hall polynomials for tame type abstract in the present paper we prove that hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line x over finite fields these polynomials are then used to define the generic ringel hall algebra of x as well as its drinfeld double combining this construction with a result of cramer we show that hall polynomials exist for tame quivers which not only refines a result of hubery but also confirms a conjecture of berenstein and greenstein

on the semi stable coha and its modules arising from smooth models abstract we study a variant of the semi stable cohomological hall algebra which we construct using equivariant chow groups this algebra we call it the semi stable chowha arises as a quotient of the coha smooth models of quiver moduli give rise to modules over the semi stable chowha we prove that these modules are cyclic and we compute a presentation using harder narasimhan methods

quantum groups via hall algebras of complexes we describe quantum enveloping algebras of symmetric kac moody lie algebras via a nite eld hall algebra construction involving z2 graded complexes of quiver representations

a direct approach toward global minimization for multiphase labeling and segmentation problems this paper intends to extend the minimization algorithm developed by bae yuan and tai ijcv 2011 in several directions first we propose a new primal dual approach for global minimization of the continuous potts model with applications to the piecewise constant mumford shah model for multiphase image segmentation different from the existing methods we work directly with the binary setting without using convex relaxation which is thereby termed as a direct approach second we provide the sufficient and necessary conditions to guarantee a global optimum moreover we provide efficient algorithms based on a reduction in the intermediate unknowns from the augmented lagrangian formulation as a result the underlying algorithms involve significantly fewer parameters and unknowns than the naive use of augmented lagrangian based methods hence they are fast and easy to implement furthermore they can produce global optimums under mild conditions

the bridgeland s ringel hall algebra associated to an algebra with global dimension at most two abstract for any finite dimensional associative algebra with global dimension 2 we show that there is an embedding from the twisted ringel hall algebra to the bridgeland s ringel hall algebra in particular this result is true for tilted algebras and canonical algebras

triangulated categories and kac moody algebras by using the ringel hall algebra approach we find a lie algebra arising in each triangulated category with t 2 1 where t is the translation functor in particular the generic form of the lie algebras determined by the root categories the 2 period orbit categories of the derived categories of finite dimensional hereditary associative algebras gives a realization of all symmetrizable kac moody lie algebras

comparing lusztig s algebras and hall algebras at v 1 abstract there are two abstract approaches to positive parts of quantized enveloping algebras of kac moody algebras lusztig s axiomatic approach leading to non degenerate objects in green categories and ringel s approach via hall algebras generically and for certain values of the quantum parameter these two approaches produce isomorphic objects as shown by green this note studies a case where these objects turn out to be different for v 1 the algebra arising from lusztig s approach is shown to be super commutative for any datum i whereas hall algebras are super commutative only in a trivial case

the hall algebra approach to drinfeld s presentation of quantum loop algebras abstract the quantum loop algebra u v l g was defined as a generalization of the drinfeld s new realization of the quantum affine algebra to the loop algebra of any kac moody algebra g it has been shown by schiffmann that the hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra u v l g for some g with a star shaped dynkin diagram in this paper we study drinfeld s presentation of u v l g in the double hall algebra setting based on schiffmann s work we explicitly find out a collection of generators of the double composition algebra dc coh x and verify that they satisfy all the drinfeld relations

triangular decomposition of the composition algebra of the kronecker algebra abstract letabe a finite dimensional hereditary algebra over a finite field and let h a and c a be respectively the ringel hall algebra and the composition algebra ofa definerdto be the element m h a where m runs over the isomorphism classes of the regulara modules with dimension vectord we prove thatrdand the exceptionala modules all lie in c a letkbe the kronecker algebra p resp i the subalgebra of c k generated by the preprojective resp preinjective k modules and t the subalgebra generated byr n n forn 0 then we prove that c k p t i and then t is just the subalgebra of c k generated by all regular elements

identification of simple representations for affine q schur algebras abstract we define an explicit action of the quantum loop algebra u q gl n on the tensor space and show that these actions are compatible with the natural embedding u q gl n u q gl n 1 as an application we identify simple representations of affine q schur algebras s n r c arising from simple polynomial representations of u q gl n frenkel and mukhin 2002 11 with those arising from simple modules of the affine hecke algebra h r c of type a rogawski 1985 26

symmetric functions and the centre of the ringel hall algebra of a cyclic quiver we obtain explicit generators for the centre of the ringel hall algebra of a cyclic quiver and define a canonical algebra monomorphism from macdonald s ring of symmetric functions to the centre which furthermore respects the comultiplication and the symmetric bilinear form

generalized green classes generalized green classes are introduced some basic properties of members in a generalized green class are studied finally we apply the results to h the ringel hall algebra of a finite dimensional hereditary algebra over a finite field in particular it is proved that h belongs to a suitable generalized green class and that there is direct decomposition of spaces h c j where c is the composition algebra of and j is a twisted hopf ideal of h which is exactly the orthogonal complement of c

the hall algebra of the category of coherent sheaves on the projective line to an abelian category a of homological dimension one satisfying certain finiteness conditions one can associate an algebra called the hall algebra kapranov studied this algebra when a is the category of coherent sheaves over a smooth projective curve defined over a finite field and observed analogies with quantum affine algebras we recover here in an elementary way his results in the case when the curve is the projective line

minimal generators of ringel hall algebras of affine quivers abstract let h kq be the ringel hall algebra of affine quiver q all indecomposable representations of q which can be generated inside h kq by smaller representations of q are classified systems of minimal homogeneous generators of h kq are explicitly written out

a realisation of the quantum supergroup u glm n abstract we reconstruct the quantum enveloping superalgebra u gl m n over q via finite dimensional quantum schur superalgebras in particular we obtain a new basis containing the standard generators for u gl m n and explicit multiplication formulas of an arbitrary basis element by a generator

on the hall algebra of an elliptic curve i these rings admit numerous algebraic and geometric realizations but one of the historically first constructions dating to the work of steinitz in 1900 completed later by hall was given in terms of what is now called the classical hall algebra h see ma chapter ii this algebra has a basis consisting of isomorphism classes of abelian q groups where q is a fixed prime power and the structure constants are defined by counting extensions between such abelian groups in fact these structure constants are polynomials in q and we can therefore consider h as a c q 1 algebra a theorem of steinitz and hall provides an isomorphism h q c q x1 x2 under this isomorphism the natural basis of h resp the natural scalar product is mapped to the basis of hall littlewood polynomials resp the hall littlewood scalar product in addition zelevinsky z endowed q with a structure of a cocommutative hopf algebra and the whole algebra q c q 1

modified ringel hall algebras green s formula and derived hall algebras abstract in this paper we define the modified ringel hall algebra m h a of a hereditary abelian category a from the category c b a of bounded z graded complexes two main results are obtained one is to give a new proof of green s formula on ringel hall numbers by using the associative multiplication of the modified ringel hall algebra the other is to show that in certain twisted cases the derived hall algebra can be embedded in the modified ringel hall algebra as a consequence of the second result we get that in certain twisted cases the modified ringel hall algebra is isomorphic to the tensor algebra of the derived hall algebra and the torus of acyclic complexes and therefore the modified ringel hall algebra is invariant under derived equivalences

bridgeland s hall algebras of tilted algebras and lusztig s symmetries abstract for each finite dimensional algebra a of finite global dimension over a finite field bridgeland considered the ringel hall algebra of 2 cyclic complexes of projective a modules and achieved an algebra called bridgeland s hall algebra by taking some localization and reduction in this paper we show that there exist homomorphisms of algebras from quantum groups to bridgeland s hall algebras of tilted algebras in particular for apr tilting this homomorphism provides a realization of lusztig s symmetries of quantum groups via bridgeland s hall algebras

triangular decomposition of tame non simply laced composition algebras let a be a finite dimensional hereditary algebra over a finite field and let c a be the composition algebra of a let p resp i be the subalgebra of c a generated by the preprojective resp preinjective a modules and let t be the subalgebra generated by rd where rd m h a m runs over the isomorphism classes of the regular a modules with dimension vector d and h a is the ringel hall algebra of a then we prove that c a p t i if a is one of the types bn cn bcn bdn cdn f41 f42 g21 g22 or a11

skew differential operator algebras of twisted hopf algebras abstract by introducing a twisted hopf algebra we unify several important objects of study skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied this generalizes results in the weyl algebra applying this investigation to the twisted ringel hall algebra we get in particular a natural realization of the non positive part of a quantized generalized kac moody algebra by identifying the canonical generators with some linear skew differential operators this also induces some algebras which are quantum group like

superspecies and their representations abstract superspecies are introduced to provide a construction of all finite dimensional superalgebras all acyclic superspecies are classified according to their graded representation types

drinfeld double and ringel green theory of hall algebras abstract as a continuation of the work of ringel and green on hall algebras the hopf algebra structure of a extended twisted hall algebra is presented here by introducing the ringel pairing of hall algebras the drinfeld double of a hall algebra is constructed the reduced form of this drinfeld double is applicable to the generic composition algebra which gives a complete realization of quantum group this ringel green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of hall algebras

representations of cohomological hall algebras and donaldson thomas theory with classical structure groups we introduce a new class of representations of the cohomological hall algebras of kontsevich and soibelman which we call cohomological hall modules or cohm for short these representations are constructed from self dual representations of a quiver with contravariant involution sigma and provide a mathematical model for the space of bps states in orientifold string theory we use the cohm to define a generalization of the cohomological donaldson thomas theory of quivers in which the quiver representations have orthogonal or symplectic structure groups the associated invariants are called orientifold donaldson thomas invariants we prove the orientifold analogue of the integrality conjecture for sigma symmetric quivers we also formulate precise conjectures regarding the geometric meaning of orientifold donaldson thomas invariants and the freeness of the cohm of a sigma symmetric quiver we prove the freeness conjecture for disjoint union quivers loop quivers and the affine dynkin quiver of type widetilde a 1 we also verify the geometric conjecture in a number of examples finally we construct explicit poincare birkhoff witt type bases of the cohm of finite type quivers

the hall polynomials for tame quiver algebras abstract let a be a finite dimensional tame quiver algebra over a finite field k we prove that hall polynomials exist for a

ringel hall algebras and lusztig s symmetries abstract let d be the double ringel hall algebra of a finite dimensional hereditary algebra the present paper shows that the bgp reflection operators of d coincide with lusztig s symmetries up to canonical isomorphisms and satisfy the braid group relations on the whole double ringel hall algebra d this answers a question of sevenhant and van den bergh j algebra 221 1999 135 160

a new approach to kac s theorem on representations of valued quivers by using the ringel hall algebra approach we first present a new proof of kac s theorem for species over a finite field the method used here is quite different from earlier techniques in sv sevenhant and van den bergh have discovered an interesting relation between a conjecture of kac on representations of quivers and the structure of the ringel hall algebras in some special cases we second show that this relation still holds for species

discriminating between nonstationary and nearly nonstationary time series models a simulation study abstract this paper describes empirical evidence that it may often be possible to very simply discriminate between homogeneous once integrated nonstationary time series models and nearly nonstationary approximations to them we show that in such a situation there can be a feature of the serial correlation structure which enables the pair of models to be distinguished with considerable confidence this observation appears to be useful in the context of the box jenkins identification of time series models and we believe it opens up a promising field for new research complementary to the currently well established tests for unit roots

indecomposable vector bundles and stable higgs bundles over smooth projective curves we prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve x dened over a nite eld is given by a polynomial depending only on r d and the genus g of x in the weil numbers of x we provide a closed formula expressed in terms of generating series for this polynomial we also show that the same polynomial computes the number of points of the moduli space of stable higgs bundles of rank r and degree d over x this entails a closed formula for the poincar e polynomial of the moduli spaces of stable higgs bundles over a compact riemann surface and hence also for the poincar e polynomials of the twisted character varieties for the groups gl r

elliptic lie algebras and tubular algebras abstract this article is to study relations between tubular algebras of ringel and elliptic lie algebras in the sense of saito yoshii using the explicit structure of the derived categories of tubular algebras given by happel ringel we prove that the elliptic lie algebra of type d 4 1 1 e 6 1 1 e 7 1 1 or e 8 1 1 is isomorphic to the ringel hall lie algebra of the root category of the tubular algebra with the same type as a by product of our proof we obtain a chevalley basis of the elliptic lie algebra following indecomposable objects of the root category of the corresponding tubular algebra this can be viewed as an analogue of the frenkel malkin vybornov theorem in which they described a chevalley basis for each untwisted affine kac moody lie algebra by using indecomposable representations of the corresponding affine quiver

on derived hall numbers for tame quivers abstract in the present paper we study the derived hall algebra for the bounded derived category of the finite dimensional nilpotent representations of a dynkin or tame quiver over a finite field we show that for any three given objects in the bounded derived category the associated derived hall numbers are given by a rational function in the cardinalities of ground fields

hall algebras hereditary algebras and quantum groups let r be an associative hereditary algebra over a finite field k and let r fin be the full subcategory of r mod whose objects are those left r modules x which are finite as sets ix oc assume also that r isfinitary in c ringel s sense i e that iext l s s c for all simple s s in r fin this condition is met for example if r is finitely generated as k algebra 4 pp 435 436 let be the set of all isomorphism classes in r fin if 2 e then u will denote an r module in class 2 the class of all zero left r modules is denoted 0 let i c be the set of all isomorphism classes of simple modules in r fin thus u i e i is a complete set of simple finite left r modules we identify the grothendieck group k0 r fin with the free abelian group e1 v i v e 7z having i as free basis so that if x e r fin then the corresponding element of k0 r fin is the dimension vector dimx y i vii where for each i e vi is the multiplicity of the simple module u in any composition series of x clearly d imx lies in the subsemigroup

on borcherds type lie algebras for each even lattice mathcal l there is a canonical way to construct an infinite dimensional lie algebra via lattice vertex operator algebra theory we call this lie algebra and its subalgebras the borcherds type lie algebras associated to mathcal l in this paper we apply this construction to even lattices arising from representation theory of finite dimensional associative algebras this is motivated by the different realizations of kac moody algebras by borcherds using lattice vertex operators and by peng xiao using ringel hall algebras respectively for any finite dimensional algebra a of finite global dimension we associate a borcherds type lie algebra mathfrak bl a to a in contrast to the ringel hall lie algebra approach mathfrak bl a only depends on the symmetric euler form or tits form but not the full representation theory of a however our results show that for certain classes of finite dimensional algebras whose representation theory is controlled by the euler bilinear forms or tits forms their borcherds type lie algebras do have close relations with the representation theory of these algebras beyond the class of hereditary algebras these algebras include canonical algebras representation directed algebras and incidence algebras of finite prinjective types

double hall algebras and derived equivalences abstract we show that the reduced drinfeld double of the ringel hall algebra of a hereditary category is invariant under derived equivalences by associating an explicit isomorphism to a given derived equivalence we also extend the results of burban and schiffmann 5 6 sevenhant and van den bergh 1999 25 and xiao and yang 2001 31

more effective time series analysis and forecasting our aim is to suggest ways of improving time domain modelling for the purpose of more effective forecasting by better interpretation of the sample autocorrelations and partial autocorrelations obtained from raw time series data for this objective we assume no specialist knowledge as we start by surveying all those standard ideas of univariate analysis which are needed for the subsequent development of our thesis

linear programming approach to nonconvex variational problems in nonconvex optimization problems in particular in nonconvex variational problems there usually does not exist any classical solution but only generalized solutions which involve young measures in this paper after reviewing briefly the relaxation theory for such problems an iterative scheme leading to a sequential linear programming slp scheme is introduced and its convergence is proved by a banach fixed point technique then an approximation scheme is proposed and analyzed and calculations of an illustrative 2d broken extremal example are presented

twisted hopf algebras ringel hall algebras and green s categories with an appendix by the referee abstract the concept and some basic properties of a twisted hopf algebra are introduced and investigated its unique difference from a hopf algebra is that the comultiplication a a a is an algebra homomorphism not for the componentwise multiplication on a a but for the twisted multiplication on a a given by lusztig s rule also it is proved that any object a in green s category has a twisted hopf algebra structure any morphism between objects is a twisted hopf algebra homomorphism the antipode s of a is self adjoint under the lusztig form on a and the green polynomials m a b t share a so called cyclic symmetry as examples the twisted ringel hall algebras ringel s twisted composition algebras lusztig s free algebras f and non degenerate algebras f the positive part u of the drinfeld jimbo quantized enveloping algebras u and rosso s quantum shuffle algebra t v all are twisted hopf algebras the antipode and its inverse for a twisted ringel hall are explicitly given and all primitive elements are determined

a note on bridgeland s hall algebra of two periodic complexes we show that the hall algebra of two periodic complexes which is recently introduced by t bridgeland coincides with the drinfeld double of the ordinary hall bialgebra

pbw bases of the twisted generic composition algebras of affine valued quivers abstract pbw type bases of the twisted generic composition algebras of the affine valued quivers are constructed by using the frobenius morphism

composition algebras of affine type abstract the aim of this paper is to study the structure of the composition algebras of affine type it turns out that they have a triangular decomposition p t i corresponding to the division of the indecomposables into the preprojectives the regulars and the preinjectives by the recent ringel green theorem the composition algebra can be twisted in order to obtain the positive partu of the drinfeld jimbo quantized enveloping algebrau u u0 u of the corresponding kac moody algebra and one obtains a corresponding triangular decomposition foru in particular a natural basis ofu in terms ofa representations

indecomposables as elements in affine composition algebras abstract letabe a path algebra of tame type over a finite field letmbe an indecomposablea module and let c a be the composition algebra ofa the main result in this paper is that m c a if and only ifmis a stone i e ext1a m m 0

toen s formula and green s formula abstract let a be a finitary hereditary abelian category in this note we show that green s formula on hall numbers of a can be deduced from the associativity of the derived hall algebra of a whose multiplication structure constants are given by the so called toen s formula

hall algebras associated to triangulated categories by counting triangles and applying the octahedral axiom we find a direct way to prove the formula of toen in t section 5 1 for a triangulated category satisfying the left homological finiteness condition

the hall module of an exact category with duality abstract we construct from a finitary exact category with duality a a module over its hall algebra called the hall module encoding the first order self dual extension structure of a we study in detail hall modules arising from the representation theory of a quiver with involution in this case we show that the hall module is naturally a module over the specialized reduced analogue of the quantum kac moody algebra attached to the quiver for finite type quivers we explicitly determine the decomposition of the hall module into irreducible highest weight modules