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Let A be a connected cochain DG algebra such that H (A) is a Noetherian graded algebra. We give some criteria for A to be homologically smooth in terms of the singularity category, the cone length of the canonical module k and the global dimension of A . For any cohomologically finite DG A -module M, we show that it is compact when A is homologically smooth. If A is in addition Gorenstein, we get CMreg M = depth A A + Ext. reg M < ∞ , where CMreg M is the Castelnuovo-Mumford regularity of M, depth A A is the depth of A and Ext. reg M is the Ext-regularity of M. [ABSTRACT FROM AUTHOR]

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras. [ABSTRACT FROM AUTHOR]

Classifications of varieties of algebras of almost polynomial growth were considered by several authors in different contexts. An algebra graded by a group G and endowed with a graded involution ∗ is called a (G,∗)-algebra. In this paper, we study (G,∗)-algebras when G is a finite abelian group and we classify all varieties generated by finite dimensional (G,∗)-algebras of almost polynomial growth. Along the way, we characterize the finite dimensional simple (G,∗)-algebras and as a consequence, we classify the finite dimensional simple (Cp,∗)-algebras, for an odd prime p, over any algebraically closed field of characteristic zero. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and let R be an algebra that admits a regular grading by an abelian group H. Moreover, we consider G a group and let A be an algebra with a grading by the group G × H, we define the R-hull of A as the G × H-graded algebra given by R (A) = ⊕ (g , h) ∈ G × H A (g , h) ⊗ R h . In this paper we provide a basis for the graded identities (resp. central polynomials) of the R-hull of A, assuming that a (suitable) basis for the graded identities (resp. central polynomials) of the G × H-graded algebra A is known. In particular, for any a, b ∈ ℕ , we find a basis for the graded identities and the graded central polynomials for the algebra Ma,b(E), graded by the group G × ℤ 2 . Here E is the Grassmann algebra of an infinite dimensional F-vector space, equipped with its natural ℤ 2 -grading and the matrix algebra Ma+b(F) is equipped with an elementary grading by the group G × ℤ 2 , so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on Ma,b(E) that arise in this way and count the isomorphism classes of such gradings. Moreover, we give an alternative proof of the fact that the tensor product Ma,b(E) ⊗ Mr,s(E) is PI-equivalent to Mar+bs,as+br(E). Finally, when the grading group is ℤ 3 × ℤ 2 (resp. ℤ × ℤ 2 ), we present a complete description of a basis for the graded central polynomials for the algebra M2,1(E) (resp. Ma,b(E) in the case b = 1). [ABSTRACT FROM AUTHOR]

We introduce an algebraic version of the Katsura C∗-algebra of a pair A,B of integer matrices and an algebraic version of the Exel–Pardo C∗-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura C∗-algebras are all isomorphic to Steinberg algebras. [ABSTRACT FROM AUTHOR]

We raise the following general question regarding a ring graded by a group: "If P is a ring-theoretic property, how does one define the graded version Pgr of the property P in a meaningful way?". Some properties of rings have straightforward and unambiguous generalizations to their graded versions and these generalizations satisfy all the matching properties of the nongraded case. If P is either being unit-regular, having stable range 1 or being directly finite, that is not the case. The first part of the paper addresses this issue. Searching for appropriate generalizations, we consider graded versions of cancellation, internal cancellation, substitution, and module-theoretic direct finiteness. In the second part of the paper, we consider graded matrix and Leavitt path algebras. If K is a trivially graded field and E is a directed graph, the Leavitt path algebra LK(E) is naturally graded by the ring of integers. If E is a finite graph, we present a property of E which is equivalent with LK(E) being graded unit-regular. This property critically depends on the lengths of paths to cycles and it further illustrates that graded unit-regularity is quite restrictive in comparison to the alternative generalization of unit-regularity from the first part of the paper. [ABSTRACT FROM AUTHOR]

Let K be a field. We characterise the row-finite weighted graphs (E, w) such that the weighted Leavitt path algebra LK(E, w) is isomorphic to an unweighted Leavitt path algebra. Moreover, we prove that if LK(E, w) is locally finite, or Noetherian, or Artinian, or von Neumann regular, or has finite Gelfand-Kirillov dimension, then LK(E, w) is isomorphic to an unweighted Leavitt path algebra. [ABSTRACT FROM AUTHOR]

In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The cases of finite dimensional algebras as well as that of graded algebras arise as subvarieties of the varieties we define. As an application we show that for algebras of global dimension two over the complex numbers, any algebra in the variety continuously deforms to a monomial algebra. [ABSTRACT FROM AUTHOR]

Let G be a group with neutral element e and let S = ⊕ g ∈ G S g be a G-graded ring. A necessary condition for S to be noetherian is that the principal component Se is noetherian. The following partial converse is well-known: If S is strongly-graded and G is a polycyclic-by-finite group, then Se being noetherian implies that S is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain characterizations of noetherian and artinian Leavitt path algebras with coefficients in a general unital ring. This extends a recent characterization by Steinberg for Leavitt path algebras with coefficients in a commutative unital ring and previous characterizations by Abrams, Aranda Pino and Siles Molina for Leavitt path algebras with coefficients in a field. Secondly, we obtain characterizations of noetherian and artinian unital partial crossed products. [ABSTRACT FROM AUTHOR]

Let A be a superalgebra with graded involution or superinvolution ∗ and let c n ∗ (A) , n = 1,2,..., be its sequence of ∗-codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599–611 2016, Linear Multilinear Algebra 64(3), 484–501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of ℤ 2 and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varieties of ∗-superalgebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth, and we give a full classification of their subvarieties which was started in Ioppolo and La Mattina (J. Algebra 472, 519–545 2017). [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field. For a graded algebra, Mori introduced a notion of cogeometric pair (E, σ), where E⊂ℙn−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=A!(E,σ) 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=ℙn−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. [ABSTRACT FROM AUTHOR]

Starting from the invariant theory of binary forms, we extend the classical notion of covariants and introduce the ring of T-covariants. This ring consists of maps defined on a ring of polynomials in one variable which commute with all the translation operators. We study this ring and we show some of its meaningful features. We state an analogue of the classical Hermite reciprocity law, and recover the Hilbert series associated with a suitable double grading via the elementary theory of partitions. Together with classical covariants of binary forms other remarkable mathematical notions, such as orthogonal polynomials and cumulants, turn out to have a natural and simple interpretation in this algebraic framework. As a consequence, a Heine integral representation for the cumulants of a random variable is obtained. [ABSTRACT FROM AUTHOR]

In this article we introduce the notion of a controlled group graded ring. Let G be a group, with identity element e, and let R = ⊕ R be a unital G-graded ring. We say that R is G- controlled if there is a one-to-one correspondence between subsets of the group G and (mutually non-isomorphic) R -sub-bimodules of R, given by G ⊇ H↦ ⊕ R . For strongly G-graded rings, the property of being G-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general G-graded ring to be G-controlled. We also give a characterization of strongly G-graded rings which are G-controlled. As an application of our main results we give a description of all intermediate subrings T with R ⊆ T ⊆ R of a G-controlled strongly G-graded ring R. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras. [ABSTRACT FROM AUTHOR]

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras L ( n, n + k) constructed by Leavitt. Using Bergman's diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular. [ABSTRACT FROM AUTHOR]

We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the latter. In particular, this applies to graded representations of the universal enveloping algebra of the Takiff Lie algebra (or the truncated current algebra) and its (super)analogues, and also to semidirect products of quantum groups with braided symmetric and exterior module algebras in case the latter are flat deformations of classical ones. [ABSTRACT FROM AUTHOR]

Let A be an associative algebra with superinvolution ∗ over a field of characteristic zero and let $c_{n}^{\ast }(A)$ be its sequence of corresponding ∗-codimensions. In case A is finite dimensional, we prove that such sequence is polynomially bounded if and only if the variety generated by A does not contain three explicitly described algebras with superinvolution. As a consequence we find out that no intermediate growth of the ∗-codimensions between polynomial and exponential is allowed. [ABSTRACT FROM AUTHOR]

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted shifted right adjoint to restriction of A-modules to B-modules. A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius graded superalgebra, B is a graded subalgebra that is also a Frobenius graded superalgebra, and A is projective as a left B-module, then A is a twisted Frobenius extension of B. [ABSTRACT FROM AUTHOR]

Decomposition maps control the representation theory of algebras obtained through the process of specialization. In this note, we study a factorization result for graded decomposition maps associated with the specializations of graded algebras. We obtain results previously known only in the ungraded setting. [ABSTRACT FROM AUTHOR]

Given a finitely generated free monoid X and a morphism ϕ : X → X, we show that one can construct an algebra, which we call an iterative algebra, in a natural way. We show that many ring theoretic properties of iterative algebras can be easily characterized in terms of linear algebra and combinatorial data from the morphism and that, moreover, it is decidable whether or not an iterative algebra has these properties. Finally, we use our construction to answer several questions of Greenfeld, Leroy, Smoktunowicz, and Ziembowski by constructing a primitive graded nilpotent algebra with Gelfand-Kirillov dimension two that is finitely generated as a Lie algebra. [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional Hopf algebra acting on an algebra A. Then A is called an H-algebra whenever H acts on products in A via the coproduct. For example, when H= K[ G] is a group Hopf algebra, then A is an H-algebra precisely when the group G acts by automorphisms on A; hence, if G acts by both automorphisms and anti-automorphisms on A, then A is not an H-algebra. In order to study such H-actions, we introduce the notion of an oriented H-algebra. An oriented Hopf algebra is a Hopf algebra with a specified $\mathbb {Z}_{2}$-grading, H = H ⊕ H, while A is an oriented H-algebra whenever H acts on products via the coproduct and H acts on products via the 'anti-coproduct'. This leads to an oriented convolution operation, ⋆. First, we characterize when ⋆ is associative. Then, when H is isomorphic to the dual of a group algebra of a finite abelian group, we provide certain duality theorems between oriented H-algebra actions on A and grading properties of the associated vector space decomposition of A. For example, a Lie algebra A is an oriented H-algebra if and only if A is quasigroup-graded with respect to ⋆. We show that there exists a quasigroup-grading of a Lie algebra that cannot be realized as a semigroup-grading, which was once thought impossible. [ABSTRACT FROM AUTHOR]

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 R finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If R≠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. [ABSTRACT FROM AUTHOR]

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element ∈ T, T/ g T is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T( d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal structure of T( d) is quite rigid. Our results generalise those of the first author in Rogalski (Advances Math. 226, 1433-1473, ). In the companion paper Rogalski et al. (), we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra. [ABSTRACT FROM AUTHOR]

Bezrukavnikov, later together with Arinkin, recovered Deligne's work defining perverse t-structures in the derived category of coherent sheaves on a projective scheme. We prove that these t-structures can be obtained through tilting with respect to torsion theories, as in the work of Happel, Reiten and Smalø. This approach allows us to define, in the quasi-coherent setting, similar perverse t-structures for certain noncommutative projective planes. [ABSTRACT FROM AUTHOR]

Suppose a finite dimensional semisimple Lie algebra $\mathfrak g$ acts by derivations on a finite dimensional associative or Lie algebra A over a field of characteristic 0. We prove the $\mathfrak g$-invariant analogs of Wedderburn-Mal'cev and Levi theorems, and the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of A. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur's conjecture for finite dimensional associative algebras with an action of a reductive affine algebraic group by automorphisms and anti-automorphisms or graded by an arbitrary Abelian group. In addition, we provide criteria for G-, H- and graded simplicity in terms of codimensions. [ABSTRACT FROM AUTHOR]

We study the category of $\mathbb Z^\ell$-graded modules with finite-dimensional graded pieces for certain $\mathbb Z^\ell_+$-graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov-Reshetikhin modules and give a recursive formula for computing their graded characters. [ABSTRACT FROM AUTHOR]

Let Mk( F) be the algebra of k × k matrices over a field F of characteristic 0. If G is any group, we endow Mk( F) with the elementary grading induced by the k-tuple (1,...,1, g) where g ∈  G, g2 ≠ 1. Then the graded identities of Mk( F) depending only on variables of homogeneous degree g and g − 1 are obtained by a natural translation of the identities of bilinear mappings (see Bahturin and Drensky, Linear Algebra Appl 369:95–112, 2003 ). Here we study such identities by means of the representation theory of the symmetric group. We act with two copies of the symmetric group on a space of multilinear graded polynomials of homogeneous degree g and g − 1 and we find an explicit decomposition of the corresponding graded cocharacter into irreducibles. [ABSTRACT FROM AUTHOR]

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group Γ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group G such that the identity component G0 is the affine line and G/ G0 ≃ Γ. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained. [ABSTRACT FROM AUTHOR]