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Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

Abstract In this paper, we study the representation theory of the Fuss–Catalan algebras, FC n (a , b). We prove that this algebra is cellular with a cellular basis and forms a tower of recollement, as defined by Cox, Martin, Parker, and Xi [7] , and hence, it is quasi-hereditary algebra if a , b are non-zero complex numbers. [ABSTRACT FROM AUTHOR]

Abstract: Quasimodules for vertex algebras are generalizations of modules for vertex algebras. These new objects arise from a generalization of locality for fields. Quasimodules tie together module theory and twisted module theory, and both twisted and untwisted modules feature Poincaré–Birkhoff–Witt-like spanning sets. This paper generalizes these spanning set results to quasimodules for certain Möbius vertex algebras. In particular this paper presents two spanning sets, one featuring a difference-zero ordering restriction on modes and another featuring a difference-one ordering restriction. [Copyright &y& Elsevier]

Abstract: The main result in the paper states the following: Let π be a set of odd primes. Let the finite group be the product of a π-decomposable subgroup and a π-subgroup B. Then ; equivalently the group G possesses Hall π-subgroups. In this case is a Hall π-subgroup of G. This result extends previous results of Berkovich (1966), Rowley (1977), Arad and Chillag (1981) and Kazarin (1980) where stronger hypotheses on the factors A and B of the group G were being considered. The results under consideration in the paper provide in particular criteria for the existence of non-trivial soluble normal subgroups for a factorized group G. [Copyright &y& Elsevier]

Abstract: In the first part of the paper we give a characterization for an associative algebra to be standardly stratified in the sense of Cline, Parshall and Scott, generalizing a theorem of V. Dlab. In the second part of the paper we construct characteristic tilting modules for standardly stratified algebras and use them to estimate the finitistic dimension of such algebras. These tilting modules give rise to the Ringel duality concept for stratified algebras. We also define and investigate a generalization of the notion of properly stratified algebras to the above setup. [Copyright &y& Elsevier]

Abstract: In this paper we study division algebras over the function fields of curves over . The first and main tool is to view these fields as function fields over nonsingular S which are projective of relative dimension 1 over the p adic ring . A previous paper showed such division algebras had index bounded by assuming the exponent was n and n was prime to p. In this paper we consider algebras of prime degree (and hence exponent) and show these algebras are cyclic. We also find a geometric criterion for a Brauer class to have index q. [Copyright &y& Elsevier]

Abstract: In this paper we use the Etingof–Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld–Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: (1) a theorem stating that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the corresponding D-J type superalgebra and (2) a super version of the Drinfeld–Kohno theorem. [Copyright &y& Elsevier]

Abstract: Rickard proved in his paper [J. Rickard, Equivalences of derived categories for symmetric algebras, J. Algebra 257 (2002) 460–481] that if Λ is a finite-dimensional symmetric k-algebra and if there is a set of objects in satisfying some conditions, then there is a derived equivalence taking these objects to the simple modules of another algebra Γ. In this paper we generalize Rickard''s results to finite-dimensional selfinjective k-algebras by adding an extra condition. We use the techniques of Rickard''s paper in this paper. [Copyright &y& Elsevier]

Abstract: Using new techniques of Zelmanov, C. Martinez improved on work of Jacobson, McCrimmon, and Parvathi to give a necessary and sufficient Ore-type condition for an arbitrary linear Jordan algebra (with no 2- or 3-torsion) to have an algebra of fractions. In this paper we extend to quadratic algebras the concept of algebras of outer fractions with respect to an Ore monad, and describe necessary and sufficient Ore-type conditions for the embedding in such an algebra of fractions. The details of the actual embedding will appear in a subsequent paper. [Copyright &y& Elsevier]

Abstract: This is the second in a series of four papers on fixed point ratios in non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either for all elements of prime order, or is one of a small number of known exceptions. In this paper we record a number of preliminary results and prove the main theorem in the case where the stabiliser is contained in a maximal non-subspace subgroup which lies in one of the Aschbacher families , where . [Copyright &y& Elsevier]

Abstract: Let A be a symmetric k-algebra over a perfect field k. Külshammer defined for any integer n a mapping on the degree 0 Hochschild cohomology and a mapping on the degree 0 Hochschild homology of A as adjoint mappings of the respective p-power mappings with respect to the symmetrising bilinear form. In an earlier paper it is shown that is invariant under derived equivalences. In the present paper we generalise the definition of to higher Hochschild homology and show the invariance of κ and its generalisation under derived equivalences. This provides fine invariants of derived categories. [Copyright &y& Elsevier]

Abstract: The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case. In this paper we study the Gelfand–Kirillov dimension of the relatively free algebras of verbally prime and related algebras. We compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras and are not PI equivalent in characteristic . Furthermore we show that the following algebras are not PI equivalent in positive characteristic: and ; and when , , and ; and finally, and . Here E stands for the infinite-dimensional Grassmann algebra with 1, and is the subalgebra of of the block matrices with blocks and on the main diagonal with entries from , and off-diagonal entries from ; is the natural grading on E. [Copyright &y& Elsevier]

Abstract: This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring R of prime characteristic p. For a given ideal of R, there is a power Q of p, depending on , such that the Qth Frobenius power of the Frobenius closure of is equal to the Qth Frobenius power of . The paper addresses the question as to whether there exists a uniform which ‘works’ in this context for all parameter ideals of R simultaneously. In a recent paper, Katzman and Sharp proved that there does exists such a uniform when R is Cohen–Macaulay. The purpose of this paper is to show that such a uniform exists when R is a generalized Cohen–Macaulay local ring. A variety of concepts and techniques from commutative algebra are used, including unconditioned strong d-sequences, cohomological annihilators, modules of generalized fractions, and the Hartshorne–Speiser–Lyubeznik Theorem employed by Katzman and Sharp in the Cohen–Macaulay case. [Copyright &y& Elsevier]

Abstract: In [I. Gelfand, V. Retakh, S. Serconek, R.L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. (N.S.) 11 (2005), math.QA/0506507] I. Gelfand and the authors of this paper introduced a new class of algebras associated to directed graphs. In this paper we show that these algebras are Koszul for a large class of layered graphs. [Copyright &y& Elsevier]

Abstract: Consider the enveloping algebra of a complex semisimple Lie algebra . The heart of a prime ideal I of is the center of the total ring of fractions of . This is an extension field of the field of fractions of the center of . Let d be the degree of this field extension. An old problem of J. Dixmier asked whether . A recent paper of the second author [R. Rentschler, A negative answer to the problem of Dixmier on hearts of prime quotients of enveloping algebras, preprint, 2004] gave a negative answer by an example in . The present paper provides many more examples, involving the so-called sheets of primitive ideals introduced and studied by A. Joseph and the first author in [W. Borho, A. Joseph, Sheets and topology of primitive spectra for semisimple Lie algebras, J. Algebra 244 (2001) 76–167]. A sheet corresponds to a prime ideal I which has a heart of degree d. The main result of this paper is that d equals the covering degree of the sheet as introduced in [W. Borho, A. Joseph, Sheets and topology of primitive spectra for semisimple Lie algebras, J. Algebra 244 (2001) 76–167, 8.7]. [Copyright &y& Elsevier]

Abstract: Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of Hopf algebras and work with multiplier Hopf algebras. Moreover, whereas in the finite-dimensional case, there is a complete symmetry between the bialgebra and its dual, this is no longer the case in infinite dimensions. Therefore we consider a direct version (with integrals) and a dual version (with cointegrals) of the Larson–Sweedler theorem. We also add some results about the antipode. Furthermore, in the process of this paper, we obtain a new approach to multiplier Hopf algebras with integrals. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate certain ideals in the center of a symmetric algebra A over an algebraically closed field of characteristic . These ideals include the Higman ideal and the Reynolds ideal. They are closely related to the p-power map on A. We generalize some results concerning these ideals from group algebras to symmetric algebras, and we obtain some new results as well. In case , these ideals detect odd diagonal entries in the Cartan matrix of A. In a sequel to this paper, we will apply our results to group algebras. [Copyright &y& Elsevier]

Abstract: In an earlier paper [T. Košir, B.A. Sethuraman, Determinantal varieties over truncated polynomial rings, J. Pure Appl. Algebra 195 (2005) 75–95] we had begun a study of the components and dimensions of the spaces of th order jets of the classical determinantal varieties: these are the varieties obtained by considering generic () matrices over rings of the form , and for some fixed r, setting the coefficients of powers of t of all minors to zero. In this paper, we consider the case where , and provide a Groebner basis for the ideal which defines the tangent bundle to the classical determinantal variety. We use the results of these Groebner basis calculations to describe the components of the varieties where r is arbitrary. (The components of and were already described in the above cited paper.) [Copyright &y& Elsevier]

All groups considered in this paper will be finite. A 2-group is called quaternion-free if it has no section isomorphic to the quaternion group of order 8. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(P). Zhang [Proc. Amer. Math. Soc. 98 (4) (1986) 579] proved that if P is a Sylow p-subgroup of G, Ω1(P)⩽Z(P) and NG(Z(P)) is p-nilpotent, then G is p-nilpotent, i.e., G has a normal Hall p′-subgroup. Recently, Ballester-Bolinches and Guo [J. Algebra 228 (2000) 491] proved that if P is a Sylow 2-subgroup G, P is quaternion-free, Ω1(P∩G′)⩽Z(P) and NG(P) is 2-nilpotent, then G is 2-nilpotent. Bannuscher and Tiedt [Ann. Univ. Sci. Budapest 37 (1994) 9] proved that if p>2, P is a Sylow p-subgroup of G, &z.sfnc;Ω1(P∩Px)&z.sfnc;⩽pp-1 for all x∈G&z.drule;NG(P) and NG(P) is p-nilpotent, then G is p-nilpotent. The object of this paper is to improve and extend these results. [Copyright &y& Elsevier]

A subsemigroup S of a semigroup Q is a local left order in Q if, for every group H-class H of Q, S∩H is a left order in H in the sense of group theory. That is, every q∈H can be written as a♯b for some a,b∈S∩H, where a♯ denotes the group inverse of a in H. On the other hand, S is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as c♯d where c,d∈S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. If one also insists that c and d can be chosen such that c R d in Q, then S is said to be a straight left order in Q.This paper investigates the close relation between local left orders and straight left orders in a semigroup Q and gives some quite general conditions for a left order S in Q to be straight. In the light of the connection between locality and straightness we give a complete description of straight left orders that improves upon that in our earlier paper. [Copyright &y& Elsevier]

Let R be a ring and let A be a finite projective R -algebra of rank n . Manjul Bhargava and Matthew Satriano have recently constructed an R -algebra G ( A / R ) , the Galois closure of A / R . Many natural questions were asked at the end of their paper. Here we address one of these questions, proving the existence of the natural constructions they call intermediate S n -closures. We will also study properties of these constructions, generalizing some of their results, and proving new results both on the intermediate S n -closures and on G ( A / R ) . [ABSTRACT FROM AUTHOR]

The central theme of the paper is the specialization of algebraic function field extensions. Our main results are Tchebotarev type theorems for Galois function field extensions, finite or infinite, over various base fields: under some conditions, we extend the classical finite field case to number fields, p -adic fields, PAC fields, function fields κ ( x ) , etc. We also compare the Tchebotarev conclusion – existence of unramified local specializations with Galois group any cyclic subgroup of the generic Galois group (up to conjugation) – to the Hilbert specialization property. For a function field extension with the Tchebotarev property, the exponent of the Galois group is bounded by the l.c.m. of the local specialization degrees. Local–global questions arise for which we provide answers, examples and counter-examples. [ABSTRACT FROM AUTHOR]

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian m -Koszul twisted Calabi–Yau or, equivalently, m -Koszul Artin–Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D ( w , i ) for a unique-up-to-scalar-multiples twisted superpotential w . By definition, D ( w , i ) is the quotient of the tensor algebra TV , where V = D ( w , i ) 1 , by ( ∂ i w ) , the ideal generated by all i -th-order left partial derivatives of w . The restriction map σ ↦ σ | V is used to identify the group of graded algebra automorphisms of D ( w , i ) with a subgroup of GL ( V ) . We show that the homological determinant of a graded algebra automorphism σ of an m -Koszul Artin–Schelter regular algebra D ( w , i ) is given by the formula hdet ( σ ) w = σ ⊗ ( m + i ) ( w ) . It follows from this that the homological determinant of the Nakayama automorphism of an m -Koszul Artin–Schelter regular algebra is 1. As an application, we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian Artin–Schelter regular algebras of dimension 3. [ABSTRACT FROM AUTHOR]

Let M be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal Q for M , a criterion for the equality g s ( Q ; M ) = hdeg Q ( M ) − e Q 0 ( M ) − T Q 1 ( M ) , where g s ( Q ; M ) , hdeg Q ( M ) , e Q 0 ( M ) , and T Q 1 ( M ) respectively denote the sectional genus, the homological degree, the multiplicity, and the homological torsion of M with respect to Q . [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate Lie bialgebra structures on the twisted Heisenberg–Virasoro algebra. With the determination of certain Lie bialgebra structures on the Virasoro algebra, we determine certain structures on the twisted Heisenberg–Virasoro algebra. Moreover, some general and useful results are obtained. With our methods and results we also can easily determine certain structures on some Lie algebras related to the twisted Heisenberg–Virasoro algebra. [Copyright &y& Elsevier]

Abstract: In this paper we give an explicit algorithm to construct the ordinary quiver of a finite EI category for which the endomorphism groups of all objects have orders invertible in the field k. We classify all finite EI categories with hereditary category algebras, characterizing them as free EI categories (in a sense which we define) for which all endomorphism groups of objects have invertible orders. Some applications on the representation types of finite EI categories are derived. [Copyright &y& Elsevier]

Abstract: In this paper we use projectivity relative to kG-modules to define groups of relatively endotrivial modules, which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules. To achieve this goal we develop the theory of projectivity relative to modules with respect to standard group operations such as induction, restriction and inflation. As a particular example, we show how these groups can generalise the Dade group. Finally, for finite groups having a cyclic Sylow p-subgroup, we determine all the different subcategories of relatively projective modules and, using the structure of the group of endotrivial modules described in Mazza and Thévenaz (2007) , the structure of all the different groups of relatively endotrivial modules. [Copyright &y& Elsevier]

Abstract: In this paper, we use the method in Rui and Si (2011) to give a necessary and sufficient condition on non-vanishing Gram determinants for cyclotomic NW and cyclotomic BMW algebras over an arbitrary field. Equivalently, we give a necessary and sufficient condition for each cell module of such algebras being equal to its simple head over an arbitrary field. [Copyright &y& Elsevier]

Abstract: We discuss an algebraic connection between two kinds of multiple zeta values or their q-analogues: the (q-)multiple zeta values and the (q-)multiple zeta-star values. These two classes of values generate the same algebra, but in this paper, we show that the translation map between these two classes has a quite interesting algebraic property in a general setting, for example, the compatibility with the harmonic product. We also study several applications of the result. [Copyright &y& Elsevier]

Abstract: In this paper we study the concept of a Cartan subalgebra in the context of locally finite and Borcherds–Kac–Moody Lie algebras. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study the structure of finite groups whose non-subnormal subgroups lie in some subclasses of the class of finite supersoluble groups. [Copyright &y& Elsevier]

Abstract: Let L be a Lie algebra. We call a linear map a near-derivation if there exists a linear map such that is a derivation for every . The paper is devoted to describing the structure of near-derivations in certain Lie algebras arising from associative ones. [Copyright &y& Elsevier]

Abstract: The n-valued Łukasiewicz–Moisil algebras, MV-algebras and Post algebras are structures developed in connection to the algebra of the n-valued Łukasiewicz logic. In this paper, we obtain categorical equivalences which allow us to represent any such structure as an algebra of decreasing Boolean sequences of length n. Moreover any algebra L belonging to one of these classes is characterized using a sequence of n Boolean ideals , which are called the Boolean nuances of L. The type of L can be deduced from set-theoretical properties of the corresponding sequence of Boolean ideals. As an application, we prove that L is σ-complete if and only if the corresponding ideals are σ-closed. [Copyright &y& Elsevier]

Abstract: We define a quiver to be rigid if all the associated truncated quiver algebras are rigid. The rigidity of quivers is then determined by the combinatorics of the set of pairs of parallel paths of the underlying quiver as follows from Cibils'' criteria for the rigidity of truncated quiver algebras. In this paper we characterize rigid quivers Δ and relate this characterization with the condensed quiver and the quiver of beads of Δ, two much simpler quivers associated to Δ. The first one is a well-known object and the second one is introduced by us to this end. [Copyright &y& Elsevier]

Abstract: Let G be a connected reductive algebraic group over an algebraically closed field of characteristic . Assume that p is good for G. Pommerening''s theorem [K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977) 525–536; K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, II, J. Algebra 65 (1980) 373–398] asserts that any distinguished nilpotent element in the Lie algebra of G is a Richardson element for a distinguished parabolic subgroup of G. This theorem implies the Bala–Carter theorem in good characteristic. In this paper we give a short proof of Pommerening''s theorem, which is a further simplification of Premet''s first uniform proof [A. Premet, Nilpotent orbits in good characteristic and the Kempf–Rousseau theory, J. Algebra 260 (2003) 338–366]. We also simplify Premet''s proof of the existence theorem for good transverse slices to the nilpotent -orbits in . [Copyright &y& Elsevier]

Abstract: In this paper, we study the problem when a finitely generated torsionless module is projective. Let Λ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless Λ-module M is projective if . For a commutative Artinian ring Λ, a finitely generated torsionless Λ-module M is projective if the following conditions are satisfied: (1) for ; and (2) for . As a consequence of this result, we have that for a commutative Artinian ring Λ, a finitely generated Gorenstein projective Λ-module is projective if and only if it is selforthogonal. [Copyright &y& Elsevier]

Abstract: In this paper we prove a formula for fusion coefficients of affine Kac–Moody algebras first conjectured by Walton [M.A. Walton, Tensor products and fusion rules, Canad. J. Phys. 72 (1994) 527–536], and rediscovered by Feingold [A. Feingold, Fusion rules for affine Kac–Moody algebras, in: N. Sthanumoorthy, Kailash Misra (Eds.), Kac–Moody Lie Algebras and Related Topics, Ramanujan International Symposium on Kac–Moody Algebras and Applications, Jan. 28–31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, in: Contemp. Math., vol. 343, American Mathematical Society, Providence, RI, 2004, pp. 53–96]. It is a reformulation of the Frenkel–Zhu affine fusion rule theorem [I.B. Frenkel, Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992) 123–168], written so that it can be seen as a beautiful generalization of the classical Parthasarathy–Ranga Rao–Varadarajan tensor product theorem [K.R. Parthasarathy, R. Ranga Rao, V.S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967) 383–429]. [Copyright &y& Elsevier]

Abstract: In this paper we consider partial actions of groups on algebras and partial skew group rings. After some general results we prove two versions of Maschke''s theorem and then we study von Neumann regularity, the prime radical and the Jacobson radical of partial skew group rings. In this way we extend many results which are known for skew group rings. [Copyright &y& Elsevier]

Abstract: The paper has its origin in an attempt to answer the following question: Given an arbitrary finite dimensional associative K-algebra A, does there exist a quasi-hereditary algebra B such that the subcategories of all A-modules and all B-modules, filtered by the corresponding standard modules are equivalent. Such an algebra will be called a quasi-hereditary approximation of A. The question is answered in the appropriate language of standardly stratified algebras: For any K-algebra A, there is a uniquely defined basic algebra such that is Δ-filtered and the subcategories and of all Δ-filtered modules are equivalent; similarly there is a uniquely defined basic algebra such that is -filtered and the subcategories and of all -filtered modules are equivalent. These subcategories play a fundamental role in the theory of stratified algebras. Since, in general, it is difficult to localize these subcategories in the category of all A-modules, the construction of and often helps to describe them explicitly. By applying consecutively the operators Σ and Ω for an algebra, we get a sequence of standardly stratified algebras which, after a finite number of steps, stabilizes in a properly stratified algebra. Thus, all standardly stratified algebras are partitioned into (generally infinite) trees, indexed by properly stratified algebras (as their roots). [Copyright &y& Elsevier]

Abstract: We conclude our classification of powerful 2-Engel groups of class three that are minimal in the sense that every proper powerful section is nilpotent of class at most two. In the predecessor to this paper we obtained three families of minimal groups. Here we get a fourth family of minimal examples that is described in terms of irreducible polynomials over the field of three elements. We also get one isolated minimal example of rank 5 and exponent 27. The last one has a related algebraic structure that we call a “symplectic alternating algebra.” To each symplectic alternating algebra over the field of three elements there corresponds a unique 2-Engel group of exponent 27. [Copyright &y& Elsevier]

Abstract: The notion of a topological Jordan decomposition of a compact element of a reductive p-adic group has proven useful in many contexts. In this paper, we generalise it to groups defined over fairly general discretely valued fields and prove the usual existence and uniqueness properties, as well as an analogue of a fixed-point result of Prasad and Yu. [Copyright &y& Elsevier]

Abstract: In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the “maximal implies prime” variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules. [Copyright &y& Elsevier]

Abstract: By a theorem due to the first author, the bounded derived category of a finite dimensional algebra over a field embeds fully faithfully into the stable category over its repetitive algebra. This embedding is an equivalence if the algebra is of finite global dimension. The purpose of this paper is to investigate the relationship between the derived category and the stable category over the repetitive algebra from various points of view for algebras of infinite global dimension. The most satisfactory results are obtained for Gorenstein algebras, especially for selfinjective algebras. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a generalization for the class of laura algebras, called almost laura. We show that this new class of algebras retains most of the essential features of laura algebras, especially concerning the important role played by the non-semiregular components in their Auslander–Reiten quivers. Also, we study more intensively the left supported almost laura algebras, showing that these are characterized by the presence of a generalized standard, convex and faithful component. Finally, we prove that almost laura algebras behave well with respect to full subcategories, split-by-nilpotent extensions and skew group algebras. [Copyright &y& Elsevier]

Abstract: In [Ch. Brouder, A. Frabetti, Renormalization of QED with planar binary trees, Eur. Phys. J. C 19 (2001) 715–741; Ch. Brouder, A. Frabetti, QED Hopf algebras on planar binary trees, J. Algebra 267 (2003) 298–322] we introduced three Hopf algebras on planar binary trees related to the renormalization of quantum electrodynamics. One of them, the algebra , is commutative, and is therefore the ring of coordinate functions of a proalgebraic group . The other two algebras, and , are free non-commutative. Therefore their abelian quotients are the coordinate rings of two proalgebraic groups and . In this paper we describe explicitly these groups. Using two monoidal structures and a set-operad structure on planar binary trees, we show that these groups can be realized on formal series expanded over trees, and that the group laws are generalizations of the multiplication and the composition of usual series in one variable. Therefore we obtain some new groups of invertible tree-expanded series and of tree-expanded formal diffeomorphisms respectively. The Hopf algebra describing the renormalization of the electric charge corresponds to the subgroup of tree-expanded formal diffeomorphisms formed of the translations, which fix the zero, by some particular tree-expanded series which remind the proper correlation functions in quantum field theory. In turn, the group of tree-expanded formal diffeomorphisms and some of its subgroups give rise to new Hopf algebras on trees. All the constructions are done in a general operad-theoretic setting, and then applied to the specific duplicial operad on trees. [Copyright &y& Elsevier]

Abstract: Let Φ be a root system of type , and let G be a group generated by non-trivial subgroups , , satisfying some generalized Steinberg relations, which are also satisfied by root subgroups corresponding to a Moufang hexagon. These relations can be considered as a generalization of the element-wise commutator relations in Chevalley groups. The Steinberg presentation specifies the groups satisfying the Chevalley commutator relations. In the present paper some sort of generalized Steinberg presentation for groups with root system of type is provided. As a main result we classify the possible structures for G. [Copyright &y& Elsevier]

Abstract: Let be a Henselian valued field satisfying the following conditions, for a given prime number p: (i) central division K-algebras of (finite) p-primary dimensions have Schur indices equal to their exponents; (ii) the value group properly includes its subgroup . The paper shows that if is the residue field of and is an intermediate field of the maximal p-extension , then the natural homomorphism of Brauer groups maps surjectively the p-component on . It proves that is divisible, if or is a nonreal field, and that is of order 2 when is formally real. We also obtain that embeds as a -subalgebra in a central division -algebra if and only if the degree divides the index of . [Copyright &y& Elsevier]

Abstract: Let be a polynomial ring and let be a graded ideal. In [T. Römer, Betti numbers and shifts in minimal graded free resolutions, arXiv: AC/070119], Römer asked whether under the Cohen–Macaulay assumption the ith Betti number can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen–Macaulay algebras when I is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds are sharp. [Copyright &y& Elsevier]

Abstract: We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.14. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra A is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that A is a double cross product, or biproduct in Madjid''s terminology. The last result is based on a universal property of double cross products which, by Theorem 2.15, works in braided monoidal categories. We also investigate the situation when the right action of the associated matched pair is trivial. [Copyright &y& Elsevier]

Abstract: For any vertex algebra and any subalgebra , there is a new subalgebra of known as the commutant of in . This construction was introduced by Frenkel–Zhu, and is a generalization of an earlier construction due to Kac–Peterson and Goddard–Kent–Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (i.e., a pair of mutual commutants) in the vertex algebra setting. [Copyright &y& Elsevier]