the green rings of the 2 rank taft algebra and its two relatives twisted abstract in the paper the representation rings or the green rings for a family of hopf algebras of tame type the 2 rank taft algebra at q 1 and its two relatives twisted by 2 cocycles are explicitly described via a representation theoretic analysis it turns out that the green rings can serve to detect effectively the twist equivalent hopf algebras here

covariant connections on bicovariant differential calculus abstract given a bicovariant differential calculus e d such that the braiding map is diagonalisable in a certain sense the bimodule of two tensors admits a direct sum decomposition into symmetric and anti symmetric tensors this is used to prove the existence of a bicovariant torsionless connection on e following heckenberger and schmudgen we study invariant metrics and the compatibility of covariant connections with such metrics a sufficient condition for the existence and uniqueness of bicovariant levi civita connections is derived this condition is shown to hold for cocycle deformations of classical lie groups

monomorphism categories cotilting theory and gorenstein projective modules abstract the monomorphism category s n x is introduced where x is a full subcategory of the module category a mod of an artin algebra a the key result is a reciprocity of the monomorphism operator s n and the left perpendicular operator for a cotilting a module t there is a canonical construction of a cotilting module m t over the upper triangular matrix algebra t n a such that s n t m t as applications s n x is a resolving contravariantly finite subcategory in t n a mod with s n x t n a mod if and only if x is a resolving contravariantly finite subcategory in a mod with x a mod for a gorenstein algebra a the category t n a g proj of gorenstein projective t n a modules can be explicitly determined as s n a also self injective algebras a can be characterized by the property t n a g proj s n a finally we obtain a characterization of those categories s n a which have finite representation type in terms of auslander s representation dimension

tame wild dichotomy of birkhoff type problems for nilpotent linear operators abstract inspired by recent results of ringel schmidmeier kussin lenzing meltzer xiao wu chen and pu zhang given a field k m 1 and a finite poset i i with a unique maximal element we study the category m o n i f m of mono representations of i over the frobenius k algebra f m k t t m of k dimension m viewed as k vector spaces u with an m nilpotent k linear operator t u u together with t invariant subspaces u i u j u for all i j in i the problem of when the krull schmidt k category m o n i f m is of wild resp tame representation type is called a wild resp tame birkhoff type problem for m nilpotent operators in case when k is algebraically closed we give a complete solution of the problem by describing all minimal pairs i m resp all pairs with m 1 such that category m o n i f m is of wild resp tame representation type we reduce the problem to a birkhoff type problem for the category fspr i f m m o n i f m of subprojective representations over f m of a larger poset i i the tame wild dichotomy for the category m o n i f m is also proved surprisingly in case when i i a b is the union of two incomparable chains i and i of length i a 1 1 and i b 1 1 with i i the problem is equivalent with the wildness resp tameness of the category coh x p of coherent sheaves over the weighted projective line x p for the weight triple p a b m with a b m 2 studied by kussin lenzing and meltzer 19 in relation with the hypersurface singularity f x 1 a x 2 b x 3 m

a frobenius category within the monomorphism category abstract given a selfinjective artin algebra we consider the category s inj of all embeddings of a left module in a finitely generated injective left module we show that s inj is a frobenius category with auslander reiten sequences such that the categories mod and s inj are stably equivalent and s inj has twice as many indecomposable injective objects as mod

logarithmic widehat s ell 2 cft models from nichols algebras i we construct chiral algebras that centralize rank 2 nichols algebras with at least one fermionic generator this gives logarithmic w algebra extensions of a fractional level algebra we discuss crucial aspects of the emerging general relation between nichols algebras and logarithmic conformal field theory cft models i the extra input beyond the nichols algebra proper needed to uniquely specify a conformal model ii a relation between the cft counterparts of nichols algebras connected by weyl groupoid maps and iii the common double bosonization u x of such nichols algebras for an extended chiral algebra candidates for its simple modules that are counterparts of the u x simple modules are proposed as a first step toward a functorial relation between u x and w algebra representation categories

invariant subspaces of nilpotent linear operators i let k be a field we consider triples v u t where v is a finite dimensional k space u a subspace of v and t v v a linear operator with t n 0 for some n and such that t u u thus t is a nilpotent operator on v and u is an invariant subspace with respect to t we will discuss the question whether it is possible to classify these triples these triples v u t are the objects of a category with the krullremak schmidt property thus it will be sufficient to deal with indecomposable triples obviously the classification problem depends on n and it will turn out that the decisive case is n 6 for n 6 we deal with what is called wild representation type so no complete classification can be expected for n 6 we will exhibit a complete description of all the indecomposable triples let k be a field and v a finite dimensional k vector space we consider a linear operator t v v which is nilpotent say satisfying t n 0 for some fixed natural number n and we are interested in the subspaces u of v which are t invariant this means that t u u to be more precise we are going to consider the corresponding triples v u t and ask for a classification of all the isomorphism classes in order to do so we consider these triples v u t as the objects of a category s n or sk n in case a reference to the base field

on semisimple hopf algebras of dimension pq2 let p and q be distinct prime numbers we prove a result on the existence of nontrivial group like elements in a certain class of semisimple hopf algebras of dimension pqr we conclude the classification of semisimple hopf algebras a of dimension pq2 over an algebraically closed field k of characteristic zero such that both a and a are of frobenius type we also complete the classification of semisimple hopf algebras of dimension pq2 100

algebraic deformations of toric varieties i general constructions abstract we construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry isospectral deformations and noncommutative geometry in braided monoidal categories our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus we develop a sheaf theory using techniques from noncommutative algebraic geometry the cases of projective varieties are studied in detail and several explicit examples are worked out including new noncommutative deformations of grassmann and flag varieties our constructions set up the basic ingredients for a thorough study of instantons on noncommutative toric varieties which will be the subject of the sequel to this paper

quantum differentials on cross product hopf algebras abstract we construct canonical strongly bicovariant differential graded algebra structures on all four flavours of cross product hopf algebras namely double cross products a a h h double cross coproducts biproducts and bicrossproducts on the assumption that the factors have strongly bicovariant calculi a h or a braided version b we use super versions of each of the constructions moreover the latter three quantum groups all coact canonically on one of their factors and we show that this coaction is differentiable in the case of the drinfeld double d a h a op h where a is dually paired to h we show that its canonical actions on a h are differentiable examples include a canonical c q g l 2 c 2 for the quantum group of affine transformations of the quantum plane and c poin c 1 1 for the bicrossproduct poincare quantum group in 2 dimensions we also show that c q g l 2 itself is uniquely determined by differentiability of the canonical coaction on the quantum plane and of the determinant subalgebra

perron vector ordering for a subclass of tournament matrices abstract we consider the class of n n tournament matrices having scor vector 1 1 2 3 n 4 n 3 n 2 n 2 1 for each such matrix we give formulas for the entries in its perron vector in terms of the corresponding perron value this leads to a discussion of the ordering of the entries in the perron vector and so yields some insight into the kendall wei method for ranking players in a round robin competition finally we characterize all matrices from this class such that the ordering of the entries in the perron vector coincides with the ordering of the entries in the score vector

quantization by cochain twists and nonassociative differentials we show that several standard associative quantizations in mathematical physics can be expressed as cochain module algebra twists in the spirit of moyal products at least to o 3 but to achieve this we twist not by a 2 cocycle but by a 2 cochain this implies a hidden nonassociativity not visible in the algebra itself but present in its deeper noncommutative differential geometry a phenomenon first seen in our previous work on semiclassicalization of differential structures the quantizations are induced by a classical group covariance and include enveloping algebras u g as quantizations of g a fedosov type quantization of the sphere s2 under a lorentz group covariance the mackey quantization of homogeneous spaces and the standard quantum groups cq g we also consider the differential quantization of rn for a given symplectic connection as part of our semiclassical analysis and we outline a proposal for the dirac operator

yetter drinfeld modules under cocycle twists abstract we give an explicit formula for the correspondence between simple yetter drinfeld modules for certain finite dimensional pointed hopf algebras h and those for cocycle twists h of h this implies an equivalence between modules for their drinfeld doubles to illustrate our results we consider the restricted two parameter quantum groups u r s sl n under conditions on the parameters guaranteeing that u r s sl n is a drinfeld double of its borel subalgebra we determine explicit correspondences between u r s sl n modules for different values of r and s and provide examples where no such correspondence can exist our examples were obtained via the computer algebra system singular plural

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

bicrossproduct approach to the connes moscovici hopf algebra abstract we give a rigorous proof that the codimension one connes moscovici hopf algebra h cm is isomorphic to a bicrossproduct hopf algebra linked to a group factorisation of the diffeomorphism group diff r we construct a second bicrossproduct u cm equipped with a nondegenerate dual pairing with h cm we give a natural quotient hopf algebra k heis of h cm and hopf subalgebra u heis of u cm which again are in duality all these hopf algebras arise as deformations of commutative or cocommutative hopf algebras that we describe in each case finally we develop the noncommutative differential geometry of k heis by studying first order differential calculi of small dimension

from submodule categories to the stable auslander algebra abstract we construct two functors from the submodule category of a representation finite self injective algebra to the module category of the stable auslander algebra of these functors factor through the module category of the auslander algebra of moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism their construction uses a recollement of the module category of the auslander algebra induced by an idempotent and this recollement determines a characteristic tilting and cotilting module if is taken to be a nakayama algebra then said tilting and cotilting module is a characteristic tilting module of a quasi hereditary structure on the auslander algebra we prove that the self injective nakayama algebras are the only algebras with this property

the auslander reiten translation in submodule categories let a be an artin algebra or more generally a locally bounded associative algebra and s the category of all embeddings a c b where b is a finitely generated a module and a is a submodule of b then s is an exact krull schmidt category which has auslander reiten sequences in this manuscript we show that the auslander reiten translation in s can be computed within mod a by using our construction of minimal monomorphisms if in addition a is uniserial then any indecomposable nonprojective object in s is invariant under the sixth power of the auslander reiten translation

from submodule categories to preprojective algebras let s n be the category of invariant subspaces of nilpotent operators with nilpotency index at most n such submodule categories have been studied already in 1934 by birkhoff they have attracted a lot of attention in recent years for example in connection with some weighted projective lines kussin lenzing meltzer on the other hand we consider the preprojective algebra pi n of type mathbb a n the preprojective algebras were introduced by gelfand and ponomarev they are now of great interest for example they form an important tool to study quantum groups lusztig or cluster algebras geiss leclerc schroer we are going to discuss the connection between the submodule category mathcal s n and the module category hbox mod pi n 1 of the preprojective algebra pi n 1 dense functors mathcal s n rightarrow hbox mod pi n 1 are known to exist one has been constructed quite a long time ago by auslander and reiten recently another one by li and zhang we will show that these two functors are full dense objective functors with index 2n thus hbox mod pi n 1 is obtained from mathcal s n by factoring out an ideal which is generated by 2n indecomposable objects as a byproduct we also obtain new examples of ideals in triangulated categories namely ideals mathcal i in a triangulated category mathcal t which are generated by an idempotent such that the factor category mathcal t mathcal i is an abelian category

a construction of gorenstein projective modules abstract we determine all the gorenstein projective modules over the t 2 extension of a gorenstein algebra and over a m 0 b where a and b are self injective algebras and m is an a b bimodule with m a and m b projective using this we obtain a criterion for the cm finiteness of the t 2 extensions of cm finite gorenstein algebras as an application we get non trivial examples of cm finite gorenstein algebras

algebraic multiplicity of the eigenvalues of a tournament matrix let f denote the set of irreducible n x n tournament matrices here arc our main results 1 for all n 3 every matrix in k has at least three distinct eigenvalues such a matrix has exactly three distinct eigenvalues if and only if it is a hadamard tournament matrix 2 for all n 2 3 there is a matrix in y having n distinct eigenvalues 3 if cr denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z then ln 21 2 6 4 if r is the minimum perron value i e spectral radius of all matrices in r then 2 8

three results on frobenius categories this paper consists of three results on frobenius categories 1 we give sufficient conditions on when a factor category of a frobenius category is still a frobenius category 2 we show that any frobenius category is equivalent to an extension closed exact subcategory of the frobenius category formed by cohen macaulay modules over some additive category this is an analogue of gabriel quillen s embedding theorem for frobenius categories 3 we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure with which the given category becomes a frobenius category several applications of the results are discussed

bimodule monomorphism categories and rss equivalences via cotilting modules abstract the monomorphism category m a m b induced by a bimodule m b a is the subcategory of mod consisting of x y such that m b y x is a monic a map where a m 0 b and a b are artin algebras in general m a m b is not the monomorphism category induced by quivers it could describe the gorenstein projective modules this monomorphism category is a resolving subcategory of mod if and only if m b is projective in this case it has enough injective objects and auslander reiten sequences and can be also described as the left perpendicular category of a unique basic cotilting module if m satisfies the condition ip see subsection 1 6 then the stable category of m a m b admits a recollement of additive categories which is in fact a recollement of singularity categories if m a m b is a frobenius category ringel schmidmeier simson equivalence between m a m b and its dual is introduced if m is an exchangeable bimodule then an rss equivalence is given by a bimodule which is a two sided cotilting module with a special property and the nakayama functor n gives an rss equivalence if and only if both a and b are frobenius algebras

twisting of quantum differentials and the planck scale hopf algebra we show that the crossed modules and bicovariant differential calculi on two hopf algebras related by a cocycle twist are in 1 1 correspondence in particular for quantum groups which are cocycle deformation quantisations of classical groups the calculi are obtained as deformation quantisations of the classical ones as an application we classify all bicovariant differential calculi on the planck scale hopf algebra this is a quantum group which has an limit as the functions on a classical but non abelian group and a limit as flat space quantum mechanics we further study the noncommutative differential geometry and fourier theory for this hopf algebra as a toy model for planck scale physics the fourier theory implements a t duality like self duality the noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

separated monic representations i gorenstein projective modules abstract for a finite acyclic quiver q an ideal i of a path algebra kq generated by monomial relations and a finite dimensional k algebra a we introduce the separated monic representations of a bound quiver q i over a they differ from the usual monic representations the category smon q i a of the separated monic representations of q i over a coincides with the category mon q i a of the usual monic representations if and only if i 0 and each vertex of q is the ending vertex of at most one arrow we give properties of the structural maps of separated monic representations and prove that smon q i a is a resolving subcategory of rep q i a we introduce the condition g let a k q i by the equivalence rep q i a mod of categories the main result claims that a module is gorenstein projective if and only if it is in smon q i a and has a local a gorenstein projective property g as consequences the separated monic modules are exactly the projective modules if and only if a is semi simple and they are exactly the gorenstein projective modules if and only if a is self injective and if and only if smon q i a is a frobenius category

exterior differentials of higher order and their covariant generalization we investigate a particular realization of generalized q differential calculus of exterior forms on a smooth manifold based on the assumption that dn 0 while dk 0 for k n it implies the existence of cyclic commutation relations for the differentials of first order and their generalization for the differentials of higher order special attention is paid to the cases n 3 and n 4 a covariant basis of the algebra of such q grade forms is introduced and the analogs of torsion and curvature of higher order are considered we also study a zn graded exterior calculus on a generalized clifford algebra

braided cyclic cocycles and nonassociative geometry we use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a drinfeld type cochain twist these are the so called quasialgebras and include the octonions as braided commutative but nonassociative coordinate rings as well as quasialgebra versions cq g of the standard q deformation quantum groups we introduce the notion of ribbon algebras in the category which are algebras equipped with a suitable generalized automorphism and obtain the required generalization of cyclic cohomology we show that this braided cyclic cohomology is invariant under a cochain twist we also extend to our generalization the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra the paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry as well as the algebraic noncommutative torus as an associative example

a note on perron vectors for almost regular tournament matrices abstract let t be an almost regular tournament matrix of order n with right perron vector w we show that if the i th row sum of t is n 2 2 and the j th row sum is n 2 then w i w j thus in the round robin competition corresponding to t the ranking schemes of kendall and wei and of ramanujacharyula agree with the ranking generated by the row sums of t

noncommutative finite dimensional manifolds i spherical manifolds and related examples we exhibit large classes of examples of noncommutative finite dimensional manifolds which are non formal deformations of classical manifolds the main result of this paper is a complete description of noncommutative three dimensional spherical manifolds a noncommutative version of the sphere s 3 defined by basic k theoretic equations we find a 3 parameter family of deformations of the standard 3 sphere s 3 and a corresponding 3 parameter deformation of the 4 dimensional euclidean space 4 for generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed 4 u only depend on two parameters and are isomorphic to the algebras introduced by sklyanin in connection with the yang baxter equation it follows that different can span the same this equivalence generates a foliation of the parameter space this foliation admits singular leaves reduced to a point these critical points are either isolated or fall in two 1 parameter families up to the simple operation of taking the fixed algebra by an involution these two families are identical and we concentrate here on c for the above isomorphism with the sklyanin algebra breaks down and the corresponding algebras are special cases of deformations a notion which we generalize in any dimension and various contexts and study in some detail here and this point is crucial the dimension is not an artifact i e the dimension of the classical model but is the hochschild dimension of the corresponding algebra which remains constant during the deformation besides the standard noncommutative tori examples of deformations include the recently defined noncommutative 4 sphere as well as m dimensional generalizations noncommutative versions of spaces and quantum groups which are deformations of various classical groups we develop general tools such as the twisting of the clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative dimensional spherical manifolds a key result is the differential self duality properties of these projections which generalize the self duality of the round instanton

on algebras of finite cohen macaulay type abstract we study artin algebras and commutative noetherian complete local rings r in connection with the following decomposition property of gorenstein projective modules any gorenstein projective module is a direct sum of finitely generated modules we show that the class of algebras enjoying coincides with the class of virtually gorenstein algebras of finite cohen macaulay type introduced in beligiannis and reiten 2007 27 beligiannis 2005 24 thus we solve the problem stated in chen 2008 33 this is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits thus generalizing a classical result of auslander 1976 9 and ringel and tachikawa 1974 63 in the commutative case if r admits a non free finitely generated gorenstein projective module then we show that r is of finite cohen macaulay type iff r is gorenstein and satisfies we also generalize a result of yoshino 2005 68 by characterizing when finitely generated modules without extensions with the ring are gorenstein projective finally we study the stable relative auslander algebra of a virtually gorenstein algebra of finite cohen macaulay type and under the presence of a cluster tilting object we give descriptions of the stable category of gorenstein projective modules in terms of the cluster category associated to the quiver of the stable relative auslander algebra in this setting we show that the cluster category is invariant under derived equivalences

abelian quotients of categories of short exact sequences abstract we mainly investigate abelian quotients of categories of short exact sequences the natural framework to consider the question is via identifying quotients of morphism categories as module categories these ideas not only can be used to recover the abelian quotients produced by cluster tilting subcategories of both exact categories and triangulated categories but also can be used to reach our goal let c e be an exact category we denote by e c the category of bounded complexes whose objects are given by short exact sequences in e and by s e c the full subcategory formed by split short exact sequences in general e c is just an exact category but the quotient e c s e c turns out to be abelian in particular if c e is frobenius we present three equivalent abelian quotients of e c and point out that the equivalences are actually given by left and right rotations the abelian quotient e c s e c admits some nice properties we explicitly describe the abelian structure projective objects injective objects and simple objects which provide a new viewpoint to understanding hilton rees theorem and auslander reiten theory furthermore we present some analogous results for triangulated versions

quantisation of twistor theory by cocycle twist we present the main ingredients of twistor theory leading up to and including the penrose ward transform in a coordinate algebra form which we can then quantise by means of a functorial cocycle twist the quantum algebras for the conformal group twistor space mathbb cp 3 compactified minkowski space mathbb cm and the twistor correspondence space are obtained along with their canonical quantum differential calculi both in a local form and in a global algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties we outline how the penrose ward transform then quantises as an example we show that the pull back of the tautological bundle on mathbb cm pulls back to the basic instanton on s 4 subset mathbb cm and that this observation quantises to obtain the connes landi instanton on deformed s4 as the pull back of the tautological bundle on our deformed mathbb cm we likewise quantise the fibration mathbb cp 3 rightarrow s 4 and use it to construct the bundle on deformed mathbb cp 3 that maps over under the transform to the deformed instanton

the entries in the lr tableau let be the littlewood richardson tableau corresponding to an embedding m of a subgroup in a finite abelian p group each individual entry in yields information about the homomorphisms from m into a particular subgroup embedding and hence determines the position of m within the category of subgroup embeddings conversely this category provides a categorification for lr tableaux in the sense that all subgroup embeddings corresponding to a given lr tableau share certain homological properties

poisson lie t duality for quasitriangular lie bialgebras we introduce a new 2 parameter family of sigma models exhibiting poisson lie t duality on a quasitriangular poisson lie group g the models contain previously known models as well as a new 1 parameter line of models having the novel feature that the lagrangian takes the simple form where the generalised metric e is constant not dependent on the field u as in previous models we characterise these models in terms of a global conserved g invariance the models on g su 2 and its dual g are computed explicitly the general theory of poisson lie t duality is also extended notably the reduction of the hamiltonian formulation to constant loops as integrable motion on the group manifold the approach also points in principle to the extension of t duality in the hamiltonian formulation to group factorisations d g m where the subgroups need not be dual or connected to the drinfeld double