Let $F$ be a field of characteristic zero, $G$ be a group and $R$ be the algebra $M_n(F)$ with a $G$-grading. Bahturin and Drensky proved that if $R$ is an elementary and the neutral component is commutative then the graded identities of $R$ follow from three basic types of identities and monomial identities of length $\geq 2$ bounded by a function $f(n)$ of $n$. In this paper we prove the best upper bound is $f(n)=n$, more generally we prove that all the graded monomial identities of an elementary $G$-grading on $M_n(F)$ follow from those of degree at most $n$. We also study gradings which satisfy no monomial identities but the trivial ones, which we call almost non-degenerate gradings. The description of non-degenerate elementary gradings on matrix algebras is reduced to the description of non-degenerate elementary gradings on matrix algebras that have commutative neutral component. We provide necessary conditions so that the grading on $R$ is almost non-degenerate and we apply the results on monomial identities to describe all almost non-degenerate $\mathbb{Z}$-gradings on $M_n(F)$ for $n\leq 5$.
In this paper we describe all gradings by abelian groups without elements of order p, where p > 2 is the characteristic of the base field, on the simple graded Cartan type Lie algebras.
We introduce grading on certain finite dimensional simple Lie superalgebras of type $P(t)$ by elementary abelian 2-group. This grading gives rise to Pauli matrices and is a far generalization of $(\mathbb Z_2\times \mathbb Z_2)$-grading on Lie algebra of $(2\times 2)$-traceless matrices.We use this grading for studying numerical invariants of polyomial identities of Lie superalgebras. In particular, we compute graded PI-exponent corresponding to Pauli grading.
Let $G$ be a group and let $S=\bigoplus_{g \in G} S_g$ be a $G$-graded ring. Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading of $S$. It is well-known that if $S$ is strongly $G$-graded, then the induced $G/N$-grading is strong for any $N$. The class of epsilon-strongly graded rings was recently introduced by Nystedt, Öinert and Pinedo as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and suffic…
We find a basis for the $G$-graded identities of the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of characteristic $p>0$ with an elementary grading such that the neutral component corresponds to the diagonal of $M_n(K)$.
We consider the algebra $E\otimes E$ over an infinite field equipped with a $\mathbb{Z}_2$-grading where the canonical basis is homogeneous and prove that in various cases the graded identites are just the ordinary ones. If the grading is a non-canonical grading obtained as a quotient grading of the natural $\mathbb{Z}_2\times\mathbb{Z}_2$-grading we exhibit a basis for the graded identities.
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…
The algebra of $n\times n$ matrices over a field $F$ has a natural $\mathbb{Z}_n$-grading. Its graded identities have been described by Vasilovsky who extended a previous work of Di Vincenzo for the algebra of $2\times 2$ matrices. In this paper we study the graded identities of block-triangular matrices with the grading inherited by the grading of $M_n(F)$. We show that its graded identities follow from the graded identities of $M_n(F)$ and from its monomial identities of degree up to $2n-2$. In the case of blocks of sizes $n-1$ and 1, we give a complete description of its monomial identities…
Let P a locally finite partially ordered set, F a field, G a group, and I(P,F) the incidence algebra of P over F. We describe all the inequivalent elementary G-gradings on this algebra. If P is bounded, F is a infinite field of characteristic zero, and A, B are both elementary G-graded incidence algebras satisfying the same G-graded polynomial identities, and the automorphisms group of P acts transitively on the maximal chains of P , we show that A and B are graded isomorphic.
Let $E$ be the infinite dimensional Grassmann algebra over a field $F$ of characteristic zero. In this paper we investigate the structures of $\mathbb{Z}$-gradings on $E$ of full support. Using methods of elementary number theory, we describe the $\mathbb{Z}$-graded polynomial identities for the so-called $2$-induced $\mathbb{Z}$-gradings on $E$ of full support. As a consequence of this fact we provide examples of $\mathbb{Z}$-gradings on $E$ which are PI-equivalent but not $\mathbb{Z}$-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent and no…
The aim of this paper is to introduce and study graded and filtered gamma rings and gamma modules. We prove that the filtered $Γ$-ring (module) is a generalization of the notion of graded ring (module). Also, we construct a graded $Γ$-ring from a filtered $Γ$-ring. We investigate some properties of graded and fileterd Gamma rings and Gamma modules. Finally we define and study the strongly graded gamma rings.
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 $A$.
A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras $H(2\colon\n;ω_2)$ (of dimension one less than a power of $p$) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of $H(2\colon\n;ω_2)$ with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).
Let G be an arbitrary group and let K be a field of characteristic different from 2. We classify the G-gradings on the Jordan algebra of upper triangular matrices of order n over K. It turns out that there are, up to a graded isomorphism, two families of gradings: the elementary gradings (analogous to the ones in the associative case), and the so called mirror type (MT) gradings. Moreover we prove that the G-gradings on this algebra are uniquely determined, up to a graded isomorphism, by the graded identities they satisfy.
Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\o…
Let $F$ be an infinite field of characteristic different from 2, and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the $\mathbb{Z}$-graded polynomial identities of $E$ with respect to certain $\mathbb{Z}$-grading such that the vector space $L$ is homogeneous in the grading. More precisely, we construct three types of $\mathbb{Z}$-gradings on $E$, denoted by $E^{\infty}$, $E^{k^\ast}$ and $E^{k}$, and we give the explicit form of the corresponding $\mathbb{Z}$-graded polynomial identities. We show that the homogeneous superalgebras $E_{…
Given a Lie algebra $L$ graded by a group $G$, if $L$ is does not contain orthogonal graded ideals and $G$ is generated by the support of $L$, then $G$ is an abelian group.
In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices UT_n. For positive integers q \leq n, we classify these images on UT_n endowed with a particular elementary Z_q-grading. As a consequence, we obtain the images of multilinear graded polynomials on UT_n with the natural Z_n-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear pol…
A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field $\mathbb{F}$ can be reduced to the following three classifications, for each finite Galois extension $\mathbb{L}$ of $\mathbb{F}$: (1) finite-dimensional central division algebras over $\mathbb{L}$, up to isomorphism; (2) twisted group algebras of finite groups…
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.
Consider a Leibniz superalgebra $\mathfrak L$ additionally graded by an arbitrary set $I$ (set grading). We show that $\mathfrak L$ decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case of ${\mathfrak L}$ being of maximal length, the simplicity of ${\mathfrak L}$ is also characterized in terms of connections.
We work to find a basis of graded identities for the octonion algebra. We do so for the $\mathbb{Z}_2^2$ and $\mathbb{Z}_2^3$ gradings, both of them derived of the Cayley-Dickson process, the later grading being possible only when the characteristic of the scalars is not two.
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…
Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D4, where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.
The Jordan algebra of the symmetric matrices of order two over a field $K$ has two natural gradings by $\mathbb{Z}_2$, the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a $\mathbb{Z}…
In a recent article with Oleg Smirnov, we defined short Peirce (SP) graded Kantor pairs. For any such pair P, we defined a family, parameterized by the Weyl group of type BC_2, consisting of SP-graded Kantor pairs called Weyl images of P. In this article, we classify finite dimensional simple SP-graded Kantor pairs over an algebraically closed field of characteristic 0 in terms of marked Dynkin diagrams, and we show how to compute Weyl images using these diagrams. The theory is particularly attractive for close-to-Jordan Kantor pairs (which are variations of Freudenthal triple systems), and we…
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.
Leavitt path algebras associate to directed graphs a $\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras $L(1,k)$. In this note, we first study this $\mathbb Z$-grading and characterize the ($\mathbb Z$-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, $C_n$-comet and multi-headed graphs. The last two type are examples of graphs whose Leavitt path algebras are strongly graded. We characterize Leavitt path algebras which are strongly graded, along the way obtaining classes of algebras which are group rings or crossed-products. In an …
We study gradings by noncommutative groups on finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It is shown that if $L$ is gradeg by a non-abelian finite group $G$ then the solvable radical $R$ of $L$ is $G$-graded and there exists a Levi subalgebra $B=H_1\oplus\cdots\oplus H_m$ homogeneous in $G$-grading with graded simple summands $H_1, \ldots, H_m$. All supports $Supp~H_i, i=1\ldots, m$, are commutative subsets of $G$.
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 E be the infinite dimensional Grassmann algebra over a finite field F of characteristic not 2. In this paper we deal with the homogeneous Z_2-gradings of E. In particular, we compute an exact value for the Z_2-graded homogeneous codimensions of E, and a lower and an upper bound for the Z_2-graded (non-homogeneous) codimensions of E for each of its Z_2-homogeneous grading.
Let $G,H$ be groups, $ϕ: G \rightarrow H$ a group morphism, and $A$ a $G$-graded algebra. The morphism $ϕ$ induces an $H$-grading on $A$, and on any $G$-graded $A$-module, which thus becomes an $H$-graded $A$-module. Given an injective $G$-graded $A$-module, we give bounds for its injective dimension when seen as $H$-graded $A$-module. Following ideas by Van den Bergh, we give an application of our results to the stability of dualizing complexes through change of grading.
Let $F$ be an infinite field. The primeness property for central polynomials of $M_n(F)$ was proved by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for $M_n(F)$ and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider $M_n(R)$, where $R$ admits a regular grading, with a grading such that $M_n(F)$ is a homogeneous subalgebra and provide sufficient conditions - s…
Let $F$ be a finite field with characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis of the $\mathbb{Z}_{2}$-graded polynomial identities for any non-trivial $\mathbb{Z}_{2}$-grading such that a basis of $V$ is homogeneous in this grading.
We consider fine G-gradings on M_n(C) (i.e. gradings of the matrix algebra over the complex numbers where each component is 1 dimensional). Groups which provide such a grading are known to be solvable. We consider the T-ideal of G-graded identities and show that it is generated by a special type of binomial identities which we call elementary. In particular we show that the ideal of graded identities is finitely generated as a T-ideal. Next, given such grading we construct a universal algebra U_{G,c} in two different ways: one by means of polynomial identities and the other one by means of a g…
In this paper we look into the structure of finite-dimensional graded superalgebras of various types such as associative, Lie and Jordan over an algebraically closed field of characteristic zero.
In this paper, we study the holonomic $D$-modules when $D$ is the ring of $k$-linear differential operators on $A = k[Γ]$, the coordinate ring of an affine monomial curve over the complex numbers $k = \mathbb C$. In particular, we consider the graded case, and classify the simple graded $D$-modules and compute their extensions. The classification over the first Weyl algebra $D = A_1(k)$ is obtained as a special case.
Identities and isomorphisms of finite graded matrix algebras
Universidade Federal de Campina Grande (UFCG)（坎皮纳格兰德联邦大学）；Universidade de São Paulo（圣保罗大学）；Universidade Federal de São Carlos (UFSCar)（圣卡洛斯联邦大学）；Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学）
When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. The following question arises naturally: when a group grading on a finite dimensional algebra is weakly equivalent to a gra…
In 2004 Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a Z-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that can not be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms.
In this paper we investigate gradings on tame blocks of group algebras whose defect group is dihedral. We classify gradings on an arbitrary dihedral block up to graded Morita equivalence. We do this by computing the group of outer automorphisms that fix the isomorphism classes of simple modules. We also show how to grade these blocks via transfer of gradings via dervied equivalences.
Let $K<X> =K<X_1,...,X_n>$ be the free $K$-algebra on $X={X_1,...,X_n}$ over a field $K$, which is equipped with a weight $\mathbb{N}$-gradation (i.e., each $X_i$ is assigned a positive degree), and let ${\cal G}$ be a finite homogeneous Gröbner basis for the ideal $I=<{\cal G}>$ of $K<X>$ with respect to some monomial ordering $\prec$ on $K<X>$. It is proved that if the monomial algebra $K<X>/<{\bf LM}({\cal G})>$ is semi-prime, where ${\bf LM}({\cal G})$ is the set of leading monomials of ${\cal G}$ with respect to $\prec$, then the $\mathbb{N}$-graded algebra $A=K<X>/I$ is semiprimitive (in…
Let $K$ be an infinite field of characteristic different from two and let $U_1$ be the Lie algebra of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$. The algebra $U_1$ admits a natural $\mathbb{Z}$-grading. We provide a basis for the graded identities of $U_1$ and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension $\leq 1$, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to…
This note is devoted to the construction of two very easy examples, of respective dimensions 4 and 6, of graded Lie algebras whose grading is not given by a semigroup, the latter one being a semisimple algebra. It is shown that 4 is the minimal possible dimension.
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…
We classify group gradings on the simple Lie algebra $L$ of type $D_4$ over an algebraically closed field of characteristic different from 2: fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism. For each $G$-grading on $L$, we also study graded $L$-modules (assuming characteristic 0).
