character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

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

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

on the stack of semistable g bundles over an elliptic curve in a recent paper ben zvi and nadler proved that the induction map from b bundles of degree 0 to semistable g bundles of degree 0 over an elliptic curve is a small map with galois group isomorphic to the weyl group of g we generalize their result to all connected components of text bun g for an arbitrary reductive group g we prove that for every degree i e topological type there exists a unique parabolic subgroup such that any semistable g bundle of this degree has a reduction to it and moreover the induction map is small with galois group the relative weyl group of the levi this provides new examples of simple automorphic sheaves which are constituents of eisenstein sheaves for the trivial local system

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

the hopf algebra of skew shapes torsion sheaves on a f1n and ideals in hall algebras of monoid representations abstract we study ideals in hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations these conditions include the property that the monoid act via partial permutations that the representation possess a compatible grading and conditions on the support of the module quotients by these ideals lead to combinatorial hopf algebras which can be interpreted as hall algebras of certain sub categories of modules in the case of the free commutative monoid on n generators we obtain a co commutative hopf algebra structure on n dimensional skew shapes whose underlying associative product amounts to a stacking operation on the skew shapes the primitive elements of this hopf algebra correspond to connected skew shapes and form a graded lie algebra by anti symmetrizing the associative product we interpret this hopf algebra as the hall algebra of a certain category of coherent torsion sheaves on a f 1 n supported at the origin where f 1 denotes the field of one element this hopf algebra may be viewed as an n dimensional generalization of the hopf algebra of symmetric functions which corresponds to the case n 1

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

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

branching rules for quantum toroidal gln abstract we construct an analog of the subalgebra u gl n u gl m u gl m n in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra

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

quantum toroidal mathfrak g mathfrak l _ 1 and bethe ansatz we establish the method of bethe ansatz for the xxz type model obtained from the r matrix associated to quantum toroidal we do this by using shuffle realizations of the modules and by showing that the hamiltonian of the model is obtained from a simple multiplication operator by taking an appropriate quotient we expect this approach to be applicable to a wide variety of models

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

the q agt w relations via shuffle algebras we construct the action of the q deformed w algebra on its level r representation geometrically using the moduli space of u r instantons on the plane and the double shuffle algebra we give an explicit ldu decomposition for the action of w algebra currents in the fixed point basis of the level r representation and prove a relation between the carlsson okounkov ext operator and intertwiners for the deformed w algebra we interpret this result as a q deformed version of the agt w relations

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

upper and lower error estimation for the tau method and related polynomial techniques abstract a new method is discussed by which estimates of upper and lower bounds of the maximum tau method approximation error are obtained it improves on the upper bound estimate of lanczos and other more recently proposed estimates we give an example of a non linear ode where our technique is applied and show how it is implemented for chebyshev series expansions collocation and spectral methods

modules over plane curve singularities in any ranks and daha abstract we generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks explicit formulas are obtained for torus knots the new feature is the definition of counterparts of jacobian factors directly related to compactified jacobians for higher ranks which is parallel to the classical passage from invertible sheaves to vector bundles over algebraic curves this is an entirely local theory connected with affine springer fibers for non reduced germs of spectral curves we conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding daha superpolynomials for any algebraic knots colored by columns

inverse eigenproblem for r symmetric matrices and their approximation let email protected c n x n be a nontrivial involution i e r r 1 i n we say that email protected c n x n is r symmetric if rgr g the set of all nxnr symmetric matrices is denoted by gsc n x n in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors x i i 1 m in c n and a set of complex numbers l i i 1 m find a matrix email protected gsc n x n such that x i i 1 m and l i i 1 m are respectively the eigenvalues and eigenvectors of a we then consider the following approximation problem given an nxn matrix email protected find email protected email protected s e such that email protected email protected email protected min a s email protected email protected email protected where s e is the solution set of iep and email protected email protected is the frobenius norm we provide an explicit formula for the best approximation solution email protected by means of the canonical correlation decomposition

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

the homflypt skein algebra of the torus and the elliptic hall algebra we give a generators and relations presentation of the homflypt skein algebra h of the torus t2 and we give an explicit description of the module corresponding to the solid torus using this presentation we show that h is isomorphic to the 1 specialization of the elliptic hall algebra of burban and schiffmann as an application for an iterated cable k of the unknot we use the elliptic hall algebra to construct a 3 variable polynomial that specializes to the colored homflypt polynomial of k we show that this polynomial also specializes to one constructed by cherednik and danilenko using the gln double affine hecke algebra this proves one of the connection conjectures in their recent work

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

the affine yangian of gl1 revisited abstract the affine yangian of gl 1 has recently appeared simultaneously in the work of maulik okounkov 11 and schiffmann vasserot 20 in connection with the alday gaiotto tachikawa conjecture while the presentation from 11 is purely geometric the algebraic presentation in 20 is quite involved in this article we provide a simple loop realization of this algebra which can be viewed as an additivization of the quantum toroidal algebra of gl 1 in the same way as the yangian y h g is an additivization of the quantum loop algebra u q l g for a simple lie algebra g we also explain the similarity between the representation theories of the affine yangian and the quantum toroidal algebras of gl 1 by generalizing the main result of 10 to the current settings

an inverse eigenvalue problem and an associated approximation problem for generalized k centrohermitian matrices a partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized k centrohermitian matrices are considered it is shown under which conditions the inverse eigenproblem has a solution an expression of its general solution is given in case a solution of the inverse eigenproblem exists the optimal approximation problem can be solved the formula of its unique solution is given

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

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

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

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

the harder narasimhan system in quantum groups and cohomology of quiver moduli methods of harder and narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations using the hall algebra approach to quantum groups an analog of the harder narasimhan recursion is constructed inside the quantized enveloping algebra of a kac moody algebra this leads to a canonical orthogonal system the hn system in this algebra using a resolution of the recursion an explicit formula for the hn system is given as an application explicit formulas for betti numbers of the cohomology of quiver moduli are derived generalizing several results on the cohomology of quotients in linear algebra type situations

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

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

derived hall algebras the purpose of this work is to define a derived hall algebra mathcal dh t associated to any dg category t under some finiteness conditions our main theorem states that mathcal dh t is associative and unital it is shown that mathcal dh t contains the usual hall algebra mathcal h t when t is an abelian category we will also prove an explicit formula for the derived hall numbers purely in terms of invariants of the triangulated category associated to t as an example we describe the derived hall algebra of an hereditary abelian category