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 established, as well as, the explicit computation of its Hilbert and Poincaré series.
Pre-Koszul and Koszul algebras were defined by Priddy. There exist some relations between these algebras and the skew PBW extensions defined. We have established conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul K-algebra then every graded skew PBW extension of R is Koszul.
We discuss certain homological properties of graded algebras whose trivial modules admit non-pure resolutions. Such algebras include both of Artin-Schelter regular algebras of types (12221) and (13431). Under certain conditions, a module with non-pure resolution is decomposed to form an extension by two modules with pure resolutions.
In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero.
In this paper we compute the set of point modules of finitely semi-graded rings. In particular, from the parametrization of the point modules for the quantum affine n-space, the set of point modules for some important examples of non $\mathbb{N}$-graded quantum algebras is computed.
We study a finite dimensional quadratic graded algebra R defined from a finite ranked poset. This algebra has been central to the study of the splitting algebra of the poset, A, as introduced by Gelfand, Retakh, Serconek and Wilson . The algebra A is known to be quadratic when the poset satisfies a combinatorial condition known as uniform, and R is the quadratic dual of an associated graded algebra of A. We prove that R is Koszul and the poset is uniform if and only if the poset is Cohen-Macaulay. Koszulity of R implies Koszulity of A. We also show that when R is Koszul, the cohomology of the …
This paper is devoted to investigating the structure theory of a class of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we completely determine the derivation algebras, the automorphism groups and the second cohomology groups of these Lie superalgebras.
To a finite ranked poset $Γ$ we associate a finite-dimensional graded quadratic algebra $R_Γ$. Assuming $Γ$ satisfies a combinatorial condition known as uniform, $R_Γ$ is related to a well-known algebra, the splitting algebra $A_Γ$. First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset $Γ$, we ask: Is $R_Γ$ Koszul? The Koszulity of $R_Γ$ is related to a combinatorial topology property of $Γ$ called Cohen-Macaulay. Kloefkorn and Shelton proved that if $Γ$ is a finite ranked cy…
We consider associative algebras with involution graded by a finite abelian group G over a field of characteristic zero. Suppose that the involution is compatible with the grading. We represent conditions permitting PI-representability of such algebras. Particularly, it is proved that a finitely generated (Z/qZ)-graded associative PI-algebra with involution satisfies exactly the same graded identities with involution as some finite dimensional (Z/qZ)-graded algebra with involution for any prime q or q = 4. This is an analogue of the theorem of A.Kemer for ordinary identities, and an extension …
$N$-derivation is the natural generalization of derivation and triple derivation. Let ${\cal L}$ be a finitely generated Lie algebra graded by a finite dimensional Cartan subalgebra. In this paper, a sufficient condition for Lie $N$-derivation algebra of ${\cal L}$ coinciding with Lie derivation algebra of ${\cal L}$ is given. As applications, any $N$-derivation of Schrödinger-Virasoro algebra, generalized Witt algebras, Kac-Moody algebras and their Borel subalgebras, is a derivation.
In this note, we prove the Koszulity of the tensor product algebra defined in the author's previous work for sl(n) and a list of fundamental weights. This is achieved by constructing a graded Morita equivalence between the modules over this algebra and a sum of blocks of category O in type A.
Let $A$ and $B$ be two connected graded algebras finitely generated in degree one. If $A$ is isomorphic to $B$ as ungraded algebras, then they are also isomorphic to each other as graded algebras.
In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.
We extend the the definition of Kumjian-Pask algebras to include algebras associated to finitely aligned higher-rank graphs. We show that these Kumjian-Pask algebras are universally defined and have a graded uniqueness theorem. We also prove the Cuntz-Kreiger uniqueness theorem; to do this, we use a groupoid approach. As a consequence of the graded uniqueness theorem, we show that every Kumjian-Pask algebra is isomorphic to the Steinberg algebra associated to its boundary path groupoid. We then use Steinberg algebra results to prove the Cuntz-Kreiger uniqueness theorem and also to characterize…
Let $UT_n(F)$ be the algebra of the $n\times n$ upper triangular matrices and denote $UT_n(F)^{(-)}$ the Lie algebra on the vector space of $UT_n(F)$ with respect to the usual bracket (commutator), over an infinite field $F$. In this paper, we give a positive answer to the Specht property for the ideal of the $\mathbb{Z}_n$-graded identities of $UT_n(F)^{(-)}$ with the canonical grading when the characteristic $p$ of $F$ is 0 or is larger than $n-1$. Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of $UT_n(F)^{(-)}$, is f…
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.
Let $F$ be a field and let $F \langle X \rangle$ be the free unital associative $F$-algebra on the free generating set $X = \{ x_1, x_2, \dots \}$. A subalgebra (a vector subspace) $V$ in $F \langle X \rangle$ is called a $T$-subalgebra (a $T$-subspace) if $ϕ(V) \subseteq V$ for all endomorphisms $ϕ$ of $F \langle X \rangle$. For an algebra $G$, its central polynomials form a $T$-subalgebra $C(G)$ in $F \langle X \rangle$. Over a field of characteristic $p > 2$ there are algebras $G$ whose algebras of all central polynomials $C (G)$ are not finitely generated as $T$-subspaces in $F \langle X \…
We show that graded Hecke algebras are PI algebras if and only if they are finitely generated over their centres if and only if the deformation parameters $t_{i}$ are zero for all $i=1,\ldots,N$. This generalises a result for symplectic reflection algebras by Etingof - Ginzburg and Brown - Gordon.
The mod-p cohomology ring of a non-trivial finite p-group is an infinite dimensional, finitely presented graded unital algebra over the field with p elements, with generators in positive degrees. We describe an effective algorithm to test if two such algebras are graded isomorphic. As application, we determine all graded isomorphisms between the mod-p cohomology rings of all p-groups of order at most 100.
In sharp contrast to the Abrams-Rangaswamy Theorem that the only von Neumann regular Leavitt path algebras are exactly those associated to acyclic graphs, here we prove that the Leavitt path algebra of any arbitrary graph is a graded von Neumann regular ring. Several properties of Leavitt path algebras, such as triviality of the Jacobson radical, flatness of graded modules and finitely generated graded right (left) ideals being generated by an idempotent element, follow as a consequence of general theory of grade von Neumann regular rings.
We consider associative algebras over a field of characteristic zero. We give a version of the proof of the Kemer's theorems concerning the Specht problem solution. It is proved that the ideal of graded identities of a finitely generated PI-superalgebra coincides with the ideal of graded identities of some finite dimensional superalgebra. This implies that the ideal of polynomial identities of any (not necessary finitely generated) PI-algebra coincides with the ideal of identities of the Grassmann envelope of a finite dimensional superalgebra, and is finitely generated as a T-ideal.
In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series.
We discuss finitely graded Iwanaga-Gorenstein (IG) algebras $A$ and representation theory of their (graded) Cohen-Macaulay (CM) modules. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where $A$ is a trivial extension algebra $A = Λ\oplus C$ with the grading $ \text{deg} Λ= 0, \ \text{deg} C = 1$. In the previous study, we gave a necessary and sufficient condition that $A$ is IG in terms of $Λ$ and $C$ by using derived tensor products and derived Homs. For simplicity, we assume that $Λ$ is of finite global dimension in the sequel. In this paper, we sho…
Let $k$ be a field and let $A=\bigoplus_{n\ge 1}A_n$ be a positively graded $k$-algebra. We recall that $A$ is graded nilpotent if for every $d\ge 1$, the subalgebra of $A$ generated by elements of degree $d$ is nilpotent. We give a method of producing grading nilpotent algebras and use this to prove that over any base field $k$ there exists a finitely generated graded nilpotent algebra that contains a free $k$-subalgebra on two generators.
Let W be an associative PI-algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. The following inequality had been conjectured by Bahturin and Zaicev: exp(W)\leq |G|^2 exp(W_e). The inequality is known in case the algebra W is affine (i.e. finitely generated). Here we prove the conjecture in general.
For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$ is rational/algebraic/transcendental, then the same is $H(R,t)$. If the growth of the coefficients of $a(t)$ is polynomial, in the same way we construct a graded algebra $R$ preserving the polynomial growth of the coefficients of its Hilbert series $H(R,t)$. Applying a classical result of Fatou from 1906 we obtain that if a finitely generated graded algebra $…
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.
We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancellative. We introduce the non-unital generalizations of graded analogues of noetherian and artinian rings, graded locally noetherian and graded l…
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 $Γ$ be a cancelation monoid with the neutral element $e$. Consider a $Γ$-graded ring $A=\oplus_{γ\inΓ}A_γ$, which is not necessarily commutative. It is proved that $A_e$, the degree-$e$ part of $A$, is a local ring in the classical sense if and only if the graded two-sided ideal $\mathfrak{M}$ of $A$ generated by all non-invertible homogeneous elements is a proper ideal. Defining a $Γ$-graded local ring $A$ in terms of this equivalence, it is proved that any two minimal homogeneous generating sets of a finitely generated $Γ$-graded $A$-module have the same number of generators, and further…
The paper is dedicated to the study of certain non commutative graded AS Gorenstein algebras $Λ$. The main result of the paper is that for Koszul algebras $Λ$ with Yoneda algebra $Γ$, such that both $Λ$ and $Γ$ are graded AS Gorenstein noetherian of finite local cohomology dimension on both sides, there are dualities of triangulated categories: \underline{$gr$}$_Λ[Ω^{-1}]$ $\cong D^{b}(Qgr_Γ)$ and \underline{$gr$}$_Γ[Ω^{-1}]$ $\cong D^{b}(Qgr_Λ)$ where, and $Qgr_Γ$ is the category of tails, this is: the category of finitely generated graded modules $gr_Γ$ divided by the modules of finite lengt…
Let $k$ be an algebraically closed field and $A$ a $\mathbb{Z}$-graded finitely generated simple $k$-algebra which is a domain of Gelfand-Kirillov dimension 2. We show that the category of $\mathbb{Z}$-graded right $A$-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive qu…
As an instance of a linear action of a Hopf algebra on a free associative algebra, we consider finite group gradings of a free algebra induced by gradings on the space spanned by the free generators. The homogeneous component corresponding to the identity of the group is a free subalgebra which is graded by the usual degree. We look into its Hilbert series and prove that it is a rational function by giving an explicit formula. As an application, we show that, under suitable conditions, this subalgebra is finitely generated if and only if the grading on the base vector space is trivial.
Differential Graded Algebras can be studied through their Differential Graded modules. Among these, the compact ones attract particular attention. This paper proves that over a suitable chain Differential Graded Algebra R, each compact Differential Graded module M satisfies amp M >= amp R, where amp denotes amplitude which is defined in a straightforward way in terms of the homology of a DG module. In other words, the homology of each compact DG module M is at least as long as the homology of R itself. Conversely, DG modules with shorter homology than R are not compact, and so in general, ther…
We define Leavitt path algebras of hypergraphs generalizing simultaneously Leavitt path algebras of finitely separated graphs and Leavitt path algebras of row-finite vertex-weighted graphs. We find linear bases for those algebras, compute their Gelfand-Kirillov dimension, obtain some results on ring-theoretic properties like simplicity, von Neumann regularity and Noetherianess and investigate their K-theory and graded K-theory. By doing so we obtain new results on the Gelfand-Kirillov dimension and graded K-theory of Leavitt path algebras of separated graphs and on the graded K-theory of weigh…
Using the E-algebraic systems, various graded irreducible representations of a Leavitt path algebra L of a graph E over a field K are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left ideals of L are generated by Laurent vertices and/or line points leading to a detailed description of the graded socle of L. Following this, a complete characterization is obtained of the Leavitt path algebras over which every graded irreducible representation is finitely presented. A useful result is that the irreducible representation V_[p] induced by infini…
Suppose $A$ is a graded associative algebra over a field, $I$ is its ideal generated by a set $α$ of homogeneous elements, and B = A/I. In this note, some inequalities between Hilbert series of algebras $A,B$ and the number of elements of the set $α$ are announced. As in the Golod--Shafarevich inequality as in our case the equality in every estimate is exact iff the set $α$ is strongly free: so we obtain some new characterizations of such sets. As a consequence it is proved that over a field of zero characteristic for the class of finitely defined graded algebras there is no algorithm to answe…
We prove that if the neutral component in a finitely-generated associative algebra graded by a finite group has a Shirshov base, then so does the whole algebra.
Let R be a strongly Z-graded ring with degree-0 subring S, and let C be a chain complex of modules over the subring P of elements of non-negative degree. We show that there are non-commutative localisations of P which detect whether the complex C is S-finitely dominated or S-contractible, respectively, and that these localisations are universal among P-rings making S-finitely dominated and S-contractible complexes contractible. This generalises known results for polynomial rings to a much wider class of rings. We show by example that in general C need not be P-homotopy finite even if C is S-fi…
In this paper, we give a complete characterization of Leavitt path algebras which are graded $Σ$-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded $Σ$-$V$ ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over $K$ or $K[x,x^{-1}]$ with appropriate matrix gradings. We also obtain a graphical characterization of such a graded $Σ$-$V$ ring $L$% . When the graph $E$ is…
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…
Kavli Institute for the Physics and Mathematics of the Universe (WPI)（宇宙物理与数学研究所）；The University of Tokyo Institutes for Advanced Study（东京大学高等研究院）；The University of Tokyo（东京大学）
This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with exponentially bounded nonnegative integer coefficients. Then we summarize some related facts on algebras with polynomial identity. Further we discuss the problem how to find series $a(t)$ which are rational/algebraic/transcendental over ${\mathbb Q}(t)$. Applying a classical result of Fatou we conclude that if a finitely generated graded algebra has a finite G…
This note begins by observing that a graded central simple algebra, graded by an abelian group, is a graded Azumaya algebra and it is free over its centre. For a graded Azumaya algebra A free over its centre $R$, we show that K_i^{\gr} (A) is "very close" to K_i^{\gr}(R), where K_i^{\gr} (R) is defined to be K_i(\Pgr (R)). Here \Pgr (R) is the category of graded finitely generated projective R-modules and K_i, \,i\geq 0, are the Quillen K-groups.
We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the afor…
This paper continues math.GR/0608302's study of amenability of affine algebras (based on the notion of almost-invariant finite-dimensional subspace), and applies it to graded algebras associated with finitely generated groups. Due to a mistake in Lemma 4.3, the main statements of the present paper should not be considered as proven. The results in Sections 2 and 3 are nevertheless valid.
For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional $Q$-graded simple Lie algebras over any algebraically closed field of characteristic $0$ based on the recent classification of gradings on finite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any $Q$-graded Lie algebra (not necessari…
Happel constructed a fully faithful functor $\mathcal{H} :\mathsf{D}^{\mathrm{b}}(\text{mod} \ Λ) \to \underline{\text{mod}}^{\Bbb{Z}} \ \text{T}(Λ)$ for a finite dimensional algebra $Λ$. He also showed that this functor $\mathcal{H}$ gives an equivalence precisely when $\text{gldim } Λ< \infty$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H} (Λ)$ of $\underline{\text{mod}}^{\mathbb{Z}} \ \text{T}(Λ)$. In this paper we generalize Happel's functor $\mathcal{H}$ in the case where $\text{T}(Λ)$ is replaced with a finitely graded IG algebra $A$…
Let E be an arbitrary graph and K be any field. For every non-graded ideal I of the Leavitt path algebra L_{K}(E), we give an explicit description of the generators of I. Using this, we show that every finitely generated ideal of L_{K}(E) must be principal. In particular, if E is a finite graph, then every ideal of L_{K}(E) must be principal ideal.
It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tenso…
Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $ϕ:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^ϕ\!M$ be the abelian group $M$ with the $R$-module structure induced by the Frobenius endomorphism. The $R$-module ${}^ϕ\!M$ has a natural grading given by $\text{deg} x=j$ if $x\in M_{jp+i}$ for some $0\le i \le p-1$. In this paper, we prove that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that $\text{reg}_R\,{}^ϕ\!M <\infty$. This result supplies …
In this paper we propose a general method for computing a minimal free right resolution of a finitely presented graded right module over a finitely presented graded noncommutative algebra. In particular, if such module is the base field of the algebra then one obtains its graded homology. The approach is based on the possibility to obtain the resolution via the computation of syzygies for modules over commutative algebras. The method behaves algorithmically if one bounds the degree of the required elements in the resolution. Of course, this implies a complete computation when the resolution is…
In this paper we investigate the class of the connected graded algebras which are finitely generated in degree 1, which are finitely presented with relations of degrees greater or equal to 2 and which are of finite global dimension D and Gorenstein. For D greater or equal to 4 we add the condition that these algebras are homogeneous and Koszul. It is shown that each such algebra is completely characterized by a multilinear form satisfying a twisted cyclicity condition and some other nondegeneracy conditions depending on the global dimension D. This multilinear form plays the role of a volume f…
