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

some hall polynomials for representation finite trivial extension algebras letkbe a finite field and assume that is a finite dimensional associativek algebra with 1 denote by mod the category of all finitely generated right modules and by ind the full subcategory in which every object is a representative of the isoclass of an indecomposable right module we are interested in the existance of the hall polynomial mnlfor anl m n mod for the definition see977113or section 1 below in case is directed 7has shown that has hall polynomials and in case is cyclic serial the same result has also been obtained by4 it has been conjectured in8that any representation finitek algebra has hall polynomials in this investigation we shall show that if is a representation finite trivial extension algebra then for anyl m n mod withnindecomposable has the hall polynomials mlnand mnl using these hall polynomials we can naturally structure the free abelian group with a basis ind denoted byk mod into a lie algebra and the universal enveloping algebra ofk mod z qis just h 1 z q where h 1is the degenerated hall algebra of see section 5 below

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

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

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

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

frobenius morphisms and representations of algebras by introducing frobenius morphisms f on algebras a and their modules over the algebraic closure f q of the finite field f q of q elements we establish a relation between the representation theory of a over f q and that of the f fixed point algebra a f over f q more precisely we prove that the category mod a f of finite dimensional a f modules is equivalent to the subcategory of finite dimensional f stable a modules and when a is finite dimensional we establish a bijection between the isoclasses of indecomposable a f modules and the f orbits of the isoclasses of indecomposable a modules applying the theory to representations of quivers with automorphisms we show that representations of a modulated quiver or a species over f q can be interpreted as f stable representations of the corresponding quiver over f q we further prove that every finite dimensional hereditary algebra over f q is morita equivalent to some a f where a is the path algebra of a quiver q over f q and f is induced from a certain automorphism of q a close relation between the auslander reiten theories for a and a f is established in particular we prove that the auslander reiten modulated quiver of a f is obtained by folding the auslander reiten quiver of a finally by taking frobenius fixed points we are able to count the number of indecomposable representations of a modulated quiver over f q with a given dimension vector and to generalize kac s theorem for all modulated quivers and their associated kac moody algebras defined by symmetrizable generalized cartan matrices

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

expansion based qbf solving versus q resolution this article introduces and studies a proof system exp res that enables us to refute quantified boolean formulas qbfs the system exp res operates in two stages it expands all universal variables through conjunctions and refutes the result by propositional resolution this approach contrasts with the q resolution calculus which enables refuting qbfs by rules similar to propositional resolution in practice q resolution enables producing proofs from conflict driven dpll based qbf solvers the system exp res can on the other hand certify certain expansion based solvers so a natural question is to ask which of the systems q resolution and exp res is more powerful the article gives several partial responses to this question on the positive side we show that exp res can p simulate tree q resolution on the negative side we show that exp res does not p simulate unrestricted q resolution in the favor of exp res we show that exp res is more powerful than a certain fragment of q resolution which is important for dpll based qbf solving

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

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

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

global several parameter bifurcation and continuation theorems a unified approach via complementing maps we use the concept of complementing map in order to present a unified approach to the continuation principle global implicit function theorem and global bifurcation results for compact vector fields depending on several parameters

multi brace cotensor hopf algebras and quantum groups we give a construction of the whole quantum group associated to a symmetrizable kac moody lie algebra as a quantum quasi symmetric algebra a particular case of multi brace cotensor hopf algebras in the spirit of the quantum shuffle construction of the plus part all highest weight irreducible representations are constructed using this machinery it also provides a systematic way to construct simple modules over the quantum double of a quantum group

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

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

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

root categories and simple lie algebras abstract by using the t 2 orbit category of the derived category of a hereditary algebra which is proved to be a triangulated category too we give a complete realization of a simple lie algebra this is a global version of ringel s work for the positive parts of simple lie algebras in this realization the grothendieck group provides the cartan subalgebra the indecomposable objects correspond to the chevalley basis and the structure constants are given by the evaluations of hall polynomials

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

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

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

understanding cutting planes for qbfs abstract we study the cutting planes system cp red for quantified boolean formulas qbf obtained by augmenting propositional cutting planes with a universal reduction rule and analyse the proof theoretic strength of this new calculus while in the propositional case cutting planes is of intermediate strength between resolution and frege our findings here show that the situation in qbf is slightly more complex while cp red is again weaker than qbf frege and stronger than the cdcl based qbf resolution systems q res and qu res it turns out to be incomparable to even the weakest expansion based qbf resolution system exp res a similar picture holds for a semantic version semcp red technically our results establish the effectiveness of two lower bound techniques for cp red via strategy extraction and via monotone feasible interpolation

the qbf gallery over the last few years much progress has been made in the theory and practice of solving quantified boolean formulas qbf novel solvers have been presented that either successfully enhance established techniques or implement novel solving paradigms powerful preprocessors have been realized that tune the encoding of a formula to make it easier to solve frameworks for certification and solution extraction emerged that allow for a detailed interpretation of a qbf solver s results and new types of qbf encodings were presented for various application problems to capture these developments the qbf gallery was established in 2013 the qbf gallery aims at providing a forum to assess qbf tools and to collect new expressive benchmarks that allow for documenting the status quo and that indicate promising research directions these benchmarks became the basis for the experiments conducted in the context of the qbf gallery 2013 and follow up evaluations in this paper we report on the setup of the qbf gallery to this end we conducted numerous experiments which allowed us not only to assess the quality of the tools but also the quality of the benchmarks

locally finite triangulated categories abstract a k linear triangulated category a is called locally finite provided x ind a dim k hom a x y for any indecomposable object y in a it has auslander reiten triangles in this paper we show that if a connected triangulated category has auslander reiten triangles and contains loops then its auslander reiten quiver is of the form l n download download full size image by using this we prove that the auslander reiten quiver of any locally finite triangulated category a is of the form z g where is a dynkin diagram and g is an automorphism group of z for most automorphism groups g the triangulated categories with z g as their auslander reiten quivers are constructed in particular a triangulated category with l n as its auslander reiten quiver is constructed

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

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

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

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

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

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

resolution for quantified boolean formulas a complete and sound resolution operation directly applicable to the quantified boolean formulas is presented if we restrict the resolution to unit resolution then the completeness and soundness for extended quantified horn formulas is shown we prove that the truth of a quantified horn formula can be decided in o rn time where n is the length of the formula and r is the number of universal variables whereas in contrast the evaluation problem for extended quantified horn formulas is conp complete for formulas with prefix further we show that the resolution is exponential for extended quantified horn formulas

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

frege systems for quantified boolean logic we define and investigate frege systems for quantified boolean formulas qbf for these new proof systems we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class c to the qbf frege system operating with lines from c such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work this leads to strong lower bounds for restricted versions of qbf frege in particular an exponential lower bound for qbf frege systems operating with ac0 p circuits in contrast any non trivial lower bound for propositional ac0 p frege constitutes a major open problem improving these lower bounds to unrestricted qbf frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity in particular proving a lower bound for qbf frege systems operating with arbitrary p poly circuits is equivalent to either showing a lower bound for p poly or for propositional extended frege which operates with p poly circuits we also compare our new qbf frege systems to standard sequent calculi for qbf and establish a correspondence to intuitionistic bounded arithmetic

global properties of buckled states of plates that can suffer thickness changes this paper treats the axisymmetric buckling of nonlinearly elastic cosserat plates which can suffer thickness changes as well as flexure midplane extension and shear the governing equations are accordingly quite complicated nevertheless it is shown that all solutions bifurcating or not have a simple detailed nodal structure that distinguishes branches globally

characterising tree like frege proofs for qbf abstract we examine the tree like versions of qbf frege and extended frege systems while in the propositional setting tree like and dag like frege are equivalent we show that this is not the case for qbf frege where tree like systems are exponentially weaker this applies to the version of qbf frege where the universal reduction rule substitutes universal variables by 0 1 constants to show lower bounds for tree like qbf frege we devise a general technique that provides lower bounds for all tree like qbf systems of the form p red where p is a propositional system the lower bound is based on the semantic measure of strategy size corresponding to the size of countermodels for false qbfs we also obtain a full characterisation of hardness for tree like qbf frege lower bounds for this system either arise from a lower bound to propositional frege from a circuit lower bound or from a lower bound to strategy size

solving qbf with counterexample guided refinement abstract this article puts forward the application of counterexample guided abstraction refinement cegar in solving the well known pspace complete problem of quantified boolean formulas qbf the article studies the application of cegar in two scenarios in the first scenario cegar is used to expand quantifiers of the formula and subsequently a satisfiability sat solver is applied first it is shown how to do that for two levels of quantification and then it is generalized for arbitrary number of levels by recursion it is also shown that these ideas can be generalized to non prenex and non cnf qbf solvers in the second scenario cegar is employed as an additional learning technique in an existing dpll based qbf solver experimental evaluation of the implemented prototypes shows that the cegar driven solver outperforms existing solvers on a number of benchmark families and that the dpll solver benefits from the additional type of learning

relations for the grothendieck groups of triangulated categories a class of triangulated categories with a finiteness condition is singled out these triangulated categories have auslander reiten triangles it is proved that the relations of the grothendieck group of a triangulated category in this class are generated by all auslander reiten triangles moreover the auslander reiten quivers of certain triangulated categories in this class are described in terms of dynkin diagrams 2002 elsevier science usa all rights reserved

derived categories tilted algebras and drinfel d doubles abstract given a finite dimensional hereditary algebra over a finite field k in the derived category db we obtain some formulae on hall numbers associated to triangles using them we prove that for any tilted algebra of there is an embedding of the twisted ringel hall algebra of into the drinfel d double d of the twisted ringel hall algebra of furthermore if is also hereditary we show that this embedding can be extended as an isomorphism between the two corresponding drinfel d doubles these formulae are also used to construct an automorphism of d directly associated to auslander reiten translation and this automorphism follows by b sevenhant m van den bergh s construction in the case of quivers

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

root vectors arising from auslander reiten quivers consider the canonical isomorphism between the ringel hall algebra h and the positive part u of the drinfield jimbo quantum group uq g where the finite dimensional hereditary algebra and the semisimple lie algebra g enjoy a common dynkin diagram we get an algorithm to decompose the root vectors into combinations of monomials of the canonical generators ei in the quantum group depending only on the structure of the auslander reiten quiver of the algorithm is deduced by using a similar method as we did in x chen and j xiao 1999 compositio math 117 no 2 161 187 particularly the derivation defined there

singular global bifurcation problems for the buckling of anisotropic plates this paper treats a variety of unexpected pathologies that arise in the global bifurcation analysis of axisymmetrie buckled states of anisotropic plates the geometrically exact plate theory used accounts for flexure extension and shear the nonlinear constitutive functions have very general form as a consequence of the anisotropy the trivial solution may depend discontinuously on the load parameter accordingly the equations for the bifurcation problem have the same character so that bifurcating branches of solutions become disconnected as the load parameter crosses values at which discontinuities occur the anisotropy furthermore implies that the governing equations have a singular behaviour much worse than that for isotropic plates consequently a variety of novel constructions are required to demonstrate the validity of the essential results upon which global bifurcation theory stands these results include the compactness of certain operators and the uniqueness of solutions of initial value problems for singular ordinary differential equations it is shown that in regions of solution parameter space in which the equations depend continuously on the load parameter there exist connected global branches of solution pairs that have detailed nodal properties inherited from eigenfunctions of the linearized problem moreover these nodal properties are preserved across gaps occurring where discontinuities occur the methodology used to show this result actually supports constructive methods for finding disconnected branches

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

hall polynomials for the representation finite hereditary algebras let k be a field let r be a finite dimensional k algebra with centre k which is representation finite and hereditary thus r is morita equivalent to the tensor algebra of a k species with underlying graph a a disjoint union of dynkin diagrams and the set of isomorphism classes of indecomposable r modules corresponds bijectively to the set of positive roots of the corresponding semisimple complex lie algebra g see g and drl consequently the grothendieck group k r mod of all finitely generated r modules modulo split exact sequences is the free abelian group with basis indexed by let h be a cartan subalgebra of g and g n h 0 n the corresponding triangular decomposition note that n is the direct sum of one dimensional complex vectorspaces indexed by the elements of so we may identify k r mod q and n as vectorspaces and we deal with the problem of how to recover the lie multiplication of n on k r mod we have shown in r2 that the grothendieck group k r mod may be considered in a natural way as a lie algebra by using as structure constants the evaluations of hall polynomials at 1 the aim of this paper is to show that this lie algebra k r mod can be identified with a chevalley h form of n in particular k r mod and n are isomorphic as lie algebras we are going to determine all possible polynomials which occur as hall polynomials p where x y z e there are precisely 16 different polynomials including the zero polynomial cpo and the absolute value of their evaluations at 1 is bounded by 3 one easily observes that q x 0 in case y z x thus let us assume y z x in this case precisely one of the two polynomials cp and cpzz is non zero the non zero polynomials cp can be written in the form irqi where c i t with 1 d r 6 3 and pi is one of the following 12 integral polynomials

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

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

noncommutative projective curves and quantum loop algebras we show that the hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the quantized enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding kac moody algebra in particular this yields a geometric realization of the quantized enveloping algebra of elliptic or 2 toroidal algebras of types d 4 1 1 e 1 1 6 e 1 1 7 and e 8 1 1 in terms of coherent sheaves on weighted projective lines of genus one or equivalently in terms of equivariant coherent sheaves on elliptic curves

minimal generating system of ringel hall algebras of affine valued quivers abstract let f q be a finite field with q elements and h f q q the ringel hall algebra of affine valued quiver q all indecomposable representations of q which can be generated inside h f q q by smaller dimension are classified and minimal homogeneous generating systems of h f q q are given explicitly

on root categories of finite dimensional algebras abstract for any finite dimensional algebra a over a field k of finite global dimension we investigate the root category r a as the triangulated hull of the 2 periodic orbit category of a via the construction of b keller in on triangulated orbit categories this is motivated by ringel hall lie algebras associated to 2 periodic triangulated categories as an application we study the ringel hall lie algebras for a class of finite dimensional k algebras of global dimension 2 which turns out to give an alternative answer to a question of gim lie algebras by slodowy in beyond kac moody algebra and inside

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

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

domestic canonical algebras and simple lie algebras for each simply laced dynkin graph we realize the simple complex lie algebra of type as a quotient algebra of the complex degenerate composition lie algebra l a 1 mathbb c of a domestic canonical algebra a of type by some ideal i of l a 1 mathbb c that is defined via the hall algebra of a and give an explicit form of i moreover we show that each root space of l a 1 mathbb c i has a basis given by the coset of an indecomposable a module m with root easily computed by the dimension vector of m

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