Authors:Gabriella Böhm; Stephen Lack Pages: 1 - 46 Abstract: The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele’s definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved — in an appropriate, multiplier-valued sense — which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved — which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid. PubDate: 2017-02-01 DOI: 10.1007/s10468-016-9630-7 Issue No:Vol. 20, No. 1 (2017)

Authors:Giovanni Cerulli Irelli; Evgeny Feigin; Markus Reineke Pages: 147 - 161 Abstract: Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that each irreducible component of the quiver Grassmannians in question is isomorphic to a Schubert variety. We give an explicit description of the set of irreducible components, identify all the Schubert varieties arising, and compute the Poincaré polynomials of these quiver Grassmannians. PubDate: 2017-02-01 DOI: 10.1007/s10468-016-9634-3 Issue No:Vol. 20, No. 1 (2017)

Authors:Magnus Engenhorst Pages: 163 - 174 Abstract: Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that a quiver has a maximal green sequence if and only if it is of Dynkin type. More generally, we study module categories for finite-dimensional algebras with finitely many bricks. PubDate: 2017-02-01 DOI: 10.1007/s10468-016-9635-2 Issue No:Vol. 20, No. 1 (2017)

Authors:Matthew Bennet; Rollo Jenkins Pages: 197 - 208 Abstract: Given a finite-dimensional module V for a finite-dimensional, complex semi-simple Lie algebra \(\mathcal {g}\) , and a positive integer m, we construct a family of graded modules for the current algebra \(\mathcal {g}[t]\) indexed by simple C \(\mathcal {S}_{m}\) -modules. These modules are free of finite rank for the ring of symmetric polynomials and so can be localized to give finite-dimensional graded \(\mathcal {g}[t]\) -modules. We determine the graded characters of these modules and show that these graded characters admit a curious duality. PubDate: 2017-02-01 DOI: 10.1007/s10468-016-9637-0 Issue No:Vol. 20, No. 1 (2017)

Authors:Sarah Scherotzke Pages: 231 - 243 Abstract: In this paper, we show that generalized Nakajima Categories provide a framework to construct a desingularization of quiver Grassmannians for self-injective algebras of finite representation type. Furthermore, we show that all standard Frobenius models of orbit categories of the bounded derived category considered in Keller, Documenta Math. 10: 551–581, 2005 are equivalent to proj ð“¡, the finitely generated projective modules of the regular Nakajima category ð“¡. PubDate: 2017-02-01 DOI: 10.1007/s10468-016-9639-y Issue No:Vol. 20, No. 1 (2017)

Authors:M. Behboodi; Z. Fazelpour Pages: 245 - 255 Abstract: A theorem due to Nakayama and Skornyakov states that “a ring R is an Artinian serial ring if and only if all left R-modules are serial” and a theorem due to Warfield state that “a Noetherian ring R is serial if and only if every finitely generated left R-module is serial”. We say that an R-module M is prime uniserial (℘-uniserial, for short) if for every pair P, Q of prime submodules of M either \(P\subseteq Q\) or \(Q\subseteq P\) , and we say that M is prime serial (℘-serial, for short) if it is a direct sum of ℘-uniserial modules. Therefore, two interesting natural questions of this sort are: “Which rings have the property that every module is ℘-serial?” and “Which rings have the property that every finitely generated module is ℘-serial?” Most recently, in our paper, Prime uniserial modules and rings (submitted), we considered these questions in the context of commutative rings. The goal of this paper is to answer these questions in the case R is a Noetherian ring in which all idempotents are central or R is a left Artinian ring. PubDate: 2017-02-01 DOI: 10.1007/s10468-016-9642-3 Issue No:Vol. 20, No. 1 (2017)

Authors:Sean Lawton; Adam S. Sikora Abstract: Let G be a connected reductive affine algebraic group. In this short note we define the variety of G-characters of a finitely generated group Γ and show that the quotient of the G-character variety of Γ by the action of the trace preserving outer automorphisms of G normalizes the variety of G-characters when Γ is a free group, free abelian group, or a surface group. PubDate: 2017-03-18 DOI: 10.1007/s10468-017-9679-y

Authors:Giuseppe Baccella; Leonardo Spinosa Abstract: If R is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) R is unit-regular, (2) every factor ring of R is directly finite, (3) the abelian group K O(R) is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right R-modules. For the class of semiartinian and unit-regular rings the canonical partial order of K O(R) is investigated. Starting from any partially ordered set I, a special dimension group G(I) is built and a large class of semiartinian and unit-regular rings is shown to have the corresponding K O(R) order isomorphic to G(P r i m R ), where P r i m R is the primitive spectrum of R. Conversely, if I is an artinian partially ordered set having a finite cofinal subset, it is proved that the dimension group G(I) is realizable as K O(R) for a suitable semiartinian and unit-regular ring R. PubDate: 2017-03-13 DOI: 10.1007/s10468-017-9682-3

Authors:F. Saeedi; S. Sheikh-Mohseni Abstract: Let L be a Lie algebra, and Der z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der z (L) is abelian if and only if L is a stem Lie algebra. PubDate: 2017-03-10 DOI: 10.1007/s10468-017-9680-5

Authors:Cristina Draper; Alberto Elduque; Mikhail Kochetov Abstract: For any grading by an abelian group G on the exceptional simple Lie algebra \(\mathcal {L}\) of type E 6 or E 7 over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple \(\mathcal {L}\) -modules, as well as necessary and sufficient conditions for a finite-dimensional \(\mathcal {L}\) -module to admit a G-grading compatible with the given G-grading on \(\mathcal {L}\) . PubDate: 2017-03-06 DOI: 10.1007/s10468-017-9675-2

Authors:Benjamin Sambale Abstract: For a block B of a finite group G there are well-known orthogonality relations for the generalized decomposition numbers. We refine these relations by expressing the generalized decomposition numbers with respect to an integral basis of a certain cyclotomic field. After that, we use the refinements in order to give upper bounds for the number of irreducible characters (of height 0) in B. In this way we generalize results from [Héthelyi-Külshammer-Sambale, 2014]. These ideas are applied to blocks with abelian defect groups of rank 2. Finally, we address a recent conjecture by Navarro. PubDate: 2017-03-06 DOI: 10.1007/s10468-017-9676-1

Authors:Adrien Deloro Abstract: We give a new, geometric proof of a theorem by Timmesfeld showing that for simple Chevalley groups, abstract modules where all roots act quadratically are direct sums of minuscule representations. Our proof is uniform, treats finite and infinite fields on an equal footing, and includes Lie rings. PubDate: 2017-03-06 DOI: 10.1007/s10468-017-9671-6

Authors:Roozbeh Hazrat; Raimund Preusser Abstract: Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras L K (n, n + k) constructed by Leavitt. Using Bergman’s diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular. PubDate: 2017-02-25 DOI: 10.1007/s10468-017-9674-3

Authors:Jan O. Kleppe; Rosa M. Miró-Roig Abstract: This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves \(\mathcal {E}\) of arbitrary high rank on a general standard (resp. linear) determinantal scheme \(X\subset \mathbb {P}^{n}\) of codimension c ≥ 1, n − c ≥ 1 and defined by the maximal minors of a t × (t + c−1) homogeneous matrix \(\mathcal {A}\) . The sheaves \(\mathcal {E}\) are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme \(X\subset \mathbb {P}^{n}\) is of wild representation type provided the degrees of the entries of the matrix \(\mathcal {A}\) satisfy some weak numerical assumptions; and (2) we determine values of t, n and n − c for which a linear standard determinantal scheme \(X\subset \mathbb {P}^{n}\) is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. X is of Ulrich wild representation type. PubDate: 2017-02-25 DOI: 10.1007/s10468-017-9673-4

Authors:Alfredo Nájera Chávez Abstract: Let ð“” be a Frobenius category, \({\mathcal P}\) its subcategory of projective objects and F : ð“” → ð“” an exact automorphism. We prove that there is a fully faithful functor from the orbit category ð“”/F into \(\operatorname {gpr}({\mathcal P}/F)\) , the category of finitely-generated Gorenstein-projective modules over \({\mathcal P}/F\) . We give sufficient conditions to ensure that the essential image of ð“”/F is an extension-closed subcategory of \(\operatorname {gpr}({\mathcal P}/F)\) . If ð“” is in addition Krull-Schmidt, we give sufficient conditions to ensure that the completed orbit category \({\mathcal E} \ \widehat {\!\! /} F\) is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras. PubDate: 2017-02-23 DOI: 10.1007/s10468-017-9672-5

Authors:Matthew Ondrus; Emilie Wiesner Abstract: This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination algebra admits a triangular decomposition in the sense of Moody and Pianzola, and thus it is natural to define Whittaker modules corresponding to a given algebra homomorphism. Among other results, we show that the standard Whittaker modules are simple under certain constraints on the corresponding algebra homomorphism. PubDate: 2017-02-22 DOI: 10.1007/s10468-016-9665-9

Authors:Tobias Kildetoft Abstract: We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G r or a finite Chevalley group \(G(\mathbb {F}_q)\) . In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G r or \(G(\mathbb {F}_q)\) . Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin’s tilting conjecture. PubDate: 2017-02-22 DOI: 10.1007/s10468-017-9670-7

Authors:Minghui Zhao Abstract: Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T i on U for all i ∈ I. Since T i (f) is not contained in f, Lusztig considered two subalgebras i f and i f of f for any i ∈ I, where i f={x ∈ f T i (x) ∈ f} and \({^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\, \,\,T^{-1}_{i}(x)\in \mathbf {f}\}\) . The restriction of T i on i f is also denoted by \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\) . The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of i f, i f and \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\) by using the language ’quiver with automorphism’ introduced by Lusztig. PubDate: 2017-02-15 DOI: 10.1007/s10468-017-9669-0

Authors:Minghui Zhao Abstract: Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T i on U for all i ∈ I. Since T i (f) is not contained in f, Lusztig considered two subalgebras i f and i f of f for any i ∈ I, where i f={x ∈ f T i (x) ∈ f} and \({^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\, \,\,T^{-1}_{i}(x)\in \mathbf {f}\}\) . The restriction of T i on i f is also denoted by \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\) . The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of i f, i f and \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\) by using the language ’quiver with automorphism’ introduced by Lusztig. PubDate: 2017-02-15 DOI: 10.1007/s10468-017-9669-0

Authors:Shahn Majid 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: 2017-02-08 DOI: 10.1007/s10468-016-9661-0

Authors:Jacob Greenstein; Volodymyr Mazorchuk 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: 2017-02-03 DOI: 10.1007/s10468-016-9660-1