Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. The coefficients of polynomial invariants are integers if is a finite Galois extension of Q, and A is a scalar extension of some finite-dimensional semisimple Hopf algebra over Q. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference between the representation category and the representation ring of A. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose representation rings are isomorphic, but whose representation categories are distinct.     

Applications of stratifying systems to the finitistic dimension

Given an Ext-injective stratifying system of Λ-modules satisfying that the projective dimension of Y is finite, we prove that the finitistic dimension of the algebra Λ is equal to the finitistic dimension of the category . Moreover, using the theory of stratifying systems we obtain bounds for the finitistic dimension of Λ. In particular, we get the optimal bound 2n-2 for the finitistic dimension of a standardly stratified algebra with n simples.     

Gabriel-Popescu type theorems and applications

In this paper we obtain a general version of Gabriel-Popescu theorem representing any Grothendieck category as a quotient category of the category of modules over a ring (not necessarily with unit) with enough idempotents to right using a family of generators (U_i)_i∈I of where U_i are not supposed to be small. Applications to locally finite categories are obtained. In particular, for a coalgebra C (over a field) we prove that C is right semiperfect if and only if the category has the AB4∗ condition.     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .     

Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n+1-power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary k-category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.     

Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes

Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category where and ?R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism φ:R→S. We characterize the case when φ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms μ^′ and ν^′ between the interval in the lattice of torsion classes in , and the lattice of all torsion classes in . We provide necessary and sufficient conditions for μ^′ and ν^′ to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ^′ and ν^′ contains all injectives.     

Ext-projectives in suspended subcategories

Let H be a hereditary abelian k-category with tilting object and denote the bounded derived category of H. This paper is devoted to a study of suspended subcategories of by means of their Ext-projectives.     

Reduction of derived Hochschild functors over commutative algebras and schemes

We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finite-type flat map of noetherian schemes, with f^!OY in place of D.     

Representation dimension of extensions of hereditary algebras

We show that if H is a hereditary finite dimensional algebra, M is a finitely generated H-module and B is a semisimple subalgebra of End_H(M)^op, then the representation dimension of is less than or equal to 3 whenever one of the following conditions holds: (i) H is of finite representation type; (ii) H is tame and M is a direct sum of regular and preprojective modules; (iii) M has no self-extensions.     

Graded polynomial identities and codimensions: Computing the exponential growth

Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of , n=1,2,…, the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G×Z₂-graded algebra related to A.     

The quiver of projectives in hereditary categories with Serre duality

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to for a so-called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form ZQ.     

Irreducible morphisms in the bounded derived category

We study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra Λ, denoted , by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from X to X if X is an indecomposable complex. In case Λ is a selfinjective Artin algebra we show that for every irreducible morphism f in either f^j is split monomorphism for all j∈Z or split epimorphism, for all j∈Z. Moreover, we prove that all the non-trivial components of the Auslander-Reiten quiver of are of the form ZA_∞.     

Mesh geometries of root orbits of integral quadratic forms

Integral quadratic forms q:Zⁿ→Z, with n≥1, and the sets R_q(d)={v∈Zⁿ;q(v)=d}, with d∈Z, of their integral roots are studied by means of mesh translation quivers defined by Z-bilinear morsifications b_A:Zⁿ×Zⁿ→Z of q, with Z-regular matrices A∈M_n(Z). Mesh geometries of roots of positive definite quadratic forms q:Zⁿ→Z are studied in connection with root mesh quivers of forms associated to Dynkin diagrams A_n,D_n,E₆,E₇,E₈ and the Auslander-Reiten quivers of the derived category D^b(R) of path algebras R=KQ of Dynkin quivers Q. We introduce the concepts of a Z-morsification b_A of a quadratic form q, a weighted Φ_A-mesh of vectors in Zⁿ, and a weighted Φ_A-mesh orbit translation quiver Γ(R_q,Φ_A) of vectors in Zⁿ, where R_q?R_q(1) and Φ_A:Zⁿ→Zⁿ is the Coxeter isomorphism defined by A. The existence of mesh geometries on R_q is discussed. It is shown that, under some assumptions on the morsification b_A:Zⁿ×Zⁿ→Z, the set admit a Φ_A-orbit mesh quiver , where Φ_A:Zⁿ→Zⁿ is the Coxeter isomorphism defined by A. Moreover, splits into three infinite connected components , , and , where are isomorphic to a translation quiver Z⋅Δ, with Δ an extended Dynkin quiver, and has the shape of a sand-glass tube.     

The phantom cover of a module

A morphism of left R-modules is a phantom morphism if for any morphism , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor₁(X,f)=0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.If is the category of finitely presented right R-modules modulo projectives, then the association M?Tor₁(−,M) is a functor from the category of left R-modules to that of the flat functors on . The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.     

Critical points between varieties generated by subspace lattices of vector spaces

We denote by the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by the class of all semilattices isomorphic to for some A∈V. Given varieties V and W of algebras, the critical point of V under W is defined as . Given a finitely generated variety V of modular lattices, we obtain an integer ?, depending on V, such that for any n≥? and any field F.In a second part, using tools introduced in Gillibert (2009) [5], we prove that:

     

On cluster algebras arising from unpunctured surfaces II

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras.Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincaré characteristic of quiver Grassmannians in Dynkin type A and affine Dynkin type .     

Codimensions of algebras and growth functions

Let A be an algebra over a field F of characteristic zero and let cn(A), , be its sequence of codimensions. We prove that if cn(A) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α>1, an F-algebra Aα such that exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.     

Finitistic dimension of properly stratified algebras

We prove that the finitistic dimension of a properly stratified algebra having a simple preserving duality and for which every tilting module is cotilting, equals twice the projective dimension of the characteristic tilting module. As a corollary, we get that the global dimension of a quasi-hereditary algebra with duality equals twice the projective dimension of the characteristic tilting module. As another corollary, we obtain an affirmative answer to the conjecture of Erdmann and Parker. Finally, we calculate the finitistic dimension of the blocks of certain parabolic generalizations of the category .