multi hamiltonian structure for the finite defocusing ablowitz ladik equation we study the poisson structure associated to the defocusing ablowitz ladik equation from a functional analytical point of view by reexpressing the poisson bracket in terms of the associated caratheodory function using this expression we are able to introduce a family of compatible poisson brackets that form a multi hamiltonian structure for the ablowitz ladik equation furthermore we show using some of these new poisson brackets that the geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the ablowitz ladik and toda hierarchies 2008 wiley periodicals inc

integral bilinear forms coxeter transformations and coxeter polynomials of finite posets abstract linear algebra technique in the study of linear representations of finite posets is developed in the paper a concept of a quadratic wandering on a class of posets i is introduced and finite posets i are studied by means of the four integral bilinear forms b i b i b i b i z i z i z 1 1 the associated coxeter transformations and the coxeter polynomials in connection with bilinear forms of dynkin diagrams extended dynkin diagrams and irreducible root systems are also studied bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of dynkin diagrams and extended dynkin diagrams are discussed a homological interpretation of the bilinear forms 1 1 is given and z bilinear equivalences between them are discussed by applying well known results of bongartz loupias and zavadskij shkabara we give several characterisations of posets i with the euler form q i x b i x x weakly positive resp with the reduced euler form q i x b i x x weakly positive and posets i with the tits form q i x b i x x weakly positive

numeric and mesh algorithms for the coxeter spectral study of positive edge bipartite graphs and their isotropy groups we develop algorithmic techniques for the coxeter spectral analysis of the class ubigr n of connected loop free positive edge bipartite graphs d with n 2 vertices i e signed graphs in particular we present numerical and graphical algorithms allowing us a computer search in the study of such graphs d by means of their gram matrix email protected d the complex spectrum specc email protected c of the coxeter matrix cox d email protected email protected email protected d t r and the geometry of weyl orbits in the set mor d d of matrix morsifications email protected m n z of a simply laced dynkin diagram email protected a n d n e 6 e 7 e 8 associated with d and mesh root systems of type email protected our algorithms construct the coxeter gram polynomials cox d t z t and mesh geometries of root orbits of small connected loop free positive edge bipartite graphs d we apply them to the study of the following coxeter spectral analysis problem does email protected email protected hold i e the email protected email protected d arez congruent for any pair of connected positive loop free edge bipartite graphs d d inubigr nsuch thatspecc d specc d the problem if any square integer matrix email protected m n z is z congruent with its transpose a t r is also discussed we present a solution for graphs in ubigr n with email protected 6

algebras with separating almost cyclic coherent auslander reiten components we prove that the class of artin algebras whose auslander reiten quiver admits a separating family of almost cyclic coherent components coincides with the class of generalized multicoil enlargements of concealed canonical algebras moreover the module category homological properties and the representation type of artin algebras with separating families of almost cyclic coherent auslander reiten components are described

periodicity in bilinear lattices and the coxeter formalism abstract we introduce and study in detail so called circulant coxeter periodic elements and circulant families in a bilinear lattice k as well as their dual versions called anti circulant we show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the coxeter polynomial k of k and in consequence of coxeter polynomials of algebras of finite global dimension we discuss the properties of quadratic forms induced by circulant and anti circulant families moreover we interpret the results in the language of representation theory of algebras and point out applications facts concerning tubular families in auslander reiten quivers and quadratic forms of algebras abstract considerations in bilinear lattices are illustrated with a collection of non trivial examples arising from module and derived categories the results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts

hereditary categories with tilting object let k be an algebraically closed field and h a connected abelian k category which is hereditary that is ext2 vanishes on h assume also that hom x y and ext1 x y are finite dimensional vector spaces over k for all x and y in h we consider such hereditary categories h which in addition have a tilting object t that is an object t such that x ext1 t x 0 fact the factors of finite direct sums of copies of t hereditary categories h with a tilting object t are of special interest in connection with the construction of the class of algebras called quasitilted algbras which was introduced in hrs1 they are by definition the algebras of the form endh t op an important property is that h and endh t op have equivalent bounded derived categories the main examples of such hereditary categories are the category modh of finitely generated modules over a finite dimensional hereditary k algebra h and the category cohx of coherent sheaves on a weighted projective line in the sense of gl1 there are also others derived equivalent to them because of the simple description of the corresponding bounded derived category db h it is possible to give a description of those in the same derived equivalence class see ls h2 it follows from hre that they automatically have a tilting object the quasitilted algebras provide common generalization for the tilted and the canonical finite dimensional algebras note that the tilted algebras are those coming from modh using an arbitrary

on schur flows for the finite schur dmkdv flows a non local poisson structure is introduced and shown to be linked via backlund darboux transformations to linear and quadratic poisson structures for the toda lattice two different lax representations for the schur flows are used one to construct backlund darboux transformations and the other to solve the cauchy problem via the trigonometric moment problem

poisson geometry of the analog of the miura maps and backlund darboux transformations for equations of toda type and periodic toda flows we show that the analog of the miura maps and backlund darboux transformations for a general class of equations of toda type and for a generalized class of periodic toda flows are isomorphisms of poisson lie groups

cmv the unitary analogue of jacobi matrices we discuss a number of properties of cmv matrices by which we mean the class of unitary matrices studied recently by cantero moral and velazquez we argue that they play an equivalent role among unitary matrices to that of jacobi matrices among all hermitian matrices in particular we describe the analogues of well known properties of jacobi matrices foliation by co adjoint orbits a natural symplectic structure algorithmic reduction to this shape lax representation for an integrable lattice system ablowitz ladik and the relation to orthogonal polynomials as offshoots of our analysis we will construct action angle variables for the finite ablowitz ladik hierarchy and describe the long time behavior of this system 2006 wiley periodicals inc

the spectrum of a coxeter transformation affine coxeter transformations and the defect map abstract the spectrum of the coxeter transformation r in a reflection group defined by a matrix a without odd cycles is determined in terms of spec a for affine a r determines the defect map which is given a simple formulation in terms of null roots the affine exponents determined by spec r are related to the coefficients of the fundamental co weight corresponding to the fork node of the coxeter dynkin diagram

nonabelian nonlinear lattice equations on finite interval we apply the inverse spectral problem method to the class of non abelian nonlinear lattice equations on the finite interval the integrable discrete nonlinear schrodinger and discrete modified korteveg de vries equations are considered as examples in the latter case the large time asymptotics for solutions are found

categorification of a linear algebra identity and factorization of serre functors we provide a categorical interpretation of a well known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras as a consequence we deduce that the serre functor of a finite dimensional triangular algebra a has always a lift up to shift to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of a

variational principles for immersed surfaces with l 2 bounded second fundamental form in this work we present new fundamental tools for studying the variations of the willmore functional of immersed surfaces into r m this approach gives for instance a new proof of the existence of a willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension we explain how the same approach can solve constraint minimization problems for the willmore functional we show in particular that for a given closed surface and a given conformal class for this surface there is an immersion in r m away possibly from isolated branched points which minimizes the willmore energy among all possible lipschitz immersions in r m having an l 2 bounded second fundamental form and realizing this conformal class this branched immersion is either a smooth conformal willmore branched immersion or an isothermic branched immersion we show that branched points do not exist whenever the minimal energy in the conformal class is less than 8 and that these immersions extend to smooth conformal willmore embeddings or global isothermic embeddings of the surface in that case finally as a by product of our analysis we establish that inside a compact subspace of the moduli space the following holds weak limit of palais smale willmore sequences are conformal willmore that weak limits of palais smale sequences of conformal willmore are either conformal willmore or global isothermic and finally we observe also that weakly converging palais smale sequences of global isothermic immersions are global isothermic the analysis developped along the paper in particular these last results opens the door to the possibility of constructing new critical saddle points of the willmore functional without or with constraints using min max methods

the strong global dimension of piecewise hereditary algebras let a be a finite dimensional piecewise hereditary algebra over an algebraically closed field this text investigates the strong global dimension of a this invariant is characterised in terms of the lengths of sequences of tilting mutations relating a to a hereditary abelian category in terms of the generating hereditary abelian subcategories of the derived category of a and in terms of the auslander reiten structure of that derived category

on the periodicity of coxeter transformations and the non negativity of their euler forms abstract we show that for piecewise hereditary algebras the periodicity of the coxeter transformation implies the non negativity of the euler form contrary to previous assumptions the condition of piecewise heredity cannot be omitted even for triangular algebras as demonstrated by incidence algebras of posets we also give a simple direct proof that certain products of reflections defined for any square matrix a with 2 on its main diagonal and in particular the coxeter transformation corresponding to a generalized cartan matrix can be expressed as a 1 a t where a a are closely associated with the upper and lower triangular parts of a

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

on spectral radii of coxeter transformations let t be a finite tree and dt the maximal branching degree of its vertices let t be the spectral radius of a coxeter transformation defined by t t hen t 4dt 6

wild canonical algebras canonical algebras p depending on a weight sequence p p1 pt of positive integers and a parameter sequence 3 t of pairwise distinct non zero elements from the base eld k were introduced and studied in 23 the nite dimensional representation theory mod is completely controlled by the category coh x of coherent sheaves on a non singular weighted projective line x x p since the derived categories d mod and d coh x are equivalent as triangulated categories 9 the curve x attached to has virtual genus gx 1 2 t 2 p t i 1 p pi where p l c m p1 pt the complexity of the classi cation problem for coh x and hence for mod is essentially determined by gx for gxi1 the algebra is concealed of extended dynkin type accordingly the classi cation problem for coh x and mod is equivalent to the classi cation of indecomposable modules over a tame hereditary algebra cf 18 9 or according to 10 to the classi cation problem for cohen macaulay modules over a simple surface singularity for gx 1 the algebra is of tubular type its representation theory is known from 23 while the classi cation problem for coh x 9 20 relates to atiyah s classi cation 1 of vector bundles over an elliptic curve very little was known on mod or coh x if gx 1 in this case 0 r is the one point extension of a wild hereditary algebra 0 accordingly the category mod 0 embeds into coh x as a full exact and extension closed subcategory implying that the classi cation problems for coh x and mod are wild from now on we assume gx 1 combining the sheaf theory 9 with the spectral investigation of wild hereditary algebras 6 we obtain a fairly complete picture on the structure of auslander reiten components for mod the asymptotic behaviour of the auslander reiten translations and

invariant theory of canonical algebras abstract based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations this is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces together with the quotient morphisms from the corresponding representation spaces for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory by means of a tilting process we extend these results to the invariant theory of concealed canonical algebras thus covering the cases of tame hereditary tame concealed and tubular algebras respectively our approach yields in particular a uniform treatment to an essential part of the invariant theory of extended dynkin quivers a topic popular over the years but stretches far beyond since also concealed canonical algebras of tubular or wild representation type are covered

killing and the coxeter transformation of kac moody algebras on etudie le spectre de la transformation de coxeter associee a une algebre de kac moody arbitraire et plus particulierement a une algebre affine

mckay correspondence for the poincare series of kleinian and fuchsian singularities we give a uniform and to a large extent geometrical proof that the poincare series of the coordinate algebra of a kleinian singularity and of a fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two coxeter elements these coxeter elements are interpreted geometrically using 2 calabi yau triangulated categories and spherical twist functors

triangle singularities ade chains and weighted projective lines abstract we study the singularity category of the triangle singularity f x a y b z c or s k x y z f attached to the weighted projective line x given by the weight triple a b c by investigating the stable category vect x of vector bundles on x obtained from the category of vector bundles by factoring out all the line bundles we show that vect x is fractional calabi yau whose cy dimension is a function of the euler characteristic of x and establish the existence of a tilting object having the shape of an a 1 b 1 c 1 cuboid the weight types 2 a b yield an explanation of happel seidel symmetry for a class of important nakayama algebras further the weight sequence 2 3 p corresponds to an ade chain the e n chain extrapolating the exceptional dynkin cases e 6 e 7 and e 8 to a whole sequence of triangulated categories

periodic coxeter matrices and their associated quadratic forms we study relations between the periodicity of a coxeter matrix and the positivity or the non negativity of a quadratic form with respect to a cartan matrix of an artinian ring we introduce the notion of the weak periodicity of a coxeter matrix induced by a non negative quadratic form and we characterize the weak periodicity condition and the periodicity condition in terms of eigenvalues of a coxeter matrix

coxeter spectral classification of almost tp critical one peak posets using symbolic and numeric computations following an idea of spectral graph theory we give several characterizations of finite one peak posets i with almost p critical quadratic tits form q i zi z by applying linear algebra methods combinatorial algorithms an old idea of s a ovsienko and recent results by v m bondarenko and m v stepochkina our study is mainly motivated by an important application of the quadratic tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of nazarova roiter matrix problems over a field k in particular our study is inspired by a well known result of ju a drozd explained in introduction one of our main results asserts that given a finite poset i 1 n t such that is its unique maximal element and q i is almost p critical we have i the form q i is non negative and the subgroup kerq i v zn 1 q i v 0 of zi zn 1 is generated by a vector h h1 hn h with h1 0 hn 0 ii 5 i 9 and the coxeter polynomial coxi t z t coincides with the coxeter polynomial f i t of a uniquely determined euclidean diagram i d 4 e 6 e 7 e 8 iii the z bilinear tits form b i zn 1 zn 1 z of i is z equivalent to the gram z bilinear form b i zn 1 zn 1 z of the euclidean diagram i iv there is a z invertible matrix c such that c itr ctr c i c and c2 e where c i mn 1 z is the tits matrix of i and v i is one of the posets l1 l132 listed in the paper in particular we get in an alternative way the list of p critical posets t in the form t i presented by v m bondarenko and m v stepochkina 6 and we determine their coxeter euclidean types there are 115 one peak posets i l1 l115 with q i p critical and 17 exceptional posets i l116 l132 with q i almost p critical but not p critical our lists are obtained by using computer algebra tools mainly symbolic computation in maple and numeric computation in c

darboux transformations for cmv matrices abstract we develop a theory of darboux transformations for cmv matrices canonical representations of the unitary operators in perfect analogy with their self adjoint version the darboux transformations of jacobi matrices they are equivalent to laurent polynomial modifications of the underlying measures we address other questions which emphasize the similarities between darboux transformations for jacobi and cmv matrices like their almost isospectrality or the relation that they establish between the corresponding orthogonal polynomials showing also that both transformations are connected by the szeg mapping nevertheless we uncover some features of the darboux transformations for cmv matrices which are in striking contrast with those of the jacobi case in particular when applied to cmv matrices the matrix realization of the inverse darboux transformations what we call darboux transformations with parameters leads to spurious solutions whose interpretation deserves future research such spurious solutions are neither unitary nor band matrices so darboux transformations for cmv matrices are much more subject to the subtleties of the algebra of infinite matrices than their jacobi counterparts a key role in our theory is played by the cholesky factorizations of infinite matrices actually the darboux transformations introduced in this paper are based on the cholesky factorizations of degree one hermitian laurent polynomials evaluated on cmv matrices these transformations are also generalized to higher degree laurent polynomials as well as to the extension of cmv matrices to quasi definite functionals what we call quasi cmv matrices furthermore we show that this cmv version of darboux transformations plays a role in integrable systems like the schur flows or the ablowitz ladik model which parallels that of darboux for jacobi matrices in the toda lattice

a gram classification of non negative corank two loop free edge bipartite graphs we continue the coxeter spectral study of finite connected loop free edge bipartite graphs with m 2 3 vertices a class of signed graphs started in simson 2013 49 by means of the non symmetric gram matrix g mm 2 z of its symmetric gram matrix g 12 g g tr mm 2 12z the gram quadratic form q zm 2 z and the coxeter spectrum specc c i e the complex spectrum of the coxeter matrix cox g g tr gl m 2 z in the present paper we study non negative edge bipartite graphs of corank two in the sense that the symmetric gram matrix g mm 2 z of is positive semi definite of rank m 1 one of our aims is to get a complete classification of all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z where z means that g btr g b for some b mm 2 z such that det b 1 by one vertex extensions of the simply laced euclidean diagrams a m m 1 d m m 4 e 6 e 7 e 8 we construct a family of connected loop free corank two diagrams a m 2 d m 2 e 6 2 e 7 2 e 8 2 called simply extended euclidean diagrams such that they classify all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z a structure of connected corank two loop free edge bipartite graphs is described it is shown that every such contains a connected positive edge bipartite subgraph that is z congruent with a simply laced dynkin diagram dyn called the dynkin type of such that is a two point extension u w of along two roots u w of the positive definite gram form q zm z this yields a combinatorial algorithm u w u w allowing us to construct all connected corank two loop free edge bipartite graphs with m 2 3 vertices and d dyn from the triples u w where is positive of the dynkin type d and u w are roots of the positive definite gram form q zm z

hereditary noetherian categories of positive euler characteristic we develop the fundamentals of hereditary noetherian categories with serre duality over an arbitrary field k where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras the proper way to view such a category is to think of coherent sheaves on a possibly non commutative smooth projective curve we define for each such category notions like function field and euler characteristic determine its auslander reiten components and study stable and semistable bundles for an appropriate notion of degree we provide a complete classification of hereditary noetherian categories for the case of positive euler characteristic by relating these to finite dimensional representations of locally bounded hereditary k algebras whose underlying valued quiver admits a positive additive function

a chebysheff recursion formula for coxeter polynomials abstract a polynomial f t z t is represented by q t z t if f t 2 q t t deg q q t t 1 f t is graphically represented if f t 2 m t for m t the characteristic polynomial of a symmetric matrix m many instances of coxeter polynomials f a t for a a finite dimensional algebra are graphically representable we study the case of extended canonical algebras a see h lenzing j a de la pena extended canonical algebras and fuchsian singularities in press show that the corresponding polynomials f a t are representable and satisfy a chebysheff type recursion formula we get consequences for the eigenvalues of the coxeter transformation of a showing for instance that at most four eigenvalues may lie outside the unit circle

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

removability of point singularities of willmore surfaces we investigate point singularities of willmore surfaces which for example appear as blowups of the willmore flow near singularities and prove that closed willmore surfaces with one unit density point singularity are smooth in codimension one as applications we get in codimension one that the willmore flow of spheres with energy less than 8 exists for all time and converges to a round sphere and further that the set of willmore tori with energy less than

extended canonical algebras and fuchsian singularities the authors introduce a new class of finite dimensional algebras called extended canonical and determine the shape of their derived categories extended canonical algebras arise from a canonical algebra by onepoint extension or coextension by an indecomposable projective module our main results concern the case of negative euler characteristic of the corresponding weighted projective line mathbb x more specifically we establish for a base field of arbitrary characteristic a link to the fuchsian singularity r of mathbb x which for the base field of complex numbers is isomorphic to an algebra of automorphic forms by means of a recent result of orlov we show that the triangulated category of the graded singularities of r in the sense of buchweitz and orlov admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra of particular interest are those cases where the attached coxeter transformation has spectral radius one a k theoretic analysis then shows that this happens exactly for 38 cases including arnold s 14 exceptional unimodal singularities the paper is related to recent independent work by kajiura saito and takahashi

methods to determine coxeter polynomials abstract a reduction formula for the characteristic polynomials of the coxeter matrices of certain split finite dimensional algebras is proved in the hereditary case this yields explicit expressions for the coxeter polynomials of large families of quivers moreover a combinatorial interpretation of the entries of the coxeter matrices of path algebras gives formulas for coxeter polynomials of some quivers which cannot be treated by the above mentioned reduction process

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

on the coxeter polynomials of wild stars abstract the spectral radius of a coxeter transformation which plays an important role in the representation theory of hereditary algebras see v dlab c m ringel eigenvalues of coxeter transformations and the gelfand kirillov dimension of the preprojective algebras proc ams 83 1990 228 232 is its important invariant this paper provides both upper and lower bounds for the spectral radii of the coxeter transformations of wild stars i e trees that have a single branching point and are neither of dynkin nor of euclidean type in addition the paper determines limit of the spectral radii of particular infinite sequences of wild stars and shows different classes of graphs with the same limit the basic idea is to reduce the study of spectral radii of trees to the spectral radii of particular valued graphs with indefinite type of associated generalized cartan matrix

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

the toda flow on a generic orbit is integrable on construit des variables action angle pour le flot de toda sur des orbites generiques de l action coadjointe du groupe triangulaire inferieur sur son algebre de lie duale

salem numbers defined by coxeter transformation abstract a real algebraic integer 1 is called a salem number if all its remaining conjugates have modulus at most 1 with at least one having modulus exactly 1 it is known j a de la pena coxeter transformations and the representation theory of algebras in v dlab et al eds finite dimensional algebras and related topics proceedings of the nato advanced research workshop on representations of algebras and related topics ottawa canada kluwer august 10 18 1992 pp 223 253 j f mckee p rowlinson c j smyth salem numbers and pisot numbers from stars number theory in progress in k gy ry et al eds proc int conf banach int math center diophantine problems and polynomials vol 1 de gruyter berlin 1999 pp 309 319 p lakatos on coxeter polynomials of wild stars linear algebra appl 293 1999 159 170 that the spectral radii of coxeter transformation defined by stars which are neither of dynkin nor of extended dynkin type are salem numbers we prove that the spectral radii of the coxeter transformation of generalized stars are also salem numbers a generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

tubes in derived categories and cyclotomic factors of the coxeter polynomial of an algebra let be a k algebra of finite global dimension we study tubular families in the auslander reiten quiver of the bounded derived category db satisfying certain natural axioms in particular we precisely describe their influence on the cyclotomic factors of the coxeter polynomial of and discuss several numerical limitations for their possible shapes moreover we show that our results provide an alternative relatively simple proof of non trivial classical facts concerning tubular families in module categories and also extend them slightly

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

critical weak immersed surfaces within sub manifolds of the teichmuller space abstract we prove that the critical points of various energies such as the area the willmore energy the frame energy for tori etc among possibly branched immersions constrained to evolve within a smooth sub manifold of the teichmuller space satisfy the corresponding constrained euler lagrange equation we deduce that critical points of the willmore energy or the frame energy for tori are smooth analytic surfaces away possibly from isolated branched points under the condition that either the genus is at most 2 or if the sub manifold does not intersect the subspace of hyper elliptic points using a compactness result from 19 we can conclude that each closed sub manifold of the teichmuller space including points under the previous assumptions possess a possibly branched smooth willmore minimizer satisfying the conformal constrained willmore equation

a family of the poisson brackets compatible with the sklyanin bracket we introduce a family of compatible poisson brackets on the space of 2 2 polynomial matrices which contains the sklyanin bracket and use it to derive a multi hamiltonian structure for a set of integrable systems that includes the xxx heisenberg magnet the open and periodic toda lattices the discrete self trapping model and the goryachev chaplygin gyrostat

periodic coxeter matrices abstract let a kq i be a finite dimensional triangular k algebra consider the cartan matrix c a and the coxeter matrix a c a t c a let t det t id a be the coxeter polynomial of a we study conditions on spec a in order that a is a periodic matrix we show that in case a is periodic then the euler quadratic form q a x xc a t x t is non negative and q a 0 if and only if 1 spec a

willmore spheres in compact riemannian manifolds abstract the paper is devoted to the variational analysis of the willmore and other l 2 curvature functionals among immersions of 2 dimensional surfaces into a compact riemannian m manifold m m h with m 2 the goal of the paper is two fold on one hand we give the right setting for doing the calculus of variations including minmax methods of such functionals for immersions into manifolds and on the other hand we prove the existence results for possibly branched willmore spheres under various constraints prescribed homotopy class prescribed area or under curvature assumptions for m m to this aim using the integrability by compensation theory we first establish the regularity for the critical points of such functionals we then prove a rigidity theorem concerning the relation between cmc and willmore spheres then we prove that for every non null 2 homotopy class there exists a representative given by a lipschitz map from the 2 sphere into m m realizing a connected family of conformal smooth possibly branched area constrained willmore spheres as explained in the introduction this comes as a natural extension of the minimal immersed spheres in homotopy class constructed by sacks and uhlenbeck 1981 in 38 in situations when they do not exist moreover for every a 0 we minimize the willmore functional among connected families of weak possibly branched immersions of the 2 sphere having prescribed total area equal to a and we prove full regularity for the minimizer finally under a mild curvature condition on m m h we minimize the sum of the area with the square of the l 2 norm of the second fundamental form among weak possibly branched immersions of the two spheres and we prove the regularity of the minimizer

moduli spaces for representations of concealed canonical algebras abstract continuing m domokos and h lenzing j algebra 228 2000 738 762 we investigate the relative invariants and moduli spaces introduced in a d king quart j math oxford ser 2 45 1994 515 530 which are related to a separating family of stable tubes in the module category of a concealed canonical algebra the class of concealed canonical algebras consists of representation infinite algebras of tame or wild representation type and contains in particular the representation theory of extended dynkin quivers we present a uniform characteristic free approach avoiding case by case analysis we introduce the notion of admissible weight for such algebras these are the weights i e elements of the dual of the grothendieck group of the module category attached to the infinite moduli spaces for families of modules from the separating subcategory by using an explicit description of the semigroup of admissible weights the corresponding notion of semistability is analyzed and is related to the formation of perpendicular categories we construct relative invariants belonging to admissible weights and prove that the constructed system is complete in the case of canonical algebras it turns out that for any concealed canonical algebra and for any admissible weight all the corresponding moduli spaces are isomorphic to some projective space moreover any such moduli space can be naturally identified with the moduli space belonging to a special weight called the rank as a consequence we show that in the case of a tame concealed algebra any infinite moduli space for families of modules is a projective space and all fields of rational invariants on irreducible components of representation spaces are purely transcendental

the coxeter polynomial for a one point extension algebra let be a finite dimensional algebra over an algebraically closed field k of finite global dimension let m be a finitely generated module and let m be the one point extension algebra we show how to compute the coxeter polynomial for from the coxeter polynomial of and homological invariants of m

on the perpendicular category of a partial tilting module abstract the perpendicular category of a partial tilting module is studied in particular that of a regular module over a hereditary algebra preferably of wild representation type the results are used to establish the existence of a unique preprojective component for a large class of tilted algebras

additive functions on quivers abstract an integral function on the set of vertices of a graph is additive if twice is value at any vertex v equals the sum of its values at all adjacent vertices counting multiple edges it is well known that among finite connected graphs exactly the extended dynkin graphs admit a positive additive function whereas the dynkin diagrams themselves only allow almost additive functions violating additivity in a single vertex in the present paper we study usually non positive additive or non additive functions on finite quivers and relate the concept of additivity to the radical of the homological euler form our main results concern the existence and construction of such functions for wild quivers our results are most specific in case the underlying graph is a tree possibly with multiple edges

on the mahler measure of the coxeter polynomial of an algebra abstract let a be a finite dimensional algebra over an algebraically closed field k assume a is a basic connected and triangular algebra with n pairwise non isomorphic simple modules we consider the coxeter transformation a t as the automorphism of the grothendieck group k 0 a induced by the auslander reiten translation in the derived category der b mod a of the module category mod a of finite dimensional left a modules we say that a is of cyclotomic type if the characteristic polynomial a of a is a product of cyclotomic polynomials equivalently if the mahler measure m a 1 in 6 we have considered many examples of algebras of cyclotomic type in the representation theory literature in this paper we study the mahler measure of the coxeter polynomial of accessible algebras in 1933 d h lehmer found that the polynomial t 10 t 9 t 7 t 6 t 5 t 4 t 3 t 1 has mahler measure 0 1 176280 and he asked whether there exist any smaller values exceeding 1 in this paper we prove that for any accessible algebra a either m a 1 or m b 0 for some convex subcategory b of a we introduce interlaced tower of algebras a m a n with m n 2 satisfying a s 1 t 1 a s t a s 1 for m 1 s n 1 we prove that if spec a n s 1 r and a n is not of cyclotomic type then m a m m a n we construct several examples

on tensor products of path algebras of type a abstract we derive a formula for the coxeter polynomial of the s fold tensor product i 1 s f a n i 1 of path algebras of linearly oriented quivers of dynkin type a n i 1 in terms of the weights n 1 n s 2 and show that conversely the weights can be recovered from the coxeter polynomial of the tensor product our results have relevance in singularity theory in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles

the structure and homological properties of generalized standard auslander reiten components abstract we describe the structure and homological properties of arbitrary generalized standard auslander reiten components of artin algebras in particular we prove that for all but finitely many indecomposable modules in such components the euler characteristic is defined and nonnegative further we provide a handy criterion for an infinite auslander reiten component of an artin algebra to be generalized standard we solve also the long standing open problem concerning the structure of artin algebras admitting a separating family of auslander reiten components