Authors:Mayu Tsukamoto Pages: 531 - 546 Abstract: We compute the Hochschild cohomology of any block of q-Schur algebras. We focus on the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of q-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all q-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derive equivalence. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9653-0 Issue No:Vol. 20, No. 3 (2017)

Authors:Mehdi Ghaffarzadeh; Mohsen Ghasemi; Mark L. Lewis; Hung P. Tong-Viet Pages: 547 - 567 Abstract: Given a finite group G, we say that G has property \(\mathcal P_{k}\) if every set of k distinct irreducible character degrees of G is setwise relatively prime. In this paper, we show that if G is a finite nonsolvable group satisfying \(\mathcal P_{4}, \) then G has at most 8 distinct character degrees. Combining with work of D. Benjamin on finite solvable groups, we deduce that a finite group G has at most 9 distinct character degrees if G has property \(\mathcal P_{4}\) and this bound is sharp. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9654-z Issue No:Vol. 20, No. 3 (2017)

Authors:Tobias Barthel Pages: 569 - 581 Abstract: In this paper, we construct a version of Auslander–Reiten sequences for the K(n)-local stable homotopy category. In particular, the role of the Auslander–Reiten translation is played by the local Brown–Comenetz duality functor. As an application, we produce counterexamples to the K(n)-local generating hypothesis for all heights n > 0 and all primes. Furthermore, our methods apply to other triangulated categories, as for example the derived category of quasi-coherent sheaves on a smooth projective scheme. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9655-y Issue No:Vol. 20, No. 3 (2017)

Authors:Jean-Yves Charbonnel Pages: 583 - 627 Abstract: For a reductive Lie algbera over an algbraically closed field of charasteristic zero, we consider a Borel subgroup B of its adjoint group, a Cartan subalgebra contained in the Lie algebra of B and the closure X of its orbit under B in the Grassmannian. The variety X plays an important role in the study of the commuting variety. In this note, we prove that X is Gorenstein with rational singularities. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9656-x Issue No:Vol. 20, No. 3 (2017)

Authors:Jacob Haley; David Hemminger; Aaron Landesman; Hailee Peck Pages: 629 - 653 Abstract: In 2003, Fomin and Zelevinsky proved that finite type cluster algebras can be classified by Dynkin diagrams. Then in 2013, Barot and Marsh defined the presentation of a reflection group associated to a Dynkin diagram in terms of an edge-weighted, oriented graph, and proved that this group is invariant (up to isomorphism) under diagram mutations. In this paper, we extend Barot and Marsh’s results to Artin group presentations, defining new generator relations and showing mutation-invariance for these presentations. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9657-9 Issue No:Vol. 20, No. 3 (2017)

Authors:Ulrich Krähmer; Angela Ankomaah Tabiri Pages: 655 - 658 Abstract: The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9658-8 Issue No:Vol. 20, No. 3 (2017)

Authors:Nan Gao; You-Qi Yin; Pu Zhang Pages: 659 - 673 Abstract: Let (F′, F, F″) be a comparison of left recollements of triangulated categories such that F′ and F″ are equivalences. We prove that if F is full then F is an equivalence; and on the other hand, we construct a class of examples via the derived categories of Morita rings, showing that there really exists such a comparison (F′, F, F″) so that F is not an equivalence. This is in contrast to the case of a recollement. We also give a class of examples of left recollements of homotopy categories, which can not sit in recollements. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9659-7 Issue No:Vol. 20, No. 3 (2017)

Authors:Jacob Greenstein; Volodymyr Mazorchuk Pages: 675 - 694 Abstract: We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9660-1 Issue No:Vol. 20, No. 3 (2017)

Authors:Shahn Majid Pages: 695 - 733 Abstract: We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ1 over a Hopf algebra A which are quotients of the augmentation ideal A + as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation provides a bicovariant differential graded algebra on A. We introduce Λ m a x providing the maximal prolongation, while the canonical braided-exterior algebra Λ min = B −(Λ1) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ♯ by super-braided Fourier transform on B −(Λ1) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S 3] with its 3D calculus and obeying the q-Hecke relation ♯2 = 1 + (q − q −1)♯ in middle degree on k q [S L 2] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra \(\mathcal {L}\subseteq A\) defines a sub-braided Lie algebra and \({\Lambda }^{1}\subseteq \mathcal {L}^{*}\) provides the required data A + → Λ1. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9661-0 Issue No:Vol. 20, No. 3 (2017)

Authors:David A. Towers Pages: 735 - 750 Abstract: The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call minimal non- \({\mathcal N}\) . To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length ≤k, and extreme Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non- \({\mathcal N}\) Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index ≤3. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9662-z Issue No:Vol. 20, No. 3 (2017)

Authors:Laura Geatti; Claudio Gorodski Pages: 751 - 764 Abstract: We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9663-y Issue No:Vol. 20, No. 3 (2017)

Authors:Marcelo Lanzilotta; Octavio Mendoza Pages: 765 - 802 Abstract: We develope the theory of \({\mathcal {E}}\) -relative Igusa-Todorov functions in an exact I T-context \(({\mathcal {C}},{\mathcal {E}})\) (see Definition 2.1). In the case when \({\mathcal {C}}={\text {mod}}\, ({\Lambda })\) is the category of finitely generated left Λ-modules, for an artin algebra Λ, and \({\mathcal {E}}\) is the class of all exact sequences in \({\mathcal {C}},\) we recover the usual Igusa-Todorov functions, Igusa K. and Todorov G. (2005). We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov’s results. Furthermore, we introduce the \({\mathcal {E}}\) -relative Igusa-Todorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions. PubDate: 2017-06-01 DOI: 10.1007/s10468-016-9664-x Issue No:Vol. 20, No. 3 (2017)

Abstract: We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups \(\mathcal {N}(G)\) has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups. PubDate: 2017-06-24

Abstract: We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations. PubDate: 2017-06-21

Authors:Didier Arnal; Olfa Khlifi Abstract: The simple \(GL(n,\mathbb {C})\) -modules are described by using semistandard Young tableaux. Any semistandard skew tableau can be transformed into a well defined semistandard tableau by a combinatorial operation, the Schützenberger jeu de taquin. Associated to the classical Lie groups \(SP(2n,\mathbb {C})\) , \(SO(2n+1,\mathbb {C})\) , there are other notions of semistandard Young tableaux and jeux de taquin. In this paper, we study these various jeux de taquin, proving that each of them has a simple and explicit formulation as a step-by-step sliding. Any of these jeux de taquin is the restriction of the orthogonal one, associated to \(SO(2n+1,\mathbb {C})\) . PubDate: 2017-06-20 DOI: 10.1007/s10468-017-9711-2

Authors:Aslak Bakke Buan; Yu Zhou Abstract: We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra A whose global dimension gl.dim A ≤ 2 and any 2-term silting complex P in the bounded derived category D b (A) of A, the global dimension of \(\text {End}_{{D^b(A)}}(\mathbf {P})\) is at most 7. We also show that for each n > 2, there is an algebra A with gl.dim A = n such that D b (A) admits a 2-term silting complex P with \(\mathrm {gl. dim~}\text {End}_{{D^b(A)}}(\mathbf {P})\) infinite. PubDate: 2017-06-14 DOI: 10.1007/s10468-017-9709-9

Authors:Guoqiang Zhao Abstract: For a class of modules \(\mathcal {X}\) , we introduce the \(\mathcal {X}\) -transpose of a module with respect to a bimodule, which unifies some well-known transposes. Let \(\mathcal {V}\) be a subclass of \(\mathcal {X}\) . The relations between \(\mathcal {X}\) -transposes and \(\mathcal {V}\) -transposes are investigated under the condition that \(\mathcal {V}\) is a generator or cogenerator of \(\mathcal {X}\) . The dual aspects of \(\mathcal {X}\) -transposes are also discussed. Then we give some applications of these results. In particular, the dual counterparts of Gorenstein transposes are established. PubDate: 2017-06-13 DOI: 10.1007/s10468-017-9708-x

Authors:Julia Sauter Abstract: Generalizing Schubert cells in type A and a cell decomposition of Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of a cyclic quiver admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation with finitely many fixpoints. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by Kato as analogues of standard modules. PubDate: 2017-06-05 DOI: 10.1007/s10468-017-9689-9

Authors:A. S. Hegazi; Hani Abdelwahab; A. J. Calderon Martin Abstract: The paper is devoted to classify all of the nilpotent Malcev algebras of dimension ≤ 6 over an arbitrary base field ð”½ of characteristic not 2. We also classify all 7-dimensional non-Lie nilpotent Malcev algebras which are not metabelian over any field of characteristic not 2. PubDate: 2017-06-01 DOI: 10.1007/s10468-017-9701-4