We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group $G$ is strongly-graded-equivalent to the skew group algebra by a product partial action of $G$. As to a more general idempotent graded algebra $B$, we point out that the Cohen-Montgomery duality holds for $B$, and $B$ is graded-equivalent to a global skew group algebra. We show that strongly-graded-equivalence preserves strong gradings and is nicely related to Morita equivalence of product partial actions. Furthermore, we prove that any product partial group action $α$ is globalizable up to Morita equivalence; if such a globalization $β$ is minimal, then the skew group algebras by $α$ and $β$ are graded-equivalent; moreover, $β$ is unique up to Morita equivalence. Finally, we show that strongly-graded-equivalent partially-strongly-graded algebras are stably isomorphic as graded algebras.
Given Z-graded rings A and B, we study when the categories gr-A and gr-B are equivalent. We relate the Morita-type results of Ahn-Marki and del Rio to the twisting systems introduced by Zhang. Using Z-algebras, we obtain a simple proof of Zhang's main result. This makes the definition of a Zhang twist extremely natural and extends Zhang's results.
Let k be a field, Q a finite directed graph, and kQ its path algebra. Make kQ an N-graded algebra by assigning each arrow a positive degree. Let I be an ideal in kQ generated by a finite number of paths and write A = kQ/I. Let QGr A denote the quotient of the category of graded right A-modules modulo the Serre subcategory consisting of those graded modules that are the sum of their finite dimensional submodules. This paper shows there is a finite directed graph Q' with all its arrows placed in degree 1 and an equivalence of categories QGr A = QGr kQ'. A result of Smith now implies that QGr A =…
Let A be a finitely generated connected graded k-algebra defined by a finite number of monomial relations. Then there is a finite directed graph, Q, the Ufnarovskii graph of A, for which the categories QGr(A) and QGr(kQ) are equivalent: QGr(A) denotes the quotient category of graded A-modules modulo the subcategory consisting of those that are the sum of their finite dimensional submodules; QGr(kQ) has a similar definition. The proof makes use of an algebra homomorphism A--->kQ that may be of independent interest.
Let $k$ be a field, $Q$ a finite directed graph, and $kQ$ its path algebra. Make $kQ$ an $\NN$-graded algebra by assigning each arrow a positive degree. Let $I$ be a homogeneous ideal in $kQ$ and write $A=kQ/I$. Let $\QGr A$ denote the quotient of the category of graded right $A$-modules modulo the Serre subcategory consisting of those graded modules that are the sum of their finite dimensional submodules. This paper shows there is a finite directed graph $Q'$ with all its arrows placed in degree 1 and a homogeneous ideal $I'\subset kQ'$ such that $\QGr A \equiv \QGr kQ'/I'$. This is an extens…
The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.
This paper generalises a result for upper triangular matrix rings to the situation of upper triangular matrix DGA's. An upper triangular matrix DGA has the form (R,S,M) where R and S are differential graded algebras and M is a DG-left-R-right-S-bimodule. We show that under certain conditions on the DG-module M and with the existance of a DG-R-module X, from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs (R,S,M) and (S,M',R'), where the DG-bimodule M' is obtained from M and X and R' is the endomorphism DGA of a K-pr…
This note uses a variation of graded Morita theory for finite dimensional superalgebras to determine explicitly the graded basic superalgebras for all real and complex Clifford superalgebras. As an application, the Grothendieck groups of the category of left $\mathbb{Z}_2$-graded modules over all real and complex Clifford superalgebras are described explicitly.
We study the structural and homological properties of graded Artinian modules over generalized Weyl algebras (GWAs), and this leads to a decomposition result for the category of graded Artinian modules. Then we define and examine a category of graded modules analogous to the BGG category O. We discover a condition on the data defining the GWA that ensures O has a system of projective generators. Under this condition, O has nice representation-theoretic properties. There is also a decomposition result for O. Next, we give a necessary condition for there to be a strongly graded Morita equivalenc…
In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. T…
Let kQ be the path algebra of a quiver Q with its standard grading. We show that the category of graded kQ-modules modulo those that are the sum of their finite dimensional submodules, QGr(kQ), is equivalent to several other categories: the graded modules over a suitable Leavitt path algebra, the modules over a certain direct limit of finite dimensional multi-matrix algebras, QGr(kQ') where Q' is the quiver whose incidence matrix is the n^{th} power of that for Q, and others. A relation with a suitable Cuntz-Krieger algebra is established. All short exact sequences in the full subcategory of f…
We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group.
We classify, up to equivalence, all finite-dimensional simple graded division algebras over the field of real numbers. The grading group is any finite abelian group.
Recently it was shown that the notion of flow equivalence of shifts of finite type in symbolic dynamics is related to the Morita theory and the Grothendieck group in the theory of Leavitt path algebras \cite{flowa}. In this paper we show that the notion of conjugacy of shifts of finite type is closely related to the {\it graded} Morita theory and consequently the {\it graded} Grothendieck group. This fits into the general framework we have in these two theories: Conjugacy yields the flow equivalence and the graded Morita equivalence can be lifted to the Morita equivalence. Starting from a fini…
Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning identity elements, inverses, existence of limits and colimits and adjointness of certain functors. In the particular instance of noncommutatively graded Lie algebras, we establish the existence of universal graded enveloping algebras and we show a graded version of the Poincaré-Birkhoff-Witt theorem.
In this work, we extend the definition of the graded prime ideals from those in commutative graded rings to the ideals over graded Lie algebras. We prove some facts about graded prime Lie ideals in arbitrary Lie algebras that are similar to those about graded prime ideals over a commutative or non-commutative ring.In addition, the ideas of graded semiprime Lie ideals and graded total prime Lie ideals will be introduced.
For an ample groupoid $\mathcal{G}$ and a unit $x$ of $\mathcal{G}$, Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra $A_R(\mathcal{G})$ and the category of modules over the isotropy group algebra $R\mathcal{G}_x$. In this paper, we prove a graded version of these functors and related results for the graded Steinberg algebra of a graded ample groupoid. As an application, the spectral simple and graded simple modules over the Leavitt path algebra $L_K(E)$ are classified. In particular, we show that many of previously known s…
Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of F and a finite dimensional Z/2ZxG-graded algebra A over K such that id_{G}(W)=id_{G}(A^{*}) where A^{*} is the Grassmann envelope of A. 2. {[G-graded Specht problem]} The T-ideal id_{G}(W) is finitely generated as a T-ideal. 3. {[G-graded PI-equivalence for affine algebras]} Let W be a G-graded affine algebra over F. Then there exists a field extension K of…
Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed fie…
Let G be a group and A be a G-graded algebra satisfying a polynomial identity. We buid up a model for the relative free G-graded algebra and we obtain, as an application, the "factoring" property for the T_G-ideals of block triangular matrices with entries from the finite dimensional Grassmann algebra E for some particular Z_2-grading.
We extend the loop algebra construction for algebras graded by abelian groups to study graded-simple algebras over the field of real numbers (or any real closed field). As an application, we classify up to isomorphism the graded-simple alternative (nonassociative) algebras and graded-simple finite-dimensional Jordan algebras of degree 2. We also classify the graded-division alternative (nonassociative) algebras up to equivalence.
We compute the graded automorphisms of the upper triangular matrices, viewed as associative, Lie and Jordan algebras. We compute also the so called self-equivalences and Weyl and diagonal groups for every grading.
We prove that if $A$ is a regular graded skew Clifford algebra and is a twist of a regular graded Clifford algebra $B$ by an automorphism, then the subalgebra of $A$ generated by a certain normalizing sequence of homogeneous degree-two elements is a twist of a polynomial ring by an automorphism, and is a skew polynomial ring. We also present an example that demonstrates that this can fail when $A$ is not a twist of $B$.
We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of characteristic zero) graded by a finite abelian group.
Let $\k$ be a field, and let $A$ and $B$ be connected $\N$-graded $\k$-algebras. The algebra $A$ is said to be a graded right-free extension of $B$ provided there is a surjective graded algebra morphism $π: A \to B$ such that $\kerπ$ is free as a right $A$-module. Suppose that $B$ is graded left coherent, and that $A$ is a graded right-free extension of $B$. We characterize when $A$ is also graded left coherent. We apply our criterion to prove graded coherence of certain non-Noetherian graded twisted tensor products.
We show that if $T$ is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative $T$-graded algebra over a field of characteristic $0$ such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic $0$ with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if $T$ is a left or right zero band and the $T$-grad…
In this paper, the so-called differential graded (DG for short) Poisson Hopf algebra is introduced, which can be considered as a natural extension of Poisson Hopf algebras in the differential graded setting. The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras are discussed.
In this paper we explore graded algebras of quotients of Lie algebras with special emphasis on the 3-graded case and answer some natural questions concerning its relation to maximal Jordan systems of quotients.
We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal $I$ of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalized hedgehog graph, which is defined based on certain sets of vertices uniquely determined by $I$. However, this isomorphism may not be graded. We show that replacing the short "spines" of the generalized hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) such that its Leavitt path…
In this paper, we introduce the notion of differential graded Poisson algebra and study its universal enveloping algebra. From any differential graded Poisson algebra $A$, we construct two isomorphic differential graded algebras: $A^e$ and $A^E$. It is proved that the category of differential graded Poisson modules over $A$ is isomorphic to the category of differential graded modules over $A^e$, and $A^e$ is the unique universal enveloping algebra of $A$ up to isomorphisms. As applications of the universal property of $A^e$, we prove that $(A^e)^{op}\cong (A^{op})^e$ and $(A\otimes_{\Bbbk}B)^e…
We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded semisimple algebra is graded symmetric. The center of a symmetric algebra is not necessarily symmetric, but we prove that the center of a finite dimensional graded division algebra is symmetric, provided that the order of the grading group is not divisible by the characteristic of the base field.
We compute point schemes of some regular algebras using (Wolfram) Mathematica. These algebras are Ore extensions of regular graded skew Clifford algebras of global dimension 3.
We determine some of the derived equivalences of a class of gentle algebras called surface algebras. These algebras are constructed from an unpunctured Riemann surface of genus 0 with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. In particular, we fix a triangulation of a surface and determine when different cuts produce derived equivalent algebras.
The motivation for this paper has been to study the relation between the zero component of the maximal graded algebra of quotients and the maximal graded algebra of quotients of the zero component, both in the Lie case and when considering Martindale algebras of quotients in the associative setting. We apply our results to prove that the finitary complex Lie algebras are (graded) strongly nondegenerate and compute their maximal algebras of quotients.
We classify a class of infinite-dimensional simple graded pre-Lie algebras on the graded vector space underlying the algebra of Laurent polynomials, with a specific form for the product.
In this paper we associate an abelian category to a finite directed graph and prove the categories arising from two graphs are equivalent if the incidence matrices of the graphs are shift equivalent. The abelian category is the quotient of the category of graded vector space representations of the quiver obtained by making the graded representations that are the sum of their finite dimensional submodules isomorphic to zero. Actually, the main result in this paper is that the abelian categories are equivalent if the incidence matrices are strong shift equivalent. That result is combined with an…
In this paper, we introduce the class of finitely semi-graded algebras which extends the connected graded algebras finitely generated in degree one. The Koszul behavior of finitely semi-graded algebras is investigated by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series and the Yoneda algebra are defined for this class of algebras. Finitely semi-graded algebras include many important examples of non $\mathbb{N}$-graded algebras finitely generated in degree one coming from mathematical physics, and for these concrete examples the Koszulity will be …
We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p>2, and we compute its Z_2-graded Gelfand-Kirillov dimension as a Z_2-graded PI-algebra.
In this paper, we study the Ext-algebras of graded skew extensions. For a connected graded algebra $A$ and a graded automorphism $σ$, we analyze the Yoneda product of the Ext-algebra of graded skew extension $A[z;σ]$, and prove this Ext-algebra is an $R$-smash product of the Ext-algebra of $A$ and the one of polynomial algebra $k[z]$ as an associative algebra.
It is shown that the methods and algorithms, developed in (A. Capani et al., Computing minimal finite free resolutions, {\it Journal of Pure and Applied Algebra}, (117& 118)(1997), 105 -- 117; M. Kreuzer and L. Robbiano, {\it Computational Commutative Algebra 2}, Springer, 2005.) for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a commutative polynomial algebra, can be adapted for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a weighted $\mathbb{N}$-graded…
Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. It is shown that if $A$ is an $\mathbb{N}$-graded algebra of $({\cal B},d(~))$-type, then $A$ has a graded monomial ordering $\prec_{gr}$. It is also shown that $A$ is an $\mathbb{N}$-filtered algebra of $({\cal B},d(~))$-type if and only if $A$ has a graded momomial ordering, where ${\cal B}$ is the PBW basis of $A$.
We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid $Γ$. First we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra $A$ we prove the existence of the graded PI-exponent, provided that $Γ$ is a commutative semigroup. If $A$ is simple in a non-graded sense the existence of the graded PI-exponent is proved without any restrictions on $Γ$.
Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of $B$. Then $A$ is isomorphic to $B$ as an $S$-graded algebra.
Let $R$ be a finite-dimensional algebra over an algebraically closed field $F$ graded by an arbitrary group $G$. We prove that $R$ is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite subgroup of $G$. If the characteristic of $F$ is zero or ${\rm char} F$ does not divide the order of any finite subgroup of $G$ then we prove that $R$ is graded simple if and only if it is a matrix algebra over a finite-dimensional graded division algebra.
We observe that a finitely generated algebraic algebra R (over a field) is finite dimensional if and only if the associated graded ring grR is right noetherian, if and only if grR has right Krull dimension, if and only if grR satisfies a polynomial identity.
