For a twisted partial action Θof a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_ΘG is proved to be associative. Given a G-graded k-algebra B = \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_ΘG for some twisted partial action of G on B_1. The equality B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.
We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizi…
We show that if a non-associative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a non-associative setting. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.
We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nastasescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appear in the class of epsilon-strongly graded…
The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let $G$ be a group, $A=\oplus_{g\in G}\,A_g$ an epsilon-strongly graded ring and ${\bf pic}{R}$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of ${\bf pic}{R}$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on ${\bf pic}{R}$ which induces a partial action $γ$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^γ$-algebra are presented in the case tha…
Let A#_{α, ω}H be a partial crossed product. In this paper, we first generalize the theorem about the existence of an enveloping action to twisted partial actions. Second, we construct a Morita context between the partial crossed product and the crossed product related to the enveloping action. Furthermore, we discuss equivalences of partial crossed products Finally, we investigate when A\subset A#_{α, ω}H becomes a separable extension.
If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb Z.$ We formulate properties of the graph $E$ which are equivalent with $L_K(E)$ being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of $L_K(E)$ are also characterized in terms of the pre-ordered group properties of the Grothendieck $\mathbb Z$-group of $L_K(E)$. If $E$ has finitely many vertices, we characterize when $L_K(E)$ is strongly graded…
We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group graded situation to the groupoid graded case, of the bijection from a certain second cohomology group, defined by the grading and the fun…
Twisted étale groupoid algebras have been studied recently in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper, we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups and we show that they are classified by Lausch's second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists…
We prove that any twisted generalized Weyl algebra satisfying certain consistency conditions can be embedded into a crossed product. We also introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's $q$-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus,…
We generalize graded Hecke algebras to include a twisting two-cocycle for the associated finite group. We give examples where the parameter spaces of the resulting twisted graded Hecke algebras are larger than that of the graded Hecke algebras. We prove that twisted graded Hecke algebras are particular types of deformations of a crossed product.
In this paper we study twisted algebras of multiplier Hopf ($^*$-)algebras which generalize all kinds of smash products such as generalized smash products, twisted smash products, diagonal crossed products, L-R-smash products, two-sided crossed products and two-sided smash products for the ordinary Hopf algebras appeared in [P-O].
We determine explicitly the center of the twisted graded Hecke algebras associated to homocyclic groups. Our results are a generalization of formulas by M. Douglas and B. Fiol in [J. High Energy Phys. 2005 (2005), no. 9, 053, 22 pages, hep-th/9903031].
We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sylvester matrix rank function rk of R preserved by S has a canonical extension to a Sylvester matrix rank function rk_F for S. In the case of twisted crossed product by an amenable group, and the tensor product with a field extension, it is also shown that rk_F depends on rk continuously. When an ame…
Let $G$ be a group of order $k$. We consider the algebra $M_k(\mathbb{C})$ of $k$ by $k$ matrices over the complex numbers and view it as a crossed product with respect to $G$ by embedding $G$ in the symmetric group $S_k$ via the regular representation and embedding $S_k$ in $M_k(\mathbb{C})$ in the usual way. This induces a natural $G$-grading on $M_k(\mathbb{C})$ which we call a crossed-product grading. We study the graded $*$-identities for $M_k(\mathbb{C})$ equipped with such a crossed-product grading and the transpose involution. To each multilinear monomial in the free graded algebra wit…
In this paper, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by crossed homomorphisms on Leibniz algebras. This allows us to define cohomology for a crossed homomorphism. Finally, we study linear deformations, formal deformations and extendibility of finite order deformations of a crossed homomorphism in terms of the cohomology theory.
We consider the algebra M_k(C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by embedding G in the symmetric group S_k via the regular representation and embedding S_k in M_k(C) in the usual way. This induces a natural G-grading on M_k(C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g_1,g_2,..., g_k) of distinct elements g_i in G. We study the graded polynomial identities for M_k(C) equipped with a crossed product grading. To each multilinear monomial in the free graded…
This is a general study of twisted Calabi-Yau algebras that are $\mathbb{N}$-graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi-Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin-Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi-Yau algebras of dimension 0 as separable $k$-algebras, and we similarly characterize graded twisted Calabi-Yau algebras of dime…
Let $\mathbb{k}$ be an algebraically closed field. Building upon previous work, we classify, up to isomorphism of graded algebras, quadratic graded twisted tensor products of $\mathbb{k}[x,y]$ and $\mathbb{k}[z]$. When such an algebra is Artin-Schelter regular, we identify its point scheme and type. We also describe which three-dimensional Sklyanin algebras contain a subalgebra isomorphic to a quantum $\mathbb{P}^1$, and we show that every algebra in this family is a graded twisted tensor product of $\mathbb{k}_{-1}[x,y]$ and $\mathbb{k}[z]$.
Let G be a group and let W be an algebra over a field K. We will say that W is a G-graded twisted algebra if W can be written as a direct sum over the elements of G of one dimensional K-vector spaces. It is also assumed that W has no monomial which is a zero divisor. We also demand that W has a multiplicative identity element. We focus in the case where G is a finite abelian group and the field K is either the real numbers or the complex numbers. In this article, using methods of group cohomology, we classify all associative G-graded twisted algebras in the case G is a finite abelian group. On…
In order to simultaneously generalize matrix rings and group graded crossed products, we introduce category crossed products. For such algebras we describe the center and the commutant of the coefficient ring. We also investigate the connection between on the one hand maximal commutativity of the coefficient ring and on the other hand nonemptyness of intersections of the coefficient ring by nonzero twosided ideals.
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 t…
A twisting system is one of the major tools to study graded algebras, however, it is often difficult to construct a (non-algebraic) twisting system if a graded algebra is given by generators and relations. In this paper, we show that a twisted algebra of a geometric algebra is determined by a certain automorphism of its point variety. As an application, we classify twisted algebras of $3$-dimensional geometric Artin-Schelter regular algebras up to graded algebra isomorphism.
We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilcoxeter graded superalgebras…
The loop algebra construction by Allison, Berman, Faulkner, and Pianzola, describes graded-central-simple algebras with split centroid in terms of central simple algebras graded by a quotient of the original grading group. Here the restriction on the centroid is removed, at the expense of allowing some deformations (cocycle twists) of the loop algebras.
Crystalline graded rings are generalizations of certain classes of rings like generalized twisted group rings, generalized Weyl algebras, and generalized skew crossed products. When the base ring is a commutative Dedekind domain, two constructions are given for producing maximal graded orders. On the way, a new concept appears, so-called, spectrally twisted group. Some general properties of it are studied. At the end of the paper several examples are considered.
In this article, we consider a twisted partial action $α$ of a group $G$ on a ring $R$ and it is associated partial crossed product $R*_α^wG$. We study necessary and sufficient conditions for the commutativity and simplicity of $R*_α^wG$. Let $R=C(X)$ the algebra of continuous functions of a topological space $X$ on the complex numbers and $C(X)*_αG$ the partial skew group ring, where $α$ is a partial action of a topological group $G$ on $C(X)$. We study some topological properties to obtain results on the algebra $C(X)$. Also, we study the simplicity of $C(X)*_αG$ using topological properties…
In this paper we work with unital twisted partial actions. We investigate ring theoretic properties of partial crossed products as artinianity, noetherianity, perfect property, semilocalproperty, semiprimary property and we also study the Krull dimension. Moreover, we consider triangular matrix representation of partial skew group rings, weak and global dimensions of partial crossed products Also we study when the partial crossed products are Frobenius and symmetric algebras.
We initiate a study of the growth and matrix-valued Hilbert series of non-negatively graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d at most 3, giving precise descriptions of their matrix-valued Hilbert series and partial results desc…
Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.
We use a representation of a graded twisted tensor product of $K[x]$ with $K[y]$ in $L(K^{\Bbb{N}_0})$ in order to obtain a nearly complete classification of these graded twisted tensor products via infinite matrices. There is one particular example and three main cases: quadratic algebras classified by Conner and Goetz, a family called $A(n,d,a)$ with the $n+1$-extension property for $n\ge 2$, and a third case, not fully classified, which contains a family $B(a,L)$ parameterized by quasi-balanced sequences.
Let $k$ be a field. Let $A$ and $B$ be connected $N$-graded $k$-algebras. Let $C$ denote a twisted tensor product of $A$ and $B$ in the category of connected $N$-graded $k$-algebras. The purpose of this paper is to understand when $C$ possesses the Koszul property, and related questions. We prove that if $A$ and $B$ are quadratic, then $C$ is quadratic if and only if the associated graded twisting map has a property we call the unique extension property. We show that $A$ and $B$ being Koszul does not imply $C$ is Koszul (or even quadratic), and we establish sufficient conditions under which $C…
We introduce a new class of graded rings extending the class of generalized Weyl algebras. These rings are orders in crossed products of the most general type, and we introduce their basic structure theory. We provide an extensive list of examples, some completely new but also some considered earlier, and highlight some specific structure results. Many new and interesting problems about the crystalline graded rings may be identified.
We study prime ideals, prime modules, and associated primes of graded modules over rings $S$ graded by a unique product monoid. We consider two situations in detail: (a) the case where $S$ is strongly group-graded and (b) the case where $S$ is a crossed product and the ideal or module is induced from the identity component $R$ of $S$. We give explicit conditions for ideals and modules of $R$ to induce prime ideals of or prime modules over $S$ in these two cases. We then describe the set of associated prime ideals of an arbitrary induced module. One of our main interests is to give necessary an…
We construct the $H$-von Neumann regular radical for $H$-module algebras and show that it is an $H$-radical property. We obtain that the Jacobson radical of twisted graded algebra is a graded ideal. For twisted $H$-module algebra $R$, we also show that $r_{j}(R#_σH)= r_{Hj}(R)#_σH$ and the Jacobson radical of $R$ is stable, when $k$ is an algebraically closed field or there exists an algebraic closure $F$ of $k$ such that $r_j(R \otimes F) = r_j(R) \otimes F$, where $H$ is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf algebra over $k$. In particular, we answ…
Let $H$ be a cocommutative weak Hopf algebra and let $(B, φ_{B})$ a weak left $H$-module algebra. In this paper, for a twisted convolution invertible morphism $σ:H\otimes H\rightarrow B$ we define its obstruction $θ_σ$ as a degree three Sweedler 3-cocycle with values in the center of $B$. We obtain that the class of this obstruction vanish in third Sweedler cohomology group $\mathcal{H}^3_{φ_{Z(B)}}(H, Z(B))$ if, and only if, there exists a twisted convolution invertible 2-cocycle $α:H\otimes H\rightarrow B$ such that $H\otimes B$ can be endowed with a weak crossed product structure with $α$ k…
We study some generalizations of the notion of regular crossed products K*G. For the case when K is an algebraically closed field, we give necessary and sufficient conditions for the twisted group ring K*G to be an n-weakly regular ring, a $ξ^* N$-ring or a ring without nilpotent elements.
If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring B(X, L, sigma), which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent interest.
We introduce the notion of the twisted Segre product $A\circ_ψB$ of $\mathbb Z$-graded algebras $A$ and $B$ with respect to a twisting map $ψ$. It is proved that if $A$ and $B$ are noetherian Koszul Artin-Schelter regular algebras and $ψ$ is a twisting map such that the twisted Segre product $A\circ_ψB$ is noetherian, then $A\circ_ψB$ is a noncommutative graded isolated singularity. To prove this result, the notion of densely (bi-)graded algebras is introduced. Moreover, we show that the twisted Segre product $A\circ_ψB$ of $A=k[u,v]$ and $B=k[x,y]$ with respect to a diagonal twisting map $ψ$ …
This paper is the result of a research project completed in the context of the first author's Undergraduate Student Research Award from the Natural Sciences and Engineering Research Council of Canada (NSERC). We prove a nesting phenomenon for twisted Frobenius extensions. Namely, suppose $R \subseteq B \subseteq A$ are graded superrings such that $A$ and $B$ are both twisted Frobenius extensions of $R$, $R$ is contained in the center of $A$, and $A$ is projective over $B$. Our main result is that, under these assumptions, $A$ is a twisted Frobenius extension of $B$. This generalizes a result o…
If $A\otimes _{R, σ}V$ and $A\otimes _{P, ν}W$ are two Brzeziński crossed products and $Q:W\otimes V\rightarrow V\otimes W$ is a linear map satisfying certain properties, we construct a Brzeziński crossed product $A\otimes _{S, θ}(V\otimes W)$. This construction contains as a particular case the iterated twisted tensor product of algebras.
We study finite-dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure module for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on it…
Let G denote a group and let W be an algebra over a commutative ring R. We will say that W is a G-graded twisted algebra (not necessarily commutative, neither associative) if there exists a G-grading W=\bigoplus_{g \in G}W_{g} where each summand W_{g} is a free rank one R -module, and W has no monomial zero divisors (for each pair of nonzero elements w_{a},w_{b} en W_{a} and W_{b} their product is not zero, w_{a}w_{b}\neq 0). It is also assumed that W has an identity element. In this article, methods of group cohomology are used to study the general problem of classification under graded isomo…
In this paper, we introduce the notions of crossed module of associative conformal algebras, 2-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the 3-th Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer-Cartan elements of a suitable differential graded Lie algebra, and prove that the Deligne groupoid of this differential graded Lie algebra corresponds one to on…
In this article, we introduce the concepts of graded $s$-prime submodules which is a generalization of graded prime submodules. We study the behavior of this notion with respect to graded homomorphisms, localization of graded modules, direct product, and idealization. We succeeded to prove the existence of graded $s$-prime submodules in the case of graded-Noetherian modules. Also, we provide some sufficient conditions for the existence of such objects in the general case, as well as, in the particular case of grading by $\mathbb{Z}$, a finite group, or a polycyclic-by-finite group, in addition…
In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a Lie algebra to the semidirect product Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we con…
We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.
In this work, the notion of a twisted partial Hopf action is introduced as a unified approach for twisted partial group actions, partial Hopf actions and twisted actions of Hopf algebras. The conditions on partial cocycles are established in order to construct partial crossed products, which are also related to partially cleft extensions of algebras. Examples are elaborated using algebraic groups.
The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the establ…
Families of codes such as group codes, constacyclic and skew cyclic codes, some of which independently suggested in the literature, turn out to be special instances of the general family of crossed product codes. Hamming-metric is a main feature of ambient code spaces which is used to evaluate the efficiency of their various codes. This note aims at classifying the ambient spaces of crossed products up to Hamming-isometry. We establish a criterion for two crossed products of a group G over a base ring R to be isometric in terms of a certain G-automorphism action on the second cohomology of G w…
