4 beta 3 32 in regime iii and mu v 5 2 beta 1 4 in regime iv in particular their results confirm the recent proposal by huse that regime iii is the first order coexistence surface separating the disordered fluid phase from the square ordered solid phases and that the regime iii regime iv boundary is a line of tricritical points

on the set theoretical yang baxter equation we propose a general way of constructing set theoretical solutions of the yangbaxter equation we study the properties of the construction we also show that our construction includes the earlier ones given by weinstein xu and etingof schedlersoloviev

graded artin algebras a graded artin algebra is a graded ring which neglecting the grading is an artin algebra although this definition has the merit of brevity it does not expose some of the basic properties of graded artin algebras for instance that there is a bound on absolute values of degrees of homogeneous elements indeed a different definition is given in section 1 and the discrepancy is rectified by theory 1 4 we admit that our chief interest in graded artin algebras lies in applications to the representation theory of artin algebras see 5 1 we feel nonetheless that graded artin algebras are interesting objects of study in their own right and some of our techniques especially in sections 5 and 6 are equally applicable to arbitrary graded rings thus this paper unlike its sequel 51 is designed to be read by a general ring theoretical audience it is in addition self contained from the standpoint of knowledge concerning graded rings required the major concern of the paper is the existence of gradings on given modules over a graded artin algebra we use the term gradable when a gradation exists our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable it follows immediately see section 3 that simple modules and projective modules over a graded artin algebra are gradable also inasmuch as we do not restrict ourselves to positively graded rings endomorphism rings of graded finitely generated modules over graded artin algebras are graded artin algebras in a similar vein duality and horn

stable equivalences of graded algebras abstract we extend the notion of stable equivalence to the class of locally finite graded algebras for such an algebra we focus on the krull schmidt category gr of finitely generated z graded modules with degree 0 maps and the stable category gr obtained by factoring out those maps that factor through a graded projective module we say that and are graded stably equivalent if there is an equivalence gr gr that commutes with the grading shift adapting arguments of auslander and reiten involving functor categories we show that a graded stable equivalence commutes with the syzygy operator where defined and preserves finitely presented modules as a result we see that if is right noetherian resp right graded coherent then so is any graded stably equivalent algebra furthermore if is right noetherian or k is artinian we use almost split sequences to show that a graded stable equivalence preserves finite length modules of particular interest in the nonartinian case we prove that any graded stable equivalence involving an algebra with soc 0 must be a graded morita equivalence

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

corner transfer matrices in statistical mechanics corner transfer matrices are a useful tool in the statistical mechanics of simple two dimensional models they can be a very effective way of obtaining series expansions of unsolved models and of calculating the order parameters of solved ones here we review these features and discuss the reason why the method fails to give the order parameter of the chiral potts model

covering theory for linear categories with application to derived categories abstract we extend the galois covering theory introduced by bongartz gabriel for skeletal linear categories to general linear categories we show that a galois covering between krull schmidt categories preserves irreducible morphisms and almost splits sequences specializing to derived categories we study when a galois covering between locally bounded linear categories induces a galois covering between the bounded derived categories of finite dimensional modules as an application we show that each locally bounded linear category with radical squared zero admits a gradable galois covering which induces a galois covering between the bounded derived categories of finite dimensional modules and a galois covering between the auslander reiten quivers of these bounded derived categories in a future paper this will enable us to obtain a complete description of the bounded derived category of finite dimensional modules over a finite dimensional algebra with radical squared zero

set theoretic solutions of the yang baxter equation and new classes of r matrices abstract we describe several methods of constructing r matrices that are dependent upon many parameters for example unitary r matrices and r matrices whose entries are functions as an application we construct examples of r matrices with prescribed singular values we characterise some classes of indecomposable set theoretic solutions of the quantum yang baxter equation qybe and construct r matrices related to such solutions in particular we establish a correspondence between one generator braces and indecomposable non degenerate involutive set theoretic solutions of the qybe showing that such solutions are abundant we show that r matrices related to involutive non degenerate solutions of the qybe have special form we also investigate some linear algebra questions related to r matrices

on the spectrum of six vertex corner transfer matrices we consider the eigenstates of the six vertex corner transfer matrix using its generator which is an operator of heisenberg xxz type in the ordered regime we construct a complete set of fundamental eigenstates and some bethe ansatz eigenstates as well in the disordered critical regime the spectrum of the finite size system collapses in the thermodynamic limit and we show how this depends on in m the size of the system and

nichols algebras over groups with finite root system of rank two iii abstract we study nichols algebras of direct sums of two absolutely simple yetter drinfeld modules over non abelian groups and arbitrary fields we construct families of new examples where the nichols algebra is finite dimensional and not of diagonal type

a covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

strongly simply connected schurian algebras and multiplicative bases abstract in this paper we define concepts of crowns and quasi crowns valid in an arbitrary schurian algebra and which generalise the corresponding concepts in an incidence algebra we show first that a triangular schurian algebra is strongly simply connected if and only if it is simply connected and contains no quasi crown we then prove that the absence of quasi crowns in a triangular schurian algebra implies the existence of a multiplicative basis

magnetization at corners in two dimensional ising models we investigate the corner spin magnetization of two dimensional ferromagnetic ising models in various wedge geometries results are obtained for triangular and square lattices by numerical studies on finite wedges using the star triangle transformation as well as analytic calculations using corner transfer matrices and a fermionic representation of the row to row transfer matrix the corner magnetizations vanish at the bulk critical temperature tc with an exponent c differing from the bulk exponent for isotropic systems with free edges we find that c is given simply by c 2 where is the angle at the corner for apex magnetizations of conical lattices we obtain the strikingly similar result a 4 these formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle eff determined by the relative strengths of the interactions

virtual yang baxter cocycle invariants we extend the yang baxter cocycle invariants for virtual knots by augmenting yang baxter 2 cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure these invariants coincide with the classical yang baxter cocycle invariants for classical knots but provide extra information about virtual knots and links in particular they provide a method for detecting non classicality of virtual knots and links

statistical mechanics a perspective on the work of r j baxter abstract the work of rodney baxter in statistical mechanics is surveyed with an emphasis on the role that his exact calculations have played in the development of the modern theory of critical phenomena

structure optimization to enhance its natural frequencies based on measured frequency response functions abstract a measured frequency response function frf based structural modification method is presented to obtain optimal structural changes to enhance its natural frequencies structural dynamics modification sdm has been widely used for improvements of built in structures however the optimum design obtained by sdm differs from the true optimal solution when large modal changes happen in this paper a substructure coupling concept is used to get system equations in order to extend its use to large modal changes frf matrix of baseline structure and those of modification structures are coupled at the connection points under the condition of force equilibrium and geometric compatibility constraints thus exact modified modal properties can be calculated even for the case of large modal changes the optimal structural modification is calculated by combining eigenvalue sensitivities and eigenvalue reanalysis technique iteratively special attention is given to the case where baseline structure has some unidentified structural parameters to enlighten the advantage of this proposed method an application to the case of beam stiffener optimization indicates that the proposed method can provide an accurate optimal structural change just based on measured frfs without any kind of numerical models

a freeness theorem for nichols algebras abstract we prove a freeness result for nichols algebras over certain nichols subalgebras this result allows us in particular to classify pointed hopf algebras of prime and prime squared index

corner transfer matrices abstract for two dimensional lattice models with interactions only between nearest and diagonally nearest neighbour spins a well known concept is the row to row transfer matrix less well known is the corner transfer matrix ctm this has some very useful properties if it is normalized so that its largest eigenvalue is unity and the eigenvalues are arranged in numerically decreasing order then each eigenvalue tends to a limit as the lattice becomes large for those models which have been solved exactly notably the ising eight vertex and hard hexagon models this limiting eigenvalue distribution is very simple being basically that of a direct product of two by two matrices from it the order parameter can easily be obtained for all models one can write down formally exact matrix relations for the ctm but the matrices are of infinite size if one uses a representation in which the ctm is diagonal and then truncates these relations to finite size then one obtains a quite accurate approximation the larger the size the greater the accuracy i g enting and i have thereby obtained comparatively long series expansions for the ising model in a field and for the hard squares model

corner transfer matrices and corner magnetization for the ising model the authors consider the corner magnetization of an ising square lattice with free edges along a diagonal direction using some remarkable properties of the corner transfer matrix they give a unified treatment for various opening angles the calculation of the magnetization is reduced to the solution of a matrix equation in the thermodynamic limit they construct an explicit form of this equation and discuss its numerical and asymptotic solution and the results obtained thereby

graphs with relations coverings and group graded algebras the paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs to include gradings of all basic finite dimensional algebras over an algebraically closed field a theory of coverings of graphs with relations is introduced the object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra the linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology group theory or combinatorics in the representation theory of artin algebras the association to each algebra of a finite directed graph called the quiver of the algebra has been a useful tool the reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra 1 gives a slight extension of these concepts to finitely generated algebras the main emphasis of the paper is to show that the theory of coverings of graphs with relations introduced by c riedtmann 9 and expanded by p gabriel 2 and the theory of group graded algebras are essentially the same although the original connection between coverings and gradings was inspired by the similarity of results for z graded artin algebras 3 4 and p gabriel s announced results 2 the context of this paper is more general and deals with all finitely generated algebras over a field we associate to each such algebra a finite directed graph which we still call a quiver of the algebra we show that for each regular covering t of a quiver t0 of an algebra a with certain prescribed restrictions we get a g grading of the algebra a where g is the automorphism group of the covering y over ro conversely given a certain type of g grading of a where g is a group we construct a regular covering y of the quiver t0 of the algebra such that g is isomorphic to the automorphism group of t over ro furthermore if y is a regular covering of t0 with automorphism group g we show that the category of representations of y satisfying a certain set of relations is equivalent to the category of finite dimensional graded g modules we

a geometry constraint handling technique for stiffener layout optimization problem abstract beam stiffeners have frequently been used for raising natural frequencies of base structures in stiffener layout optimization problems most of the previous researches considering the position and or the length of the stiffener as design variables dealt with structures having just simple convex shapes such as a square or rectangle the reason was because concave shape structures have difficulties in formulating geometry constraints in this paper a new geometry constraint handling technique which can define both convex and concave feasible regions and measure a degree of geometry constraint violation was proposed evolution strategies ess was utilized as an optimization tool in addition the constraint handling technique of evoslinoc evolution strategies for scalar optimization with linear and nonlinear constraints was utilized to solve constrained optimization problems from numerical examples the proposed geometry constraint handling technique was verified and proved that the technique can easily be applied to structures in not only convex but also concave shapes even with a protrusion or interior holes

quandle cohomology and state sum invariants of knotted curves and surfaces the 2 twist spun trefoil is an example of a sphere that is knotted in 4 dimensional space a proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed our proof is based on a state sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles also known as distributive groupoids a quandle is a set with a binary operation the axioms of which model the reidemeister moves in classical knot theory colorings of diagrams of knotted curves and surfaces by quandle elements together with cocycles of quandles are used to define state sum invariants for knotted circles in 3 space and knotted surfaces in 4 space cohomology groups of various quandles are computed herein and applied to the study of the state sum invariants non triviality of the invariants is proved for a variety of knots and links and conversely knot invariants are used to prove non triviality of cohomology for a variety of quandles

the classifying space of a bound quiver abstract we associate to a bound quiver q i a cw complex which we denote by b q i and call the classifying space of q i we show that the fundamental group of b q i is isomorphic to the fundamental group of q i moreover we show that this construction behaves well with respect to coverings on the other hand we study the co homology groups of b q i and compare them with the simplicial and the hochschild co homology groups of the algebra a kq i more precisely we give sufficient conditions for these groups to be isomorphic this generalizes a theorem due to gerstenhaber and schack j pure appl algebra 30 1983 143 156

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

finite dimensional pointed or copointed hopf algebras over affine racks abstract we study the pointed or copointed liftings of nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal yd realization of these racks in the copointed case we complete the classification for the six affine racks whose nichols algebra is known to be of finite dimension in the pointed case our method allows us to finish four of them in all of the cases the hopf algebras obtained turn out to be cocycle deformations of their associated graded hopf algebras all of them are new examples of finite dimensional copointed or pointed hopf algebras over non abelian groups

on the classification of finite dimensional pointed hopf algebras we classify finite dimensional complex hopf algebras a which are pointed that is all of whose irreducible comodules are one dimensional and whose group of group like elementsg a is abelian such that all prime divisors of the order of g a are 7 since these hopf algebras turn out to be deformations of a natural class of generalized small quantum groups our result can be read as an axiomatic description of generalized small quantum groups

classification of arithmetic root systems arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property they can also be considered as generalizations of ordinary root systems with rich structure and many new examples on the other hand nichols algebras are fundamental objects in the construction of quantized enveloping algebras in the noncommutative differential geometry of quantum groups and in the classification of pointed hopf algebras by the lifting method of andruskiewitsch and schneider in the present paper arithmetic root systems are classified in full generality as a byproduct many new finite dimensional pointed hopf algebras are obtained

on hopf algebras with positive bases we show that if a finite dimensional hopf algebra h over c has a basis with respect to which all the structure constants are nonnegative then h is isomorphic to the bi cross product hopf algebra constructed by takeuchi and majid from a finite group g and a unique factorization g g g of g into two subgroups we also show that hopf algebras in the category of finite sets with correspondences as morphisms are classified in a similar way our results can be used to explain some results on hopf algebras from the set theoretical point of view

divisibility relations for the dimensions and hilbert series of nichols algebras of non abelian group type abstract we present a divisibility relation for the dimensions and hilbert series of certain classes of nichols algebras of non abelian group type which generalizes nichols algebras over coxeter groups with constant cocycle 1 for this we introduce three groups of isomorphisms acting on nichols algebras which generalizes the exchange operator introduced by milinski and schneider 2000 18 for coxeter groups

homology groups of symmetric quandles and cocycle invariants of links and surface links we introduce the notion of a quandle with a good involution and its homology groups carter et al defined quandle cocycle invariants for oriented links and oriented surface links by use of good involutions quandle cocyle invariants can be defined for links and surface links which are not necessarily oriented or orientable the invariants can be used in order to estimate the minimal triple point numbers of non orientable surface links

finite quantum groups and cartan matrices abstract we consider a class of hopf algebras having as an invariant a generalized cartan matrix for a hopf algebra of this kind we prove that it is finite dimensional if and only if its generalized cartan matrix is actually a finite cartan matrix under some mild hypothesis these results allow us to classify all the finite dimensional coradically graded pointed hopf algebras whose coradical has odd prime dimension p we also characterize coradically graded pointed hopf algebras of order p 4

coverings of graded pointed hopf algebras we introduce the concept of a covering of a graded pointed hopf algebra the theory developed shows that coverings of a bosonized nichols algebra can be concretely expressed by biproducts using a quotient of the universal coalgebra covering group of the nichols algebra if there are enough quadratic relations the universal coalgebra covering is given by the bosonization by the enveloping group of the underlying rack

corner transfer matrices of the chiral potts model ii the triangular lattice we consider a two dimensional edge interaction model satisfying the star triangle relations for the triangular lattice the corner transfer matrices are functions of three rapidities we show that they possess various factorization properties and satisfy certain equations we indicate how these equations can be solved for the ising model we then consider the three state chiral potts model and obtain low temperature solutions to the equations the conjectured formula for the order parameter the spontaneous magnetization is verified to one more order in a series expansion

the emerging role of number theory in exactly solvable models in lattice statistical mechanics abstract we review the baxter model the hard hexagon model and their multistate generalizations from a point of view that stresses the connection with modular functions and additive number theory it is shown for example that various physical quantities in the hard hexagon model are all expressible in terms of modular functions the use of rogers ramanujan type identities in solvable models is also reviewed

representation theory of graded artin algebras having initiated the study of graded artin algebras a in 91 here we initiate the study of their representation theory our point of view is to study graded i modules which we believe to be somewhat more tractable than ungraded ones in order to obtain information about all i modules this point of view leads to the introduction in section 1 of the full subcategory mod of mod i consisting of the gradable objects of mod i that is the finitely generated i modules which support a gradation in these terms one of the major results of the paper asserts that mod has finite representation type if and only if mod i has finite representation type thus in section 3 we introduce a number g g 4 designed to measure the size of mod i if g co we show that i has infinite representation type if g co we show that there is a graded artin algebra q of a certain specified form such that i has infinite representation type precisely when 0 has infinite representation type the artin algebra fl has in particular desirable diagrammatic properties and these we will exploit elsewhere we speculate that when g is finite every finitely generated i module is gradable now in case n has finite representation type it is obvious that g is finite we show indeed that for every gradation of a given artin algebra of finite representation type every module is gradable this result and the others cited are chiefly consequences of a result proved in section 4 if a component of the auslander reiten graph of a graded artin algebra contains a gradable module then the component

on galois coverings and tilting modules abstract let a be a basic connected finite dimensional algebra over an algebraically closed field k let t be a basic tilting a module with arbitrary finite projective dimension for a fixed group g we compare the set of isoclasses of connected galois coverings of a with group g and the set of isoclasses of connected galois coverings of end a t with group g using the hasse diagram k a see d happel l unger on a partial order of tilting modules algebr represent theory 8 2 2005 147 156 and c riedtmann a schofield on a simplicial complex associated with tilting modules comment math helv 66 1 1991 70 78 of basic tilting a modules we give sufficient conditions on t under which there is a bijection between these two sets these conditions are always verified when a is of finite representation type then we apply these results to study when the simple connectedness of a implies the one of end a t see i assem e n marcos j a de la pena the simple connectedness of a tame tilted algebra j algebra 237 2 2001 647 656 finally using an argument due to w crawley boevey we prove that the type of any simply connected tilted algebra is a tree and that its first hochschild cohomology group vanishes

the weyl groupoid of a nichols algebra of diagonal type the theory of nichols algebras of diagonal type is known to be closely related to that of semi simple lie algebras in this paper the connection between both theories is made closer for any nichols algebra of diagonal type invertible transformations are introduced which remind one of the action of the weyl group on the root system associated to a semi simple lie algebra they give rise to the definition of a groupoid as an application an alternative proof of classification results of rosso andruskiewitsch and schneider is obtained without using any technical assumptions on the braiding

homology operations on homology of quandles abstract we introduce the concept of the quandle partial derivatives and use them to define extreme chains that yield homological operations we apply this to a large class of finite and infinite quandles to show in particular that they have nontrivial elements in the third and fourth quandle homology

the structure of medial quandles abstract medial quandles are represented using a heterogeneous affine structure as a consequence we obtain numerous structural properties including enumeration of isomorphism classes of medial quandles up to 13 elements

ring theoretic aspects of quandles abstract we associate to every quandle x and an associative ring with unity k a nonassociative ring k x following 3 the basic properties of such rings are investigated in particular under the assumption that the inner automorphism group i n n x acts orbit 2 transitively on x a complete description of right or left ideals is provided the complete description of right ideals for the dihedral quandles r n is given it is also shown that if for two quandles x and y the inner automorphism groups act 2 transitively and k x is isomorphic to k y then the quandles are of the same partition type however we provide examples when the quandle rings k x and k y are isomorphic but the quandles x and y are not isomorphic these examples answer some open problems in 3

group gradings on full matrix rings we study g gradings of the matrix ring mn k k a field and give a complete description of the gradings where all the elements ei j are homogeneous called good gradings among these we determine the ones that are strong gradings or crossed products if g is a finite cyclic group and k contains a primitive g th root of 1 we show how all g gradings of mn k can be produced in particular we give a precise description of all c2 gradings of m2 k and show that for algebraically closed k any such grading is isomorphic to one of the two good gradings

evolutionary constrained optimization using an addition of ranking method and a percentage based tolerance value adjustment scheme one of the most important issues in developing an evolutionary optimization algorithm is the proper handling of any constraints on the problem one must balance the objective function against the degree of constraint violation in such a way that neither is dominant common approaches to strike this balance include implementing a penalty function and the more recent stochastic ranking method but these methods require an extra tuning parameter which must be chosen by the user the present paper demonstrates that a proper balance can be achieved using an addition of ranking method through the ranking of the individuals with respect to the objective function and constraint violation independently we convert these two properties into numerical values of the same order of magnitude this removes the requirement of a user specified penalty coefficient or any other tuning parameters direct addition of the ranking terms is then performed to integrate all information into a single decision variable this approach is incorporated into a m l evolution strategy and tested on thirteen benchmark problems one engineering design problem and five difficult problems with a high dimensionality or many constraints the performance of the proposed strategy is similar to that of the stochastic ranking method when dealing with inequality constraints but it has a much simpler ranking procedure and does not require any tuning parameters a percentage based tolerance value adjustment scheme is also proposed to enable feasible search when dealing with equality constraints

homology of left non degenerate set theoretic solutions to the yang baxter equation abstract this paper deals with left non degenerate set theoretic solutions to the yang baxter equation lnd solutions a vast class of algebraic structures encompassing groups racks and cycle sets to each such solution there is associated a shelf i e a self distributive structure which captures its major properties we consider two co homology theories for lnd solutions one of which was previously known in a reduced form for biracks only an explicit isomorphism between these theories is described for groups and racks we recover their classical co homology whereas for cycle sets we get new constructions for a certain type of lnd solutions including quandles and non degenerate cycle sets the co homologies split into the degenerate and the normalized parts we express 2 cocycles of our theories in terms of group cohomology and in the case of cycle sets establish connexions with extensions this leads to a construction of cycle sets with interesting properties

galois coverings of pointed coalgebras we introduce the concept of a galois covering of a pointed coalgebra the theory developed shows that galois coverings of pointed coalgebras can be concretely expressed by smash coproducts using the coaction of the automorphism group of the covering thus the theory of galois coverings is seen to be equivalent to group gradings of coalgebras an advantageous feature of the coalgebra theory is that neither the grading group nor the quiver is assumed finite in order to obtain a smash product coalgebra

corner transfer matrices for the ising model we consider corner transfer matrices for the regular ising model and show that finding a diagonal representation is a soluble problem in interacting fermions we show how the requirement of having an invertible diagonalising transformation determines which representation must be used and explain why baxter s original solution of the problem did not produce such an invertible transformation

pointed hopf algebras over some sporadic simple groups abstract we show that every finite dimensional complex pointed hopf algebra with group of group likes isomorphic to a sporadic group is a group algebra except for the fischer group fi 22 the baby monster and the monster for these three groups we give a short list of irreducible yetter drinfeld modules whose nichols algebra is not known to be finite dimensional

nichols algebras of group type with many quadratic relations abstract we classify nichols algebras of irreducible yetter drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two all such nichols algebras are finite dimensional and all known finite dimensional nichols algebras of nonabelian group type appear in the result of our classification we find a new finite dimensional nichols algebra over fields of characteristic two

a simple algorithm for localized construction of non matching structural interfaces a simple and effective algorithm for the modular construction of non matched interfaces is presented for the partitioned solution of large scale structural problems the formulation is based on a recently developed four field variational principle which introduces a connection frame between the interfaced partitions a key result of the present study is a frame nodal placement criterion that uniquely determines the frame discretization into piecewise linear elements so that the interface patch test condition is satisfied a priori the method is demonstrated with several 2d and 3d example problems

on nichols algebras over sl 2 fq and gl 2 fq we compute necessary conditions on yetter drinfeld modules over the groups sl 2 fq and gl 2 fq to generate finite dimensional nichols algebras this is a first step toward a classification of pointed hopf algebras with a group of grouplikes isomorphic to one of these groups

the off critical integrable ashkin teller model the exact solution of an ashkin teller model with layer and spatial anisotropy and z2 z2 symmetry is presented the model is related to the eight vertex model by a sublattice or orbifold duality and it has the same free energy and magnetisation the exact solution manifold is a three dimensional submanifold of the full six dimensional thermodynamic space it admits a z3 symmetry relating three regimes each exhibiting different partial order in the spatially isotropic case these regimes are separated by three lines of continuously varying critical behaviour which coalesce at the four state potts critical point the magnetisations and polarisation are derived using corner transfer matrices giving rise to the appearance of chiral and virasoro characters the associated critical exponents confirm the known values in particular one obtains the value xsigma 1 8 for the magnetic scaling dimension and identify the next to leading thermal field

on a factorization of graded hopf algebras using lyndon words abstract we find a generalization of the restricted pbw basis for pointed hopf algebras over abelian groups constructed by kharchenko we obtain a factorization of the hilbert series for a wide class of graded hopf algebras these factors are parametrized by lyndon words and they are the hilbert series of certain graded hopf algebras