Authors:Julia Yael Plavnik; Sarah Witherspoon Pages: 259 - 276 Abstract: We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9713-0 Issue No:Vol. 21, No. 2 (2018)

Authors:Shmuel Zelikson Pages: 277 - 307 Abstract: Let \(\mathfrak {g}\) be a simple complex Lie algebra of types A n , D n , E n , and Q a quiver obtained by orienting its Dynkin diagram. Let λ be a dominant weight, and E(λ) the corresponding simple highest weight representation. We show that the weight multiplicities of E(λ) may be recovered by playing a numbers game Λ Q (λ), generalizing the well known Mozes game, constructing the orbit of λ under the action of the Weyl group W. The game board is provided by the Auslander-Reiten quiver Γ Q of Q. The game moves are obtained by constructing Nakajima’s monomial crystal M(λ) directly out of Γ Q . As an application, we consider Kashiwara’s parameterizations of the canonical basis. Let w 0 be a reduced expression of the longest element w 0 of W, adapted to a quiver Q of type A n . We show that a set of inequalities defining the string (Kashiwara) cone with respect to w 0, may be obtained by playing subgames of the numbers games Λ Q (ω i ) associated to fundamental representations. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9714-z Issue No:Vol. 21, No. 2 (2018)

Authors:Alberto Facchini; Andrea Lucchini Pages: 309 - 329 Abstract: We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups \(\mathcal {N}(G)\) has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9715-y Issue No:Vol. 21, No. 2 (2018)

Authors:Daniel Bissinger Pages: 331 - 358 Abstract: We investigate the generalized Kronecker algebra ð’¦ r = kΓ r with r ≥ 3 arrows. Given a regular component ð’ž of the Auslander-Reiten quiver of ð’¦ r , we show that the quasi-rank rk(ð’ž) ∈ ℤ≤1 can be described almost exactly as the distance ð’²(ð’ž) ∈ ℕ0 between two non-intersecting cones in ð’ž, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality $$-\mathcal{W}(\mathcal{C}) \leq \text{rk}(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.$$ Utilizing covering theory, we construct for each n ∈ ℕ0 a bijection φ n between the field k and {ð’ž∣ð’ž regular component, ð’²(ð’ž) = n}. As a consequence, we get new results about the number of regular components of a fixed quasi-rank. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9716-x Issue No:Vol. 21, No. 2 (2018)

Authors:V. V. Bavula Pages: 359 - 373 Abstract: Criteria are given for a ring to have a left Noetherian largest left quotient ring. It is proved that each such a ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian ring has only finitely many maximal left denominator sets. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9717-9 Issue No:Vol. 21, No. 2 (2018)

Authors:Charles F. Doran; Michael G. Faux; Sylvester J. Gates; Tristan Hübsch; Kevin Iga; Gregory D. Landweber Pages: 375 - 397 Abstract: An off-shell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1 N}\) , correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincaré algebra \(\mathfrak {p}^{1,1 p,q}\) , correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9718-8 Issue No:Vol. 21, No. 2 (2018)

Authors:Paul Balmer; Jon F. Carlson Pages: 399 - 417 Abstract: We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9719-7 Issue No:Vol. 21, No. 2 (2018)

Authors:Gena Puninski Pages: 419 - 446 Abstract: We describe the Cohen–Macaulay part of the Ziegler spectrum and calculate Ringel’s quilt of the category of finitely generated Cohen–Macaulay modules over the A-infinity plane curve singularity. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9720-1 Issue No:Vol. 21, No. 2 (2018)

Authors:Ralf Schiffler; Khrystyna Serhiyenko Pages: 447 - 470 Abstract: Every cluster-tilted algebra B is the relation extension \(C\ltimes \textup {Ext}^{2}_{C}(DC,C)\) of a tilted algebra C. A B-module is called induced if it is of the form M⊗ C B for some C-module M. We study the relation between the injective presentations of a C-module and the injective presentations of the induced B-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced B-module. In the case where the C-module, and hence the B-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9721-0 Issue No:Vol. 21, No. 2 (2018)

Authors:Saeed Nasseh; Ryo Takahashi Pages: 471 - 485 Abstract: The primary goal of this paper is to investigate the structure of irreducible monomorphisms to and irreducible epimorphisms from finitely generated free modules over a noetherian local ring. Then we show that over such a ring, self-vanishing of Ext and Tor for a finitely generated module admitting such an irreducible homomorphism forces the ring to be regular. PubDate: 2018-04-01 DOI: 10.1007/s10468-017-9722-z Issue No:Vol. 21, No. 2 (2018)

Authors:Paul-Emile Paradan Abstract: In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds. PubDate: 2018-04-16 DOI: 10.1007/s10468-018-9785-5

Authors:David Kazhdan; Tamar Ziegler Abstract: Let V be a vector space over a field k, P : V → k, d ≥ 3. We show the existence of a function C(r, d) such that rank(P) ≤ C(r, d) for any field k, char(k) > d, a finite-dimensional k-vector space V and a polynomial P : V → k of degree d such that rank(∂P/∂t) ≤ r for all t ∈ V − 0. Our proof of this theorem is based on the application of results on Gowers norms for finite fields k. We don’t know a direct proof even in the case when k = ℂ. PubDate: 2018-04-16 DOI: 10.1007/s10468-018-9783-7

Authors:Ming Lu Abstract: For given bound quiver algebras A and B, we obtain a new algebra Λ, called the simple gluing algebra, by identifying two vertices. We investigate the Gorenstein homological property, the singularity category, the Gorenstein defect category and the Cohen-Macaulay Auslander algebra of Λ in terms of that of A and B. Finally, we give applications to cluster-tilted algebras. PubDate: 2018-04-14 DOI: 10.1007/s10468-018-9782-8

Authors:Valentin Ovsienko; Serge Tabachnikov Abstract: We investigate a general method that allows one to construct new integer sequences extending existing ones. We apply this method to the classic Somos-4 and Somos-5, and the Gale-Robinson sequences, as well as to more general class of sequences introduced by Fordy and Marsh, and produce a great number of new sequences. The method is based on the notion of “weighted quiver”, a quiver with a \(\mathbb {Z}\) -valued function on the set of vertices that obeys very special rules of mutation. PubDate: 2018-03-19 DOI: 10.1007/s10468-018-9779-3

Authors:A. N. Sergeev Abstract: Let \(\mathcal {D}_{n,m}\) be the algebra of quantum integrals of the deformed Calogero-Moser-Sutherland problem corresponding to the root system of the Lie superalgebra \(\frak {gl}(n,m)\) . The algebra \(\mathcal {D}_{n,m}\) acts naturally on the quasi-invariant Laurent polynomials and we investigate the corresponding spectral decomposition. Even for general value of the parameter k the spectral decomposition is not multiplicity free and we prove that the image of the algebra \(\mathcal {D}_{n,m}\) in the algebra of endomorphisms of the generalised eigenspace is k[ε]⊗r where k[ε] is the algebra of dual numbers. The corresponding representation is the regular representation of the algebra k[ε]⊗r. PubDate: 2018-03-16 DOI: 10.1007/s10468-018-9778-4

Authors:Kenichi Shimizu Abstract: We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category \(\mathcal {C}\) by using a certain adjunction between \(\mathcal {C}\) and its Drinfeld center \(\mathcal {Z}(\mathcal {C})\) . These notions can be identified with integrals and cointegrals of a finite-dimensional Hopf algebra H if \(\mathcal {C}\) is the representation category of H. We generalize basic results on integrals and cointegrals of a finite-dimensional Hopf algebra (such as the existence, the uniqueness, and the Maschke theorem) to finite tensor categories. Motivated by results of Lorenz, we also investigate relations between categorical integrals and morphisms factoring through projective objects. Finally, we extend the n-th indicator of a finite-dimensional Hopf algebra introduced by Kashina, Montgomery and Ng to finite tensor categories. PubDate: 2018-03-13 DOI: 10.1007/s10468-018-9777-5

Authors:Yury A. Neretin Abstract: Consider a restriction of an irreducible finite dimensional holomorphic representation of \(\text {GL}(n + 1,\mathbb {C})\) to the subgroup \(\text {GL}(n,\mathbb {C})\) . We write explicitly formulas for generators of the Lie algebra \(\mathfrak {g}\mathfrak {l}(n + 1)\) in the direct sum of representations of \(\text {GL}(n,\mathbb {C})\) . Nontrivial generators act as differential-difference operators, the differential part has order n − 1, the difference part acts on the space of parameters (highest weights) of representations. We also formulate a conjecture about unitary principal series of \(\text {GL}(n,\mathbb {C})\) . PubDate: 2018-03-13 DOI: 10.1007/s10468-018-9774-8

Authors:Andreas Bächle; Leo Margolis Abstract: In this article the study of the Prime Graph Question for the integral group ring of almost simple groups which have an order divisible by exactly 4 different primes is continued. We provide more details on the recently developed “lattice method” which involves the calculation of Littlewood-Richardson coefficients. We apply the method obtaining results complementary to those previously obtained using the HeLP-method. In particular the “lattice method” is applied to infinite series of groups for the first time. We also prove the Zassenhaus Conjecture for four more simple groups. Furthermore we show that the Prime Graph Question has a positive answer around the vertex 3 provided the Sylow 3-subgroup is of order 3. PubDate: 2018-03-12 DOI: 10.1007/s10468-018-9776-6

Authors:Ayako Itaba Abstract: Let k be an algebraically closed field. For a graded algebra, Mori introduced a notion of cogeometric pair (E, σ), where \(E\subset \mathbb {P}^{n-1}\) is a subscheme and σ ∈Aut E. On the other hand, for a finite-dimensional algebra, Erdmann et al. defined the finiteness condition (Fg). In this paper, we show the following results which state relationships between cogeometric pairs and (Fg). Let \(A=\mathcal {A}^{!}(E,\sigma )\) be a cogeometric self-injective Koszul k-algebra such that the complexity of k is finite. (1) If A satisfies (Fg), then the order of σ is finite. (2) In the case of \(E=\mathbb {P}^{n-1}\) , A satisfies (Fg) if and only if the order of σ is finite. (3) If A satisfies (rad A)4 = 0, then A satisfies (Fg) if and only if the order of σ is finite. PubDate: 2018-02-28 DOI: 10.1007/s10468-018-9775-7

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