Authors:Sean Lawton; Adam S. Sikora Pages: 1133 - 1141 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: 2017-10-01 DOI: 10.1007/s10468-017-9679-y Issue No:Vol. 20, No. 5 (2017)

Authors:Torkil Stai Pages: 1239 - 1247 Abstract: We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. These considerations show that a stably hereditary or gentle one-cycle algebra is piecewise hereditary if and only if the orbit category of its bounded derived category with respect to a positive power of the shift functor is triangulated. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9684-1 Issue No:Vol. 20, No. 5 (2017)

Authors:S. K. Prajapati; M. R. Darafsheh; M. Ghorbani Pages: 1289 - 1303 Abstract: P. Hall introduced the concept of isoclinism of groups to classify p-groups. It is well-known that two isoclinic nilpotent groups have the same nilpotency class. In this paper using the classification of James of p-groups of order at most p 5 via their isoclinism classes, the degrees of irreducible characters with their frequencies are found. To do this we use the concept of generalized Camina pairs. We also investigate that whether a nonlinear irreducible character can be obtained as a product of two other nonlinear irreducible characters of same degree. PubDate: 2017-10-01 DOI: 10.1007/s10468-017-9687-y Issue No:Vol. 20, No. 5 (2017)

Authors:Henry Kvinge; Monica Vazirani Abstract: We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of “trivial” modules. The nodes of the fundamental crystal correspond to simple modules of the corresponding cyclotomic KLR algebra. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs. PubDate: 2017-11-24 DOI: 10.1007/s10468-017-9747-3

Authors:Petter Andreas Bergh; David A. Jorgensen Abstract: We prove a generalized Dade’s Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient (and necessary) condition for the vanishing of all higher Ext or Tor of the modules. This condition involves the vanishing of all higher Ext or Tor of the modules over all quotients by a minimal generator of the ideal generated by the regular sequence. PubDate: 2017-11-23 DOI: 10.1007/s10468-017-9751-7

Authors:Kei Yuen Chan Abstract: In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen filtrations. Our approach involves the use of information from the Langlands classification as well as some deeper understanding on some structure of some modules. Such module arises from the image of a Knapp-Stein type intertwining operator and is a quotient of a generalized standard module. As an application, we compute the Ext-groups for irreducible modules in a block for the graded Hecke algebra of type C 3, assuming the truth of a version of Jantzen conjecture. PubDate: 2017-11-22 DOI: 10.1007/s10468-017-9742-8

Authors:M. Behboodi; A. Daneshvar; M. R. Vedadi Abstract: We say that an R-module M is virtually semisimple if each submodule of M is isomorphic to a direct summand of M. A nonzero indecomposable virtually semisimple module is then called a virtually simple module. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules . Our study provides several natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring R: (i) Every finitely generated left (right) R-module is virtually semisimple; (ii) Every finitely generated left (right) R-module is a direct sum of virtually simple R-modules; (iii) \(R\cong {\prod }_{i = 1}^{k} M_{n_{i}}(D_{i})\) where k,n 1,…,n k ∈ ℕ and each D i is a principal ideal V-domain; and (iv) Every nonzero finitely generated left R-module can be written uniquely (up to isomorphism and order of the factors) in the form R m 1 ⊕… ⊕ R m k where each R m i is either a simple R-module or a virtually simple direct summand of R. PubDate: 2017-11-20 DOI: 10.1007/s10468-017-9748-2

Authors:Yury Volkov; Alexandra Zvonareva Abstract: We generalize the notion of Külshammer ideals to the setting of a graded category. This allows us to define and study some properties of Külshammer type ideals in the graded center of a triangulated category and in the Hochschild cohomology of an algebra, providing new derived invariants. Further properties of Külshammer ideals are studied in the case where the category is d-Calabi-Yau. PubDate: 2017-11-20 DOI: 10.1007/s10468-017-9746-4

Authors:Jingcheng Dong; Sonia Natale Abstract: Let \(\mathcal {C}\) be a modular category of Frobenius-Perron dimension d q n , where q > 2 is a prime number and d is a square-free integer. We show that \(\mathcal {C}\) must be integral and nilpotent and therefore group-theoretical. In the case where q = 2, we describe the structure of \(\mathcal {C}\) in terms of equivariantizations of group-crossed braided fusion categories. PubDate: 2017-11-16 DOI: 10.1007/s10468-017-9750-8

Authors:Igor Klep Abstract: Let R be a noncommutative ring. Two epimorphisms $$\alpha_{i}:R\to (D_{i},\leqslant_{i}),\quad i = 1,2 $$ from R to totally ordered division rings are called equivalent if there exists an order-preserving isomorphism ϕ : (D 1, ⩽ 1) → (D 2, ⩽ 2) satisfying ϕ ∘ α 1 = α 2. In this paper we study the real epi-spectrum of R, defined to be the set of all equivalence classes (with respect to this relation) of epimorphisms from R to ordered division rings. We show that it is a spectral space when endowed with a natural topology and prove a variant of the Artin-Lang homomorphism theorem for finitely generated tensor algebras over real closed division rings. PubDate: 2017-11-14 DOI: 10.1007/s10468-017-9745-5

Authors:Alessandro Ardizzoni; Isar Goyvaerts; Claudia Menini Abstract: We give a description of the category of restricted Lie algebras over a field \(\Bbbk \) of prime characteristic by means of monadic decomposition of the functor that computes the \(\Bbbk \) -vector space of primitive elements of a \(\Bbbk \) -bialgebra. PubDate: 2017-11-02 DOI: 10.1007/s10468-017-9734-8

Authors:Reuven Hodges; Venkatramani Lakshmibai Abstract: Let L w be the Levi part of the stabilizer Q w in G L N (for left multiplication) of a Schubert variety X(w) in the Grassmannian G d,N . For the natural action of L w on \(\mathbb {C}[X(w)]\) , the homogeneous coordinate ring of X(w) (for the Plücker embedding), we give a combinatorial description of the decomposition of \(\mathbb {C}[X(w)]\) into irreducible L w -modules; in fact, our description holds more generally for the action of the Levi part L of any parabolic subgroup Q that is contained in Q w . This decomposition is then used to show that all smooth Schubert varieties, all determinantal Schubert varieties, and all Schubert varieties in G2,N are spherical L w -varieties. PubDate: 2017-10-28 DOI: 10.1007/s10468-017-9744-6

Authors:Liqian Bai; Xueqing Chen; Ming Ding; Fan Xu Abstract: We define a quantum analog of a class of generalized cluster algebras which can be viewed as a generalization of quantum cluster algebras defined in Berenstein and Zelevinsky (Adv. Math. 195(2), 405–455 2005). In the case of rank two, we extend some structural results from the classical theory of generalized cluster algebras obtained in Chekhov and Shapiro (Int. Math. Res. Notices 10, 2746–2772 2014) and Rupel (2013) to the quantum case. PubDate: 2017-10-24 DOI: 10.1007/s10468-017-9743-7

Authors:Haijun Tan; Kaiming Zhao Abstract: In this paper, by using the “twisting technique” we obtain a class of new modules A b over the Witt algebras \(\mathcal {W}_{n}\) from modules A over the Weyl algebras \(\mathcal {K}_{n}\) (of Laurent polynomials) for any \(b\in \mathbb {C}\) . We give necessary and sufficient conditions for A b to be irreducible, and determine necessary and sufficient conditions for two such irreducible \(\mathcal {W}_{n}\) -modules to be isomorphic. Since \(\mathfrak {sl}_{n+1}(\mathbb {C})\) is a subalgebra of \(\mathcal {W}_{n}\) , all the above irreducible \(\mathcal {W}_{n}\) -modules A b can be considered as \(\mathfrak {sl}_{n+1}(\mathbb {C})\) -modules. For a class of such \(\mathfrak {sl}_{n+1}(\mathbb {C})\) -modules, denoted by Ω1−a (λ 1, λ 2, ⋯ ,λ n ) where \(a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}\) , we determine necessary and sufficient conditions for these \(\mathfrak {sl}_{n+1}(\mathbb {C})\) -modules to be irreducible. If the \(\mathfrak {sl}_{n+1}(\mathbb {C})\) -module Ω1−a (λ 1, λ 2,⋯ ,λ n ) is reducible, we prove that it has a unique nontrivial submodule W 1−a (λ 1, λ 2,...λ n ) and the quotient module is the finite dimensional \(\mathfrak {sl}_{n+1}(\mathbb {C})\) -module with highest weight mΛ n for some non-negative integer \(m\in \mathbb {Z}_{+}\) . We also determine necessary and sufficient conditions for two \(\mathfrak {sl}_{n+1}(\mathbb {C})\) -modules of the form Ω1−a (λ 1, λ 2,⋯ ,λ n ) or of the form W 1−a (λ 1, λ 2,...λ n ... PubDate: 2017-10-19 DOI: 10.1007/s10468-017-9738-4

Authors:Huanhuan Li Abstract: For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally \(\mathbb {Z}\) -graded and viewed as a differential graded algebra with trivial differential. PubDate: 2017-10-13 DOI: 10.1007/s10468-017-9741-9

Authors:Leonid Positselski Abstract: We present the definition of a dedualizing complex of bicomodules over a pair of cocoherent coassociative coalgebras \(\mathcal {C}\) and \(\mathcal {D}\) . Given such a complex \(\mathcal {B}^{\bullet }\) , we construct an equivalence between the (bounded or unbounded) conventional, as well as absolute, derived categories of the abelian categories of left comodules over \(\mathcal {C}\) and left contramodules over \(\mathcal {D}\) . Furthermore, we spell out the definition of a dedualizing complex of bisemimodules over a pair of semialgebras, and construct the related equivalence between the conventional or absolute derived categories of the abelian categories of semimodules and semicontramodules. Artinian, co-Noetherian, and cocoherent coalgebras are discussed as a preliminary material. PubDate: 2017-10-12 DOI: 10.1007/s10468-017-9736-6

Authors:Pamela Suarez Abstract: Let A be the one point extension of an algebra B by a projective B-module. We prove that the extension of a given support τ-tilting B-module is a support τ-tilting A-module; and, conversely, the restriction of a given support τ-tilting A-module is a support τ-tilting B-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support τ-tilting modules. PubDate: 2017-10-07 DOI: 10.1007/s10468-017-9737-5

Authors:Yu Zhou; Bin Zhu Abstract: We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type A n or A ∞ is given via rotation of Ptolemy diagrams. PubDate: 2017-10-06 DOI: 10.1007/s10468-017-9740-x

Authors:Hernán Giraldo Abstract: We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra. PubDate: 2017-10-05 DOI: 10.1007/s10468-017-9733-9

Authors:M. Aaghabali; S. Akbari; M. H. Bien Abstract: Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a 0 + a 1 x + ⋯ + a n x n (resp. right polynomial a 0 + x a 1 + ⋯ + x n a n ) over K such that a 0 + a 1 a + ⋯ + a n a n = 0 (resp. a 0 + a a 1 + ⋯ + a n a n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions. PubDate: 2017-10-03 DOI: 10.1007/s10468-017-9739-3