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In this paper, we study representations of the algebra \(\mathcal {A}\) generated by all arithmetic functions, determined by fixed primes (or prime numbers). The main purposes of this paper are (i) to establish nice representational models of \(\mathcal {A}\) under primes, (ii) to study fundamental properties of such representations, (iii) to investigate how \( \mathcal {A}\) is acting as operators in representations, and (iv) to consider new free probability models on Krein-space operator algebras.

In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables., 20 pages. This paper generalizes a previous version where the result was proven for a power series ring in two variables

Let G be a group and assume that (Ap)p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δp,q : Apq → Ap ⊗ Aq satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δp,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras.

In a previous paper, Auslander–Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincare duality space, each component of the Auslander–Reiten quiver is isomorphic to \(\mathbb{Z}A_{\infty }\).

In an earlier paper, Raphael Rouquier and the author introduced the group of self-equivalences of a derived category. In the case of a Brauer tree algebra, we determined a nontrivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism. Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( $$\mathfrak{g}$$ ) containing u( $$\mathfrak{g}$$ ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G r ) u( $$\mathfrak{g}$$ ) have also been constructed, and the results are stated in this setting.

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.

Classification problems of subcategories have been deeply considered so far. In this paper, we discuss classifying dense resolving and dense coresolving subcategories of exact categories via their Grothendieck groups. This study is motivated by the classification of dense triangulated subcategories of triangulated categories due to Thomason.

This paper describes the module categories for a family of generic Hecke algebras, called Yokonuma-type Hecke algebras. Yokonuma-type Hecke algebras specialize both to the group algebras of the complex reflection groups G(r,1,n) and to the convolution algebras of (B\(^{\prime }\),B\(^{\prime }\))-double cosets in the group algebras of finite general linear groups, for certain subgroups B\(^{\prime }\) consisting of upper triangular matrices. In particular, complete sets of inequivalent, irreducible modules for semisimple specializations of Yokonuma-type Hecke algebras are constructed.

In this paper, firstly, we characterize some rings by strict Mittag-Leffler conditions. Then, we investigate when Gorenstein projective modules are Gorenstein flat by employing tilting modules and cotorsion pairs. Finally, we study the direct limits of Gorenstein projective modules.

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field k. Let H=kG(χ,a,δ) be a Hopf-Ore extension of kG and H′ a rank one quotient Hopf algebra of H, where k is a field, G is a group, a is a central element of G and χ is a k-valued character for G with χ(a)≠1. We first show that the simple weight modules over H and H′ are finite dimensional. Then we describe the structures of all simple weight modules over H and H′, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over H′ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over H and H′, and classify them. Finally, when χ(a) is a primitive n-th root of unity for some n≥2, we determine all finite dimensional indecomposable projective objects in the category of weight modules over H′.

In this paper we provide some results regarding affine, prime, \(\mathbb {Z}\)-graded algebras \(R=\bigoplus _{i\in \mathbb {Z}}R_{i}\) generated by elements with degrees 1,−1 and 0, with R0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If Rk≠0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand Kirillov dimension less than three and any other algebra is Brown-McCoy radical.

In this paper, we use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence classification of the cluster-tilted algebra of type $A$, $D$ and $E$ respectively., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1012.4661 by other authors

In the paper we prove the double centralizer theorem for semiprime algebras. To be precise, let R be a closed semiprime algebra over its extended centroid F, and let A be a closed semiprime subalgebra of R, which is a finitely generated module over F. Then CR(A) is also a closed semiprime algebra and CR(CR(A)) = A. In addition, if CR(A) satisfies a polynomial identity, then so does the whole ring R. Here, for a subset T of R, we write CR(T): = {x ∈ R|xt = tx ∀ t ∈ T}, the centralizer of T in R.

In this paper we investigate the derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension is decomposed into three non-intersected families. We found sufficient conditions under which filiform Leibniz algebras of the first family are characteristically nilpotent. Moreover, for the first family we classify non-characteristically nilpotent algebras by means of Catalan numbers. In addition, for the rest two families of filiform Leibniz algebras we describe non-characteristically nilpotent algebras, i.e., those filiform Leibniz algebras which lie in the complementary set to those characteristically nilpotent.

In this paper, we characterize all the finite dimensional algebras that are derived equivalent to an m−cluster tilted algebra of type $\mathbb{A}$ . These algebras are gentle, and we show that the derived class of such an algebra is completely determined by the rank of its Grothendieck group and the dimension of its first Hochschild cohomology group.

Let $(\mathfrak{g}, [p]) $ be a restricted Lie superalgebra over an algebraically closed field k of characteristic p > 2. Let $\mathfrak{u}(\mathfrak{g})$ denote the restricted enveloping algebra of $\mathfrak{g}$ . In this paper we prove that the cohomology ring $\operatorname{H}^\bullet(\mathfrak{u}(\mathfrak{g}), k)$ is finitely generated. This allows one to define support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ -supermodules. We also show that support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ - supermodules satisfy the desirable properties of a support variety theory.

We study the homomorphism spaces between Specht modules for the Hecke algebras $\h$ of type $A$. We prove a cellular analogue of the kernel intersection theorem and a $q$-analogue of a theorem of Fayers and Martin and apply these results to give an algorithm which computes the homomorphism spaces $\Hom_{\h}(S^\mu,S^\lambda)$ for certain pairs of partitions $\lambda$ and $\mu$. We give an explicit description of the homomorphism spaces $\Hom_\h(S^\mu,S^\lambda)$ where $\h$ is an algebra over the complex numbers, $\lambda=(\lambda_1,\lambda_2)$ and $\mu$ is an arbitrary partition with $\mu_1 \geq \lambda_2$., Comment: 32 pages. This third version of the paper contains some comments on homomorphisms between the Specht modules defined by Dipper and James and has a more rigorous proof of the result following Proposition 4.1

In this paper we prove that the maximal algebra of quotients of a nondegenerate Lie algebra with a short ℤ-grading is ℤ-graded with the same support. As a consequence, we introduce a notion of Martindale-like quotients for Kantor pairs and Lie triple systems and construct their maximal systems of quotients.

We present in this paper a family of algebraic quantum hypergroups. Fix a natural number n ∈ ℕ. Let Hn be an infinite-dimensional vector space with a basis {Xp,i, Yq,j | p, q ∈ ℤ, i, j ∈ {0, 1, 2, ⋯ n}}. Then we consider the new multiplication on Hn with structure constants \(\{c^k_{pij}\mid p\in \mathbb{Z}, i, j, k\in \{0, 1, 2, \dots n\}\}\) and present a new method of constructing algebraic quantum hypergroups which is very different from the one in Van Daele and Wang (Math Scand 108(2):198–222, 2011). Finally, we give an explicit example to explain our procedure.

Using the determination of conjugacy classes in an earlier paper, we study the center of the Brauer algebra. In the case of finite groups, conjugacy class sums determine the center of the group algebra. In the case of the Brauer algebra the corresponding class sums only yield a basis of the centralizer of the symmetric group in the Brauer algebra. However, we exhibit an explicit algorithm to determine conditions for a centralizer element to be central and show how to compute a basis for the center using these methods. We will outline how this can be used to compute blocks over fields of arbitrary characteristic. We will also show that similar methods can be applied for computing a basis of the center of the walled Brauer algebra.

Let A be a tame quasi-tilted algebra and d the dimension vector of an indecomposable A-module. In the paper we prove that each irreducible component of the variety of A-modules of dimension vector d is regular in codimension one.

This paper deals with the maximal abelian dimension of a Lie algebra, that is, the maximal value for the dimensions of its abelian Lie subalgebras. Indeed, we compute the maximal abelian dimension for every nilpotent Lie algebra of dimension less than 7 and for the Heisenberg algebra $\mathfrak{H}_k$ , with $k\in\mathbb{N}$ . In this way, an algorithmic procedure is introduced and applied to compute the maximal abelian dimension for any arbitrary nilpotent Lie algebra with an arbitrary dimension. The maximal abelian dimension is also given for some general families of nilpotent Lie algebras.

In this paper we establish a decomposition theorem for an ordinary representation of a finite group G in any category ${\mathcal C}$ which expresses a suitable irreducible representation of G as the tensor product of two projective ones. The celebrated theorem due to Clifford for a linear representation turns out to be a particular case of it. For that purpose, a definition of projective extension of an ordinary representation of a normal subgroup of G is introduced, as well as a tensor product between two of them.

In this paper we consider a partial action α of a polycyclic by finite group G on a ring R. We prove that if R is right noetherian, then the partial skew group ring R ⋆ αG is also right noetherian. Extending the methods of Passman in Passman (Trans Am Math Soc 301:737–759, 1987), we obtain a description of the prime spectrum of R ⋆ αG. The results obtained are applied to get bounds for the Krull dimension and the classical Krull dimension of R ⋆ αG.

Let $k$ be a commutative ring and let $R$ be a commutative $k-$algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) $R-$algebra $A. $ We focus on the scheme $\ran//\GL_n$ of the $n-$dimensional representations of $A, $ on the Hilbert scheme $\Hilb_A^n$ parameterizing the left ideals of codimension $n$ of $A$ and on the affine scheme Spec $\Gamma_R^n(A)^{ab} $ of the abelianization of the divided powers of order $n$ over $A. $ We give a generalization of the Grothendieck-Deligne norm map from $\Hilb_A^n$ to Spec $\Gamma_R^n(A)^{ab} $ which specializes to the Hilbert Chow morphism on the geometric points when $A$ is commutative and $k$ is an algebraically closed field. Describing the Hilbert scheme as the base of a principal bundle we shall factor this map through the moduli space $\ran//\GL_n$ giving a nice description of this Hilbert-Chow morphism, and consequently proving that it is projective., Comment: 18 pages

In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X *, we consider the set $J_{X^*}=\{i \in \mathbb{Z}\, |\, H^i(X^*)\neq 0\}$ and we define the application $l(X^*)=\text{max}J_{X^*}-\text{min}J_{X^*}+1$ . We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras.

The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a.

Let C be a coalgebra over a field k. The aim of this paper is to study the following problem : (P) If C is a k-coalgebra such that C is a generator for the category of left comodules, is C a left quasi-co-Frobenius coalgebra ? The converse always holds. We show that if C has a finite coradical series, the answer is positive.

Let R be a commutative integral domain with field of fractions F and let Q be a finite-dimensional central simple F-algebra. If R is a Prufer domain then it is still unknown whether or not R can be extended to a Prufer order in Q in the sense of Alajbegovic and Dubrovin (J. Algebra, 135: 165–176, 1990). In this paper we investigate a more general class of rings which we call rings of Prufer type and we will prove an extension theorem for these rings. Under special assumptions this result also leads to an extension theorem for certain Prufer domains.

In this paper, we consider a finite dimensional semisimple cosemisimple quasitriangular Hopf algebra \((H,R\,)\) with \(R^{\,21}R\in C(H\otimes H\,)\) (we call this type of Hopf algebras almost-quasitriangular) over an algebraically closed field \(k\). We denote by \(B\) the vector space generated by the left tensorand of \(R^{\,21}R\). Then \(B\) is a sub-Hopf algebra of \(H\). We proved that when \(\dim B\) is odd, \(H\) has a triangular structure and can be obtained from a group algebra by twisting its usual comultiplication [14]; when \(\dim B\) is even, \(H\) is an extension of an abelian group algebra and a triangular Hopf algebra, and may not be triangular. In general, an almost-triangular Hopf algebra can be viewed as a cocycle bicrossproduct.

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois groups associated to the Gorenstein flat cover of a ℤG-module.

The aim of this paper is twofold. On the one hand, we show that the dual Goldie dimension codim(End(MR)) of the endomorphism ring End(MR) of a module MR can be used as a measure of the dimension of the module MR. On the other hand, we prove under suitable hypotheses the validity of the Krull–Schmidt Theorem for infinite direct sums of modules with homogeneous semilocal endomorphism rings.

Noncommutative projective geometry studies noncommutative graded rings by replacing the variety by a suitable Grothendieck category. One way of studying the resulting category is to examine the full subcategories which behave like curves on a commutative variety. Smith and Zhang initiated such a study by considering the subcategory generated by a particular type of module they called a ‘pure curve module in good position’. This paper generalizes their construction by allowing more general modules. The resulting category is shown to be categorically equivalent to a quotient of the category of graded modules over a graded ring. In the course of defining the category equivalence, several dimensions, including projective, injective and Krull dimensions, are calculated. In particular, this extension allows examination of the category created from a line module over more general AS-regular rings than those considered by Smith and Zhang. For instance, suppose that C is a generic line module over R d , Stafford's Sklyanin-like algebra. Let C denote the category C generates. Then C is equivalent to the category of graded k[x, y]/(x2 − y2) modules under the Z × Z/2Z-grading where deg (x) = (−1, 0) and deg (y) = (−1,1).

For a finite group G, the group Outcent(ZpG) of outer central automorphisms of ZpG only depends on the Morita equivalence class of ZpG, which allows reduction to a basic order for its calculation. If the group ring is strongly related to a graduated order, it is often possible to give an explicit description of the basic order. In this paper, we show that Outcent(B)=1 for a block B of ZpG with cyclic defect group. We also prove that Outcent(B0(3)(A6))= 1 for the principal block B0(3)(A6) of Z3A6; this allows us to verify a conjecture of Zassenhaus for the perfect group of order 1080.

In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension 22n+1 for n ≥ 2 over an algebraically closed field of characteristic 0 with the group of group-like elements isomorphic to $\mathbb {Z}_{2^{n}}\times \mathbb {Z}_{2^{n}}$ . Moreover we classify all such nonisomorphic Hopf algebras of dimension 32 and show that they are not twist-equivalent to each other. More generally, given an abelian group of order 2 m−1 we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension 2 m which have this group as their group of group-like elements.

Up to derived equivalence, the representation-finite self-injective algebras of class A n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Mobius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Mobius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Mobius algebras.

The ‘plus part’ U+ of a quantum group Uq(g) has been identified by M. Rosso with a subalgebra Gsym of an algebra G which is a quantized version of R. Ree's shuffle algebra. Rosso has shown that Gsym and G – and hence also Hopf algebras which are analogues of quantum groups – can be defined in a much wider context. In this paper we study one of Rosso's quantizations, which depends on a family of parameters tij. Gsym is determined by a family of matrices Ωα whose coefficients are polynomials in the tij. The determinants of the Ωα factorize into a number of irreducible polynomials, and our main Theorem 5.2a gives strong information on these factors. This can be regarded as a first step towards the (still very distant!) goal, the classification of the symmetric algebras Gsym which can be obtained by giving special values to the parameters tij.

The present paper is devoted to the study of semi-perfect semi-distributive rings (SPSD-rings). In particular, the concept of a prime quiver of a semi-perfect ring and a quiver of an SPSD-ring is widely used. The description of semi-hereditary SPSD-rings is reduced to the case of prime semi-hereditary serial rings and finite posets without rhombuses. For semi-hereditary, semi-perfect, semi-distributive rings we prove the existence of classical quotient rings.