Abstract: 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: 2019-04-01

Abstract: 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: 2019-04-01

Abstract: 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: 2019-04-01

Abstract: 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: 2019-04-01

Abstract: Abstract A morphism \(f: M\rightarrow N\) of left R-modules is called an n-phantom morphism (resp. a Torn-epimorphism) if the induced morphism Torn(A, f) = 0 (resp. Torn(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. Extn-monomorphism) if the induced morphism Extn(B, g) = 0 (resp. Extn(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 Torn-epimorphism; A morphism g is an n-Ext-phantom morphism if and only if the pushout of any monomorphism along g is an Extn-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: 2019-04-01

Abstract: 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: 2019-04-01

Abstract: 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: 2019-04-01

Abstract: 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: 2019-04-01

Abstract: 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: 2019-04-01

Abstract: 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: 2019-04-01

Abstract: Abstract The module category of any artin algebra is filtered by the powers of its radical, thus defining an associated graded category. As an extension of the degree of irreducible morphisms, this text introduces the degree of morphisms in the module category. When the ground ring is a perfect field, and the given morphism behaves nicely with respect to covering theory (as do irreducible morphisms with indecomposable domain or indecomposable codomain), it is shown that the degree of the morphism is finite if and only if its induced functor has a representable kernel. This gives a generalisation of Igusa and Todorov result, about irreducible morphisms with finite left degree and over an algebraically closed field. As a corollary, generalisations of known results on the degrees of irreducible morphisms over perfect fields are given. Finally, this study is applied to the composition of paths of irreducible morphisms in relationship to the powers of the radical. PubDate: 2019-04-01

Abstract: Abstract A finite dimensional Lie algebra L with magic number c(L) is said to satisfy Rentschler’s property if it admits an abelian Lie subalgebra H of dimension at least c(L) − 1. We study the occurrence of this new property in various Lie algebras, such as nonsolvable, solvable, nilpotent, metabelian and filiform Lie algebras. Under some mild condition H gives rise to a complete Poisson commutative subalgebra of the symmetric algebra S(L). Using this, we show that Milovanov’s conjecture holds for the filiform Lie algebras of type Ln, Qn, Rn, Wn and also for all filiform Lie algebras of dimension at most eight. For the latter the Poisson center of these Lie algebras is determined. PubDate: 2019-03-20

Abstract: Abstract In his seminal paper “Formality conjecture”, M. Kontsevich introduced a graph complex GC1ve closely connected with the problem of constructing a formality quasi-isomorphism for Hochschild cochains. In this paper, we express the cohomology of the full directed graph complex dfGC explicitly in terms of the cohomology of GC1ve. Applications of our results include a recent work by the first author which completely characterizes homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the stable setting. PubDate: 2019-03-20

Abstract: Abstract Let \((R, \mathfrak {m}, k)\) denote a local Cohen-Macaulay ring such that the category of maximal Cohen-Macaulay R-modules mcmR contains an n-cluster tilting object L. In this paper, we compute the Quillen K-group G1(R) := K1(modR) explicitly as a direct sum of a finitely generated free abelian group and an explicit quotient of AutR(L)ab when R is a k-algebra and k is algebraically closed with characteristic not two. Moreover, we compute AutR(L)ab and G1(R) for certain hypersurface singularities. PubDate: 2019-03-14

Abstract: Abstract We classify Morita equivalence classes of indecomposable self-injective cellular algebras which have polynomial growth representation type, assuming that the characteristic of the base field is different from two. This assumption on the characteristic is for the cellularity to be a Morita invariant property. PubDate: 2019-03-12

Abstract: If \(\mathfrak {g}\) is a Lie algebra then the semi-centre of the Poisson algebra \(S(\mathfrak {g})\) is the subalgebra generated by \(\operatorname {ad}(\mathfrak {g})\) -eigenvectors. In this paper we abstract this definition to the context of integral Poisson algebras. We identify necessary and sufficient conditions for the Poisson semi-centre Asc to be a Poisson algebra graded by its weight spaces. In that situation we show the Poisson semi-centre exhibits many nice properties: the rational Casimirs are quotients of Poisson normal elements and the Poisson Dixmier–Mœglin equivalence holds for Asc. PubDate: 2019-03-12

Abstract: Abstract We construct realizations of quantum generalized Verma modules for \(U_{q}(\mathfrak {sl}_{n}(\mathbb {C}))\) by quantum differential operators. Taking the classical limit \(q \rightarrow 1\) provides a realization of classical generalized Verma modules for \(\mathfrak {sl}_{n}(\mathbb {C})\) by differential operators. PubDate: 2019-03-11

Abstract: Abstract We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank = 6, and spin modular categories up to rank = 11. In particular, we show that, up to fusion rules, there is exactly one non-split super-modular category of rank 2, 4 and 6, namely PSU(2)4k+ 2 for k = 0,1 and 2. This classification is facilitated by adapting and extending well-known constraints from modular categories to super-modular categories, such as Verlinde and Frobenius-Schur indicator formulae. PubDate: 2019-03-09

Abstract: Abstract This paper is investigative work into the properties of a family of graded algebras recently defined by Varagnolo and Vasserot, which we call VV algebras. We compare categories of modules over KLR algebras with categories of modules over VV algebras, establishing various Morita equivalences. Using these Morita equivalences we are able to prove several properties of certain classes of VV algebras such as (graded) affine cellularity and affine quasi-heredity. PubDate: 2019-03-02

Abstract: Abstract Let R be the pullback of two surjective homomorphisms of algebras A → B and C → B. Here we consider a particular class, the so-called linearly oriented pullback, where the injective and projective R-modules can be determined by those ones over A and C. For this class of pullbacks, we study the relationship between the left and the right parts of the category of modules (see Happel et al. 13) over the involved algebras in order to find relationships between some classes of these algebras. PubDate: 2019-02-28