Authors:Igor Klep Pages: 1 - 17 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: 2018-02-01 DOI: 10.1007/s10468-017-9745-5 Issue No:Vol. 21, No. 1 (2018)

Authors:A. S. Hegazi; Hani Abdelwahab; A. J. Calderon Martin Pages: 19 - 45 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: 2018-02-01 DOI: 10.1007/s10468-017-9701-4 Issue No:Vol. 21, No. 1 (2018)

Authors:Wee Liang Gan; John Watterlond Pages: 47 - 60 Abstract: Let VI be the category whose objects are the finite dimensional vector spaces over a finite field of order q and whose morphisms are the injective linear maps. A VI-module over a ring is a functor from the category VI to the category of modules over the ring. A VI-module gives rise to a sequence of representations of the finite general linear groups. We prove that the sequence obtained from any finitely generated VI-module over an algebraically closed field of characteristic zero is representation stable - in particular, the multiplicities which appear in the irreducible decompositions eventually stabilize. We deduce as a consequence that the dimension of the representations in the sequence {V n } obtained from a finitely generated VI-module V over a field of characteristic zero is eventually a polynomial in q n . Our results are analogs of corresponding results on representation stability and polynomial growth of dimension for FI-modules (which give rise to sequences of representations of the symmetric groups) proved by Church, Ellenberg, and Farb. PubDate: 2018-02-01 DOI: 10.1007/s10468-017-9703-2 Issue No:Vol. 21, No. 1 (2018)

Authors:María Julia Redondo; Lucrecia Román Pages: 61 - 86 Abstract: We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH∗(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures. PubDate: 2018-02-01 DOI: 10.1007/s10468-017-9704-1 Issue No:Vol. 21, No. 1 (2018)

Authors:T. Geetha; Amritanshu Prasad Pages: 131 - 143 Abstract: Young’s orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young’s orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software. PubDate: 2018-02-01 DOI: 10.1007/s10468-017-9706-z Issue No:Vol. 21, No. 1 (2018)

Authors:Mirko Rösner Pages: 145 - 161 Abstract: Parahoric restriction is the parahoric analogue of Jacquet’s functor. The group GSp(4, F) of symplectic similitudes of genus two over a local number field F/ℚ p has five conjugacy classes of parahoric subgroups. For each we determine the parahoric restriction of the non-cuspidal irreducible smooth representations of GSp(4, F) in terms of explicit character values. PubDate: 2018-02-01 DOI: 10.1007/s10468-017-9707-y Issue No:Vol. 21, No. 1 (2018)

Authors:Guoqiang Zhao Pages: 163 - 179 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: 2018-02-01 DOI: 10.1007/s10468-017-9708-x Issue No:Vol. 21, No. 1 (2018)

Authors:Aslak Bakke Buan; Yu Zhou Pages: 181 - 194 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: 2018-02-01 DOI: 10.1007/s10468-017-9709-9 Issue No:Vol. 21, No. 1 (2018)

Authors:Yury Volkov; Alexandra Zvonareva Pages: 195 - 217 Abstract: Let k be a commutative ring, \(\mathcal {A}\) and \(\mathcal {B}\) – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) , where \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) is the homotopy category of finitely generated projective \(\mathcal {A}\) -complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) and a map from the set of standard G-equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)\) to \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)\) , where \(\mathcal {A}/G\) denotes the orbit category. We investigate the properties of these maps and apply our results to the case where \(\mathcal {A}=\mathcal {B}=R\) is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field. PubDate: 2018-02-01 DOI: 10.1007/s10468-017-9710-3 Issue No:Vol. 21, No. 1 (2018)

Authors:Didier Arnal; Olfa Khlifi Pages: 219 - 237 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: 2018-02-01 DOI: 10.1007/s10468-017-9711-2 Issue No:Vol. 21, No. 1 (2018)

Authors:Dijana Jakelić; Adriano Moura Pages: 239 - 258 Abstract: We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for \(\widehat {\mathfrak {sl}}_{2}\) in terms of partitions with bounded parts, which subsequently lead to certain partition identities. PubDate: 2018-02-01 DOI: 10.1007/s10468-017-9712-1 Issue No:Vol. 21, No. 1 (2018)

Authors:Jeong-Ah Kim; Dong-Uy Shin Abstract: In this paper, we introduce an new combinatorial model, which we call generalized Young walls for classical Lie algebras, and we give two realizations of the crystal B(∞) over classical Lie algebras using generalized Young walls. Also, we construct natural crystal isomorphisms between generalized Young wall realizations and other realizations, for example, monomial realization, polyhedral realization and tableau realization. Moreover, as applications, we obtain a crystal isomorphism between two different polyhedral realizations of B(∞). PubDate: 2018-02-23 DOI: 10.1007/s10468-018-9770-z

Authors:Li Qiao; Xing Gao; Li Guo Abstract: In this paper we study Rota-Baxter modules with emphasis on the role played by the Rota-Baxter operators and the resulting difference between Rota-Baxter modules and the usual modules over an algebra. We introduce the concepts of free, projective, injective and flat Rota-Baxter modules. We give the construction of free modules and show that there are enough projective, injective and flat Rota-Baxter modules to provide the corresponding resolutions for derived functors. PubDate: 2018-02-22 DOI: 10.1007/s10468-018-9769-5

Authors:Lixin Mao Abstract: A morphism \(f: M\rightarrow N\) of left R-modules is called an n-phantom morphism (resp. a Tor n -epimorphism) if the induced morphism Tor n (A, f) = 0 (resp. Tor n (A, f) is an epimorphism) for every (finitely presented) right R-module A. Analogously, a morphism \(g: X\rightarrow Y\) of left R-modules is said to be an n-Ext-phantom morphism (resp. Ext n -monomorphism) if the induced morphism Ext n (B, g) = 0 (resp. Ext n (B, g) is a monomorphism) for every finitely presented left R-module B. It is proven that a morphism f is an n-phantom morphism if and only if the pullback of any epimorphism along f is a Tor n -epimorphism; A morphism g is an n-Ext-phantom morphism if and only if the pushout of any monomorphism along g is an Ext n -monomorphism. We also prove that every module has an object-special n-phantom precover. In addition, we introduce and investigate n-phantomless and n-Ext-phantomless rings. PubDate: 2018-02-20 DOI: 10.1007/s10468-018-9773-9

Authors:Karin Erdmann; Andrzej Skowroński Abstract: We introduce and study the higher tetrahedral algebras, an exotic family of finite-dimensional tame symmetric algebras over an algebraically closed field. The Gabriel quiver of such an algebra is the triangulation quiver associated to the coherent orientation of the tetrahedron. Surprisingly, these algebras occurred in the classification of all algebras of generalized quaternion type, but are not weighted surface algebras. We prove that a higher tetrahedral algebra is periodic if and only if it is non-singular. PubDate: 2018-02-20 DOI: 10.1007/s10468-018-9772-x

Authors:Malyha Nazari; Reza Sazeedeh Abstract: Let A be a noetherian ring and \(\frak a\) be an ideal of A. We define a condition P \(_{n}(\frak a)\) for \(\frak a\) -cofiniteness of modules and we show that if A is of dimension d satisfying P \(_{d-1}(\frak a)\) for all ideals of dimension d − 1, then it satisfies P \(_{d-1}(\frak a)\) for all ideals \(\frak a\) . Let M be an A-module and let n be a non-negative integer such that \({\text {Ext}_{A}^{i}}(A/\frak a,M)\) is finite for all i ≤ n + 1. We show that if \(\dim \mathrm {A}/\frak a = 1\) , then \(H_{\frak a}^{i}(M)\) is \(\frak a\) -cofinite for all i ≤ n and if A is local with \(\dim \mathrm {A}/\frak a = 2\) , then \(H_{\frak a}^{i}(M)\) is \(\frak a\) -cofinite for all i < n if and only if \(\text {Hom}_{\mathrm {A}}(\mathrm {A}/\frak a,\mathrm {H}_{\frak a}^{i}(\mathrm {M}))\) is finite for all i ≤ n. Finally we prove that if M is an A-module of dimension d such that \((0:_{H_{\frak a}^{d}(M)}\frak a)\) is finite, then \(H_{\frak a}^{d}(M)\) is artinian. PubDate: 2018-02-13 DOI: 10.1007/s10468-018-9771-y

Authors:Piotr Dowbor; Hagen Meltzer Abstract: Formulas for the dimension vectors of all objects M in the category \(\mathcal {S}(\tilde {6})\) of nilpotent operators with nilpotency degree bounded by 6, acting on finite dimensional vector spaces with invariant subspaces in a graded sense, are given (Theorem 2.3). For this purpose we realize a tubular algebra Λ, controlling the category \(\mathcal {S}(\tilde {6})\) , as an endomorphism algebra of a suitable tilting bundle over a weighted projective line of type (2,3,6) (Theorem 3.6). Using this description and a concept of mono-epi type, the interval multiplicity vector of an object in \(\mathcal {S}(\tilde {6})\) is introduced and determined (Theorem 2.8). This is a much finer invariant than the usual dimension vector. PubDate: 2018-02-10 DOI: 10.1007/s10468-017-9759-z

Authors:Hao Su; Dong Yang Abstract: Given an elementary simple-minded collection in the derived category of a non-positive dg algebra with finite-dimensional total cohomology, we construct a silting object via Koszul duality. PubDate: 2018-02-10 DOI: 10.1007/s10468-018-9763-y

Authors:Aaron J. Feickert; Sean Sather-Wagstaff Abstract: Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective R-modules need not be closed under tensor products, even when R is local and artinian; (2) the class of Gorenstein injective R-modules need not be closed under torsion products, even when R is a local, complete hypersurface; and (3) the filtrations given in our main theorem do not yield direct sum decompositions, even when R is a local, complete hypersurface. PubDate: 2018-02-07 DOI: 10.1007/s10468-018-9768-6