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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]

Let H be an atomic monoid. The set of distances Δ ( H ) of H is the set of all d ∈ N with the following property: there are irreducible elements u 1 , … , u k , v 1 … , v k + d such that u 1 ⋅ … ⋅ u k = v 1 ⋅ … ⋅ v k + d but u 1 ⋅ … ⋅ u k cannot be written as a product of ℓ irreducible elements for any ℓ ∈ N with k < ℓ < k + d . It is well-known (and easy to show) that, if Δ ( H ) is nonempty, then min ⁡ Δ ( H ) = gcd ⁡ Δ ( H ) . In this paper we show conversely that for every finite nonempty set Δ ⊂ N with min ⁡ Δ = gcd ⁡ Δ there is a finitely generated Krull monoid H such that Δ ( H ) = Δ . [ABSTRACT FROM AUTHOR]

Abstract: This paper offers an exhaustive study of certain 2-signalizers in known non-sporadic finite simple groups. The main result of this paper is relevant to the Generation-2 proof of the Classification of Finite Simple Groups. [Copyright &y& Elsevier]

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: 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 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: 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: Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras under injections which preserve the root spaces. It is known that a root-reductive Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of finite-dimensional simple Lie algebras as well as copies of the three simple infinite-dimensional root-reductive Lie algebras , , and . As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra were introduced and studied in [K.-H. Neeb, I. Penkov, Cartan subalgebras of , Canad. Math. Bull. 46 (2003) 597–616]. In the present paper we refine and extend the results of [K.-H. Neeb, I. Penkov, Cartan subalgebras of , Canad. Math. Bull. 46 (2003) 597–616] to the case of a general root-reductive Lie algebra . We prove that the Cartan subalgebras of are the centralizers of maximal toral subalgebras and that they are nilpotent and self-normalizing. We also give an explicit description of all Cartan subalgebras of the simple Lie algebras , , and . We conclude the paper with a characterization of the set of conjugacy classes of Cartan subalgebras of the Lie algebras , , , and with respect to the group of automorphisms of the natural representation which preserve the Lie algebra. [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 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 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 compute the Poisson cohomology groups for any Poisson Del Pezzo surface. [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: 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: Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle on X. In this paper, in order to investigate the dimension of more systematically, we introduce the invariant for every integer i with . Furthermore, we study this invariant for the case where L is ample and spanned by global sections. As applications we get a lower bound (resp. an upper bound) for the dimension of if L is ample and spanned by global sections (resp. very ample). [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 introduce the notions of weak Yang–Baxter operator and weak braided Hopf algebra. We prove that it is possible to obtain examples of these notions working with Yetter–Drinfeld modules associated to a weak Hopf algebra H with invertible antipode. Finally, we complete the study of the structure of weak Hopf algebras with a projection obtaining a categorical equivalence between the category of weak Hopf algebra projections associated to H and the category of Hopf algebras in the non-strict braided monoidal category of left–left Yetter–Drinfeld modules over H. [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: The exponent of a PI-algebra A in characteristic zero is an integer and measures the exponential rate of growth of the sequence of codimensions of A [A. Giambruno, M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998) 145–155; A. Giambruno, M. Zaicev, Exponential codimension growth of P.I. algebras: An exact estimate, Adv. Math. 142 (1999) 221–243]. In this paper we study the exponential rate of growth of the sequences of proper codimensions and Lie codimensions of an associative PI-algebra. We prove that the corresponding proper exponent exists for all PI-algebras, except for some algebras of exponent two strictly related to the Grassmann algebra. We also prove that the Lie exponent exists for any finitely generated PI-algebra. The value of both exponents is always equal to or . [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: The main result of the paper is a Borel type description of the -equivariant cohomology ring of the manifold of all complete flags in . To prove this, we obtain a Goresky–Kottwitz–MacPherson type description of that ring. [Copyright &y& Elsevier]

Abstract: Let R be a commutative ring and denote its zero-divisor graph. In this paper we investigate the crosscap number of the non-orientable compact surface which can be embedded and illustrate all finite commutative rings R (up to isomorphism) such that is projective. [Copyright &y& Elsevier]

Abstract: Interpreting entwining structures as special instances of J. Beck''s distributive law, the concept of entwining module can be generalized for the setting of arbitrary monoidal category. In this paper, we use the distributive law formalism to extend in this setting basic properties of entwining 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: The purpose of the present paper is two-fold: on the one hand, to show the existence of a correspondence unifying Brauer''s and Glauberman''s ones (see Theorem 4.6), and, on the other hand, to give an alternative proof of Watanabe''s equivalence in [Atumi Watanabe, The Glauberman character correspondence and perfect isometries for blocks of finite groups, J. Algebra 216 (1999) 548–565]. By the way, we give a short proof of the coincidence of the Clifford extensions associated with a pair of Glauberman correspondent irreducible representations (see Corollary 4.16), a question that, surprisingly enough, has only been partially solved recently (see [Morton Harris, Markus Linckelmann, On the Glauberman and Watanabe correspondences for blocks of finite p-solvable groups, Trans. Amer. Math. Soc. 354 (2002) 3435–3453] and [Shigeo Koshitani, Gerhard Michler, Glauberman correspondence of p-blocks of finite groups, J. Algebra 243 (2001) 504–517]). [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: 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 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: Suppose G is a subgroup of , M a finite-dimensional vectorspace over the field k with char , generated by quadratic elements σ satisfying for all . Then one can define root-subgroups of G intrinsically, i.e. just in terms of the quadratic elements. In this paper we determine such groups G generated by three root-subgroups, which do not contain a pair of commuting root-subgroups. This is a further step of the determination of groups G, when is a quadratic pair. [Copyright &y& Elsevier]

Abstract: In this paper we prove that the non-cyclic generic abelian crossed product p-algebras constructed by Amitsur and Saltman in [S.A. Amitsur, D. Saltman, Generic Abelian crossed products and p-algebras, J. Algebra 51 (1) (1978) 76–87] remain non-cyclic after tensoring by any prime to p extension of their centers. We also prove that an example due to Saltman of an indecomposable generic abelian crossed product with exponent p and degree remains indecomposable after any prime to p extension. [Copyright &y& Elsevier]

Abstract: A perfect crystal of any level is constructed for the Kirillov–Reshetikhin module of corresponding to the middle vertex of the Dynkin diagram. The actions of Kashiwara operators are given explicitly. It is also shown that this family of perfect crystals is coherent. A uniqueness problem solved in this paper can be applied to other quantum affine algebras. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and Gorenstein. For D greater or equal to 4 we add the condition that these algebras are homogeneous and Koszul. It is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension D. This multilinear form plays the role of a volume form and canonically identifies in the quadratic case to a nontrivial Hochschild cycle of maximal degree. Several examples including the Yang–Mills algebra and the extended 4-dimensional Sklyanin algebra are analyzed in this context. Actions of quantum groups are also investigated. [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]

Abstract: In this paper, we observe that a certain local property on highest weight crystal graphs forces a more global property. In type A, this statement says that if a node has a single parent and single grandparent, then there is a unique walk from the highest weight node to it. This crystal observation was motivated by certain representation-theoretic behavior of the affine Hecke algebra of type A. In other classical types, there is a similar statement. This walk is obtained from the associated level 1 perfect crystal, . (It is unique unless the Dynkin diagram contains that of as a subdiagram.) [Copyright &y& Elsevier]