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: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

Authors:L. M. Camacho; A. Kh. Khudoyberdiyev; B. A. Omirov Abstract: In this paper we study subalgebras of complex finite dimensional evolution algebras. We obtain the classification of nilpotent evolution algebras whose any subalgebra is an evolution subalgebra with a basis which can be extended to a natural basis of algebra. Moreover, we formulate three conjectures related to the description of such non-nilpotent algebras. PubDate: 2018-01-31 DOI: 10.1007/s10468-018-9767-7

Authors:Oliver Lorscheid; Thorsten Weist Abstract: In this text, we exhibit the quiver Plücker relations for a quiver Grassmannian and show that they describe the quiver Grassmannian as a closed subscheme of a product of usual Grassmannians. PubDate: 2018-01-30 DOI: 10.1007/s10468-017-9762-4

Authors:Alexandru Chirvasitu; Ivan Penkov Abstract: We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices \(\mathfrak {gl}^{M}\left (V,V_{*}\right )\) . Here \(\mathfrak {gl}^{M}\left (V,V_{*}\right )\) is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces \(V_{*}\otimes V\to \mathbb {K}\) , where \(\mathbb {K}\) is the base field. Tensor representations of \(\mathfrak {gl}^{M}\left (V,V_{*}\right )\) are defined as arbitrary subquotients of finite direct sums of tensor products (V∗)⊗m ⊗ (V∗)⊗n ⊗ V⊗p where V∗ denotes the algebraic dual of V. The category \(\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}\) which they comprise, extends a category \(\mathbb {T}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}\) previously studied in Dan-Cohen et al. Adv. Math. 289, 205–278, (2016), Penkov and Serganova (2014) and Sam and Snowden Forum Math. Sigma 3(e11):108, (2015) . Our main result is that \(\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}\) is a finite-length, Koszul self-dual, tensor category with a certain universal property that makes it into a “categorified algebra” defined by means of a handful of generators and relations. This result uses essentially the general properties of ordered Grothendieck categories, which yield also simpler proofs of some facts about the category \(\mathbb {T}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}\) established in Penkov and Serganova (2014). Finally, we discuss the extension of \(\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}\) obtained by adjoining the algebraic dual (V∗)∗ of V∗. PubDate: 2018-01-30 DOI: 10.1007/s10468-018-9765-9

Authors:Natasha Rozhkovskaya Abstract: We deduce from a determinant identity on quantum transfer matrices of generalized quantum integrable spin chain model their generating functions. We construct the isomorphism of Clifford algebra modules of sequences of transfer matrices and the boson space of symmetric functions. As an application, tau-functions of transfer matrices immediately arise from classical tau-functions of symmetric functions. PubDate: 2018-01-29 DOI: 10.1007/s10468-018-9766-8

Authors:Indranil Biswas; Pierre-Emmanuel Chaput; Christophe Mourougane Abstract: Let G be an almost simple simply-connected affine algebraic group over an algebraically closed field k of characteristic p > 0. If G has type B n , C n or F4, we assume that p > 2, and if G has type G2, we assume that p > 3. Let P ⊂ G be a parabolic subgroup. We prove that the tangent bundle of G/P is Frobenius stable with respect to the anticanonical polarization on G/P. PubDate: 2018-01-24 DOI: 10.1007/s10468-018-9764-x

Authors:Alexander Tsymbaliuk Abstract: In this paper, we relate the well-known Fock representations of \(\ddot {U}_{q,d}(\mathfrak {sl}_{n})\) to the vertex, shuffle, and ‘L-operator’ representations of \(\ddot {U}_{q,d}(\mathfrak {sl}_{n})\) . These identifications generalize those for the quantum toroidal algebra of \(\mathfrak {gl}_{1}\) , which were recently established in Feigin et al. (J. Phys. A 48(24), 244001, 2015). PubDate: 2018-01-06 DOI: 10.1007/s10468-017-9761-5

Authors:Anil Khairnar; B. N. Waphare Abstract: The concept of a central strict ideal in a principally quasi-Baer (p.q.-Baer) ∗-ring is introduced. It is proved that the set of all prime central strict ideals in a p.q.-Baer ∗-ring is an anti-chain with respect to set inclusion. We obtain a separation theorem, which ensures an existence of prime central strict ideals in a p.q.-Baer *-ring. It is proved that the set of all prime central strict ideals (not necessarily prime ideals) of a p.q.-Baer ∗-ring carries the hull-kernel topology. We investigate the Hausdorffness and the compactness of this topology. As an application of spectral theory, it is proved that p.q.-Baer ∗-rings have a sheaf representation with injective sections. The class of p.q.-Baer ∗-rings which have a sheaf representation with stalks to be p.q.-Baer ∗-rings is determined. PubDate: 2018-01-02 DOI: 10.1007/s10468-017-9758-0