branching rules for conformal embeddings we give explicit formulas for the branching rules of the conformal embeddingssu n n 1 2 1 su n n 2 su n n 1 2 1 su n n 2 sp n 1 so n 4 su 2 n andso m n 1 so m 1 so n 1 withm andn odd

on spacelike hypersurfaces with constant mean curvature in the de sitter space soit n n n 1 c g une variete de lorentz a n 1 dimensions de courbure constante positive c soit m une hypersurface du genre espace complete a courbure moyenne constante h dans n on suppose que 1 h c 1 2 quand n 2 2 h 2 n 1 c 1 2 n quand n 3 alors m est totalement ombilicale

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

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

prinjective modules reflection functors quadratic forms and auslander reiten sequences let a b be artinian rings and let a m b be an a b bimodule which is a finitely generated left a module and a finitely generated right b module a right a m b prinjective module is a finitely genrated module x r x a x b x a a m b x b over the triangular matrix ring r a o a m b b such that x a is a projective a module x b is an injective b module and is a b homomorphism we study the category prin r b a of right a m b prinjective modules it is an additive krull schmidt subcategory of mod r closed under extensions

cluster concealed algebras abstract the cluster tilted algebras have been introduced by buan marsh and reiten they are the endomorphism rings of cluster tilting objects t in cluster categories we call such an algebra cluster concealed in case t is obtained from a preprojective tilting module for example all representation finite cluster tilted algebras are cluster concealed if c is a representation finite cluster tilted algebra then the indecomposable c modules are shown to be determined by their dimension vectors for a general cluster tilted algebra c we are going to describe the dimension vectors of the indecomposable c modules in terms of the root system of a quadratic form the roots may have both positive and negative coordinates and we have to take absolute values

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

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

the branching rules of conformal embeddings after recalling the main properties of a conformal embedding of lie algebras g id p which is defined by the equality of the sugawara central charges on both sides we launch a systematic study of their branching rules the bulk of the paper is devoted to the proof of a general formula in the case suim su m n su m at the end we give some applications to the con struction of modular invariant partition functions

degenerations in a class of matrix varieties and prinjective modules abstract let k be an algebraically closed field and let i be a finite partially ordered set to any vector v n i we associate the irreducible variety mat v i with an action of the algebraic group g v i see section 6 the main aim of this paper is to give a description of degenerations of orbits of this action in terms of homomorphisms and extensions of prinjective ki modules

socle reductions and socle projective modules a useful and simple idea in determining the representation type of an artinian ring a is to relate its module category mod a with a module category of another ring a whose representation type is known or it is easier to study this was successfully used by several authors let us mention the coxeter functors connection 9 10 3 38 stable equivalences 2 231 the connection between radical square zero rings and hereditary rings 11 221 connections between representations of algebras and representations of partially ordered sets 19 20 24 253 or more generally with representations of bows 29 and the covering technique 131 an idea close to the one above is to find some subcategory 9 of mod a whose representations type is known and to look at the factor category mod a w of mod a modulo the two sided ideal i in mod a generated by maps in sk if mod a sy zmod a 8 and the representation type of a is known then we get a lot of information about mod a 2 22 23 on the other hand there are several situations when mod is equivalent to a nice subcategory 55 in some mod r and there are constructive methods for studying indecomposable objects in v it follows from 6 s 31 32 34 361 that the matrix problems technique and the differentiation procedure 19 201 lead to this scheme in connection with these ideas the notion of a right peak ring r is introduced and socle projective modules mod r over r are studied in 31 32f it is shown there that several factor categories of mod a where a is an artinian ring are of the form mod r where r is a right peak ring in particular if a t y where f is a division pi ring and t is a

space like submanifolds in de sitter spaces let m n be a space like submanifold in a de sitter space mpn p c with flat normal bundle this paper gives some intrinsic conditions for m n to be totally umbilical

a reduction functor tameness and tits form for a class of orders abstract let d be a complete discrete valuation domain which is an algebra over an algebraically closed field k we study in the paper a class of suborders of tiled d orders by means of a rational quadratic form q associated to and a finite poset i having exactly two maximal elements and criteria for finite lattice type and for tame lattice type of are given in terms of the form q and of the category i spr of socle projective k linear representations of the poset i the shape of auslander reiten quiver latt is described in corollary 3 2 a reduction functor h latt i spr preserving representation types is constructed it is shown in corollary 2 10 that for any poset i having exactly two maximal elements there exists a d order in our class an additive functor latt i spr preserving representation types and a poset isomorphism i i

schurian vector space categories hereditary algebras and roiter s norm abstract we associate a positive real number c to any vector space k category c over a field k generalizing a result of nazarova and roiter we show that a schurian vector space k category c is representation finite if and only if ind c is finite and c 1 4 such vector space categories are quasilinear i e its indecomposables are simple modules over their endomorphism ring recently nazarova and roiter introduced the concept of p faithful poset in order to clarify the structure of critical posets their conjecture on the precise form of p faithful posets was established by zeldich we generalize these results and characterize p faithful quasilinear vector space k categories in terms of a class of hereditary algebras h d parametrized by a skew field d and a rational number 1

conformal embeddings rank level duality and exceptional modular invariants we compute the branching rules of the conformal embeddingsso 4nk 1 sp 2n k sp 2k n andso rq 1 so r q so q r forrq even using this we prove that the affine algebrassp 2n k andsp 2k n have the sames matrix and modular invariants as a second application we show how the triality ofso 8 leads to an exceptional modular invariant forsu 2 at level 16 and for allso q 4 at level 8

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

partial coxeter functors and right pure semisimple hereditary rings an important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type see 2 7 16 201 we recall that a ring r is of finite representation type if r is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated r modules the main tools for studying the representation theory of hereditary artinian rings are partial coxeter functors and coxeter functors see 4 8 10 141 they are successfuly applied in the investigation of hereditary rings of finite representation type 8 9 12 141 in the present paper we study partial coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings which are close to rings of finite representation type we recall that for any ring r the right pure global dimension r p gl dim r and the left pure global dimension l p gl dim r are defined 19 21 261 there is an interesting problem how to characterize rings r with r p gl dim r 0 which are exactly those right artinian rings r all of whose right r modules are algebraically compact 19 20 261 or equivalently all of whose right r modules are direct sums of modules of finite length a ring r with r p gl dim r 0 is called right pure semisimple 25 271 it is well known that any ring of finite representation type is both left and right pure semisimple 12 20 23 251 and the converse implication also holds true 2 15 201 however it is still an open question if a right pure semisimple ring r is of finite representation type or equivalently if r p gl dim r 0 implies l p gl dim r 0 see 29 a positive solution of this problem is given by auslander 3 for artin algebras he also discusses the problem for arbitrary left artinian rings see also 1181 we recall that a 195 0021 8693 81 070195 24 02 00 0

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

a peak reduction functor for socle projective representations throughout this paper k is a fixed algebraically closed field given a finite poset i e partially ordered set j with the partial order u we denote by max j the set of all maximal elements of j with respect to the order u and we set jys j max j the poset j is said to be an r peak poset provided max j s r throughout this paper we suppose that j s 4 4 1 n p p and max j s p p without loss of generality 1 r 1 r we can assume that the partial order u in j is such that i u j implies 4 i f j in the natural order for all i j g 1 n w x we recall from 18 that the category j spr of peak j spaces or socle projective representations of j over the field k is defined as follows the objects of j spr are systems m s m of finite dimensional k vector j jg j spaces m such that j

a reduced tits quadratic form and tameness of three partite subamalgams of tiled orders let d be a complete discrete valuation domain with the unique maximal ideal p we suppose that d is an algebra over an algebraically closed field k and d p k subamalgam d suborders of a tiled d order are studied in the paper by means of the integral tits quadratic form q zn1 2n3 2 z a criterion for a subamalgam d order to be of tame lattice type is given in terms of the tits quadratic form q and a forbidden list 1 17 of minor d suborders of presented in the tables

a narrow over ring adjustment functor such that xi is finitely generated xl is finitely cogenerated 9 is surjective and the a homomorphism cp xl hom an xg adjoint to rp is injective the category of adjusted right r modules will be denoted by adj r note that if r is an artin algebra then x is in adj r if and only if x is finitely generated sot x is a b module and top x is an a module via the natural epimorphisms a t r b in this case adj i r is one of the classes of modules considered by auslander and smal l and therefore adj has the auslander reiten sequences 1 proposition 6 51 the category adjg r was introduced in 25 26 in order to give a module theoretical interpretation of arbitrary bimodule matrix problems in the sense of drozd and roiter it was shown in 26 that if n kop x k mod k is an additive k functor k is a field k and l are additive krull schmidt k categories and mat jv is the category of matrices then there are a faithfully right b traced ring r and a full dense functor 0 mat wl adj r preserving the representation type and such that 0 vanishes only on a small set of trivial idecomposable matrices see also 14 this allows to show in 14 261 that the category mat n has auslander reiten sequences see also 30 moreover a coxeter map is defined in terms of 0 which provides us a useful computa

decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of z2 graded lie algebras we give uniform formulas for the branching rules of level 1 modules over orthogonal affine lie algebras for all conformal pairs associated to symmetric spaces we also provide a combinatorial interpretation of these formulas in terms of certain abelian subalgebras of simple lie algebras

new characterizations of hyperbolic cylinders in semi riemannian space forms abstract through the study of the so called linear weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector in a semi riemannian space form q p n p c of index p and constant sectional curvature c 1 0 1 our aim in this paper is establish new characterizations for the hyperbolic cylinders of q p n p c our approach is based on the use of a simons type formula related to an appropriated cheng yau modified operator jointly with the application of suitable generalized maximum principles

on weakly non negative unit forms and tame algebras in this paper we consider integral quadratic forms q ip z of the form q x 2i x i jqijx xj and call them unit forms such a unit form q is said to be weakly non negative if q x q whenever jt2 0 that is whenever x o for i 1 n our main purpose is to formulate a criterion for weak non negativity of unit forms this question arises in various areas of representation theory in connection with the characterization of tame representation type in the representation theory of quivers 17 6 partially ordered sets 18 19 certain classes of bimodules 8 23 1 3 and certain classes of algebras 20 14 24 16 for each of these structures q there is a unit form qn naturally attached to q such that the weak non negativity of qn is an equivalent or at least a necessary condition for q to be of tame representation type a class of algebras for which tame representation type is equivalent to the weak non negativity of the associated unit form is described in the last section of this paper a general criterion for weak non negativity using determinants was given by zeldich 28 see also 21 another criterion was obtained by de la pena 21 our approach is similar to that in 11 where we gave a characterization of weakly positive unit forms based on the classification of the critical forms let us recall some notions given a unit form q ip tl and a non empty subset ix im of 1 n we denote by qt the quadratic form qd where d z it maps the cth natural basis vector ek onto eik such a unit form q is called the restriction of q to we call a unit form q on ip a hypercritical form if q is weakly non negative for every proper subset 1 n though q itself is not in these terms a unit form q is weakly non negative if and only if q admits no hypercritical restriction so we have to classify the hypercritical forms in 1 we show that they can be obtained by certain elementary transformations from the unit forms naturally attached to the hyperbolic graphs of fig 1 on p 50 in particular there are only finitely many hypercritical forms however their number is large by a simple process we reduce the problem to the classification of the stretched hypercritical forms in 2 we show that except for a few of them these are of the shape q x q x xn xn ixn where q q n d is critical in a similar way the unit forms attached to the hyperbolic graphs am dm em correspond to those

the existence of hall polynomials for posets of finite prinjective type abstract we prove the existence of hall polynomials for prinjective representations of finite partially ordered sets of finite prinjective type in section 4 we shortly discuss consequences of the existence of hall polynomials in particular we are able to define a generic ringel hall algebra for prinjective representations of posets of finite prinjective type

tame prinjective type and tits form of two peak posets ii throughout this paper we use the terminology and notation introduced w x in 5 7 11 13 in particular k is an algebraically closed field and we denote by i u a finite poset i e partially ordered set with respect to the partial order u we shall write i j if i u j and i j for the sake of simplicity we write i instead of i u we denote by max i the set of all maximal elements of i and i will be called an n peak poset if max i s n let ki be the incidence k algebra of the poset i that is ki is the k subalgebra of the full i i matrix algebra m k consisting of all i w x matrices l in m k such that l s 0 if i uu j in i u we recall from i j i i j w x 13 that the socle soc ki of the right ki module ki is projective and the global dimension of the algebra ki is finite we denote by mod ki the category of finitely generated right ki mod ules and by prin ki the full subcategory of mod ki consisting of all prinjective modules that is modules x in mod ki having a projective resolution of the form

the max plus algebra of exponent matrices of tiled orders abstract an exponent matrix is an n n matrix a a i j over n 0 satisfying 1 a i i 0 for all i 1 n and 2 a i j a j k a i k for all pairwise distinct i j k 1 n in the present paper we study the set e n of all non negative n n exponent matrices as an algebra with the operations of component wise maximum and of component wise addition we provide a basis of the algebra e n 0 and give a row and a column decompositions of a matrix a e n with respect to this basis this structure result determines all n n tiled orders over a fixed discrete valuation domain we also study automorphisms of e n with respect to each of the operations and and prove that aut e n 0 aut e n aut e n s n c 2 n 2

quantum field theories in one and two dimensions axiomes pour les theories de champ quantiques theories quantiques du champ a une dimension theories quantiques du champ chirales a 2 dimensions

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

exchange property and the natural preorder between simple modules over semi artinian rings abstract we prove that if r is a right semi artinian ring then r is an exchange ring and every irredundant set simp r of representatives of simple right r modules carries a canonical structure of an artinian poset which is a morita invariant we investigate several basic features of this order structure and for a wide class of right semi artinian rings which we call nice we establish a link between those two sided ideals which are pure as left ideals and some upper subsets of simp r if r is nice and simp r does not contain infinite antichains then that link realizes an anti isomorphism from the lattice of upper subsets of simp r to the set of all ideals which are pure as left ideals further we show that every artinian poset possibly after adding a suitable maximal element if it is infinite is order isomorphic to simp r for some nice right semi artinian ring r

bipartite posets of finite prinjective type one of the main results of this paper is theorem 1 2 which contains a characterization of finite bipartite posetsi i i for which the category prin ki of prinjective modules over the incidencek algebrakiofiis of finite representation type wherekis a field in particular it is shown that for any such poseti the auslander reiten quiver of the category prin ki has no oriented cycle and there is a bijection between the isomorphism classes of indecomposable objects in prin ki and the positive roots of the quadratic formqiassociated withi an existence of a preprojective component in prin ki is proved for faithful posetsiwhich area free

quadratic bimodules and quadratic orders in some branches of mathematics especially the representation theory of artin algebras and orders it has been an important problem to study the geometric quotient gln mn m for a module m g m gmg 1 for g gln and m mn m an effective formulation of such problems is a bimodule problem 4 9 which is a study of the matrix category mat c m associated to an additive category c and a c c module m section 2 1 the bimodule problems have nice module theoretical interpretation by means of prinjective modules 22 29 30 they have already many useful applications in the representation theory of finite dimensional algebras artinian rings orders vector space categories and bocses see 6 7 9 16 19 25 28 29 in this paper we introduce the concept of quadratic extensions of rings section 1 2 which are closely related to many rings known in representation theory for example bass orders 10 17 backstrom orders 28 green orders 27 special biserial algebras 31 and clannish algebras 5 see sections 1 3 3 4 and 3 5 using quadratic extensions we introduce the concept of quadratic bimodules section 3 1 where we can easily formulate self reproducing systems 21 and clans 5 in terms of quadratic bimodules then we study their matrix category mat c m associated to a quadratic c c module m especially we give a description of all indecomposable objects in terms of walks sections 3 3 6 1 which can be regarded as a generalization of green walks 13 section 3 4 strings and bands 5 31 section 3 5 as an application we give a description of finitely generated indecomposable modules over quadratic orders section 1 3 2 in section 7 3 in forthcoming papers our results will be applied in construction of large new classes of orders of tame representation type some of the results of this paper were presented at the conference and workshop constanta 2000 and were announced in 19 without proof