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)

Authors:Yao Ma; Liangyun Chen Abstract: In this paper, we investigate the Leibniz triple system T and its universal Leibniz envelope U(T). The involutive automorphism of U(T) determining T is introduced, which gives a characterization of the \(\mathbb {Z}_{2}\) -grading of U(T). We show that the category of Leibniz triple systems is equivalent to a full subcategory of the category of \(\mathbb {Z}_{2}\) -graded Leibniz algebras. We give the relationship between the solvable radical R(T) of T and R a d(U(T)), the solvable radical of U(T). Further, Levi’s theorem for Leibniz triple systems is obtained. Moreover, the relationship between the nilpotent radical of T and that of U(T) is studied. Finally, we introduce the notion of representations of a Leibniz triple system, which can be described by using involutive representations of its universal Leibniz envelope. PubDate: 2017-05-17 DOI: 10.1007/s10468-017-9700-5

Authors:M. Behboodi; S. Heidari Abstract: A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and only if every R-module is serial. This motivated us to study commutative rings for which every proper ideal is serial. In this paper, we determine completely the structure of commutative rings R of which every proper ideal is serial. It is shown that every proper ideal of R is serial, if and only if, either R is a serial ring, or R is a local ring with maximal ideal \({\mathcal {M}}\) such that there exist a uniserial module U and a semisimple module T with \({\mathcal {M}}=U\oplus T\) . Moreover, in the latter case, every proper ideal of R is isomorphic to \(U^{\prime }\oplus T^{\prime }\) , for some \(U^{\prime }\leq U\) and \(T^{\prime }\leq T\) . Furthermore, it is shown that every proper ideal of a commutative Noetherian ring R is serial, if and only if, either R is a finite direct product of discrete valuation domains and local Artinian principal ideal rings, or R is a local ring with maximal ideal \({\mathcal {M}}\) containing a set of elements {w 1,…,w n } such that \({\mathcal {M}}=\bigoplus _{i=1}^{n} Rw_{i}\) with at most one non-simple summand. Moreover, another equivalent condition states that: there exists an integer n ≥ 1 such that every proper ideal of R is a direct sum of at most n uniserial R-modules. Finally, we discuss some examples to illustrate our results. PubDate: 2017-05-16 DOI: 10.1007/s10468-017-9699-7

Authors:Nohra Hage; Philippe Malbos Abstract: We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators. PubDate: 2017-05-12 DOI: 10.1007/s10468-017-9686-z

Authors:Onofrio M. Di Vincenzo; Vincenzo Nardozza Abstract: Let F be a field of characteristic zero. We study two minimal superalgebras A and B having the same superexponent but such that T 2 (A) ⫋ T 2 (B), thus providing the first example of a minimal superalgebra generating a non minimal supervariety. We compare the structures and codimension sequences of A and B. PubDate: 2017-05-12 DOI: 10.1007/s10468-017-9698-8

Authors:Johan Öinert Abstract: In this article we introduce the notion of a controlled group graded ring. Let G be a group, with identity element e, and let R = ⊕ g∈G R g be a unital G-graded ring. We say that R is G-controlled if there is a one-to-one correspondence between subsets of the group G and (mutually non-isomorphic) R e -sub-bimodules of R, given by G ⊇ H↦ ⊕ h∈H R h . For strongly G-graded rings, the property of being G-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general G-graded ring to be G-controlled. We also give a characterization of strongly G-graded rings which are G-controlled. As an application of our main results we give a description of all intermediate subrings T with R e ⊆ T ⊆ R of a G-controlled strongly G-graded ring R. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras. PubDate: 2017-05-10 DOI: 10.1007/s10468-017-9696-x

Authors:Karel Casteels; Siân Fryer Abstract: We study the commutative algebras Z J K appearing in Brown and Goodearl’s extension of the \(\mathcal {H}\) -stratification framework, and show that if A is the single parameter quantized coordinate ring of M m,n , G L n or S L n , then the algebras Z J K can always be constructed in terms of centres of localizations. The main purpose of the Z J K is to study the structure of the topological space s p e c(A), which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real m × n matrices in terms of its restricted permutation. PubDate: 2017-05-08 DOI: 10.1007/s10468-017-9668-1

Authors:Louis Rowen; Yoav Segev Abstract: The purpose of this note is to obtain precise information about associative or Jordan algebras generated by two idempotents. PubDate: 2017-05-06 DOI: 10.1007/s10468-017-9697-9

Authors:Ken A. Brown; Marjory J. Macleod Abstract: Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres. PubDate: 2017-05-02 DOI: 10.1007/s10468-017-9694-z