Georgia Benkart, Thomas Gregory, Alexander Premet
The Recognition Theorem for graded Lie algebras is an essential ingredient in the classification of finite-dimensional simple Lie algebras over an algebraically closed field of characteristic p > 3. The main goal of this monograph is to present the first complete proof of this fundamental result.
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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.
The purpose of this article is to define and examine graded almost prime ideals over a non-commutative graded ring, and consider some cases where all graded right ideals of a non-commutative graded ring are graded almost prime.
In this article, we consider the structure of graded rings, not necessarily commutative nor with unity, and study the graded weakly prime ideals. We investigate the graded rings in which all graded ideals are graded weakly prime. Several properties are given, and several examples to support given propositions are constructed. We initiate the study of graded weakly total prime ideals and investigate graded rings for which every proper graded ideal is graded weakly total prime.
Graded Classical Weakly Prime Submodules over Non-Commutative Graded Rings
本文提出并研究了非交换分次环上的分次经典弱素子模，推广了交换情形，并考察了其在各类分次模中的结构。
Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. We consider graded Verbally prime $T$-ideals in the free $G$-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime $G$-graded $T$-ideals and strongly verbally prime $T$-ideals. At first, following Kemer's ideas, we classify $G$-graded verbally prime $T$-ideals. The main bulk of the paper is devoted to the stronger notion. We classify $G$-graded strongly verbally prime $T$-ideals which…
Nonunital prime rings graded by ordered groups
该研究将有序群分次素环的经典定理推广至非单位环情形，建立分次素理想与单位分支素理想的双射对应，并给出Leavitt路径环素性新证明及群环分次子环素性条件。
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…
For commutative graded rings, the concept of graded $2$-absorbing (graded weakly $2$-absorbing) ideals was introduced and examined by Al-Zoubi, Abu-Dawwas and Çeken (Hacettepe Journal of Mathematics and Statistics, 48 (3) (2019), 724-731) as a generalization of the concept of graded prime (graded weakly prime) ideals. Up to now, research on these topics mainly concentrated on commutative graded rings. On the other hand, graded prime ideals over non-commutative graded rings have been introduced and examined by Abu-Dawwas, Bataineh and Al-Muanger (Vietnam Journal of Mathematics, 46 (3) (2018), 6…
In this article we study graded Clifford algebras with a gradation preserving action of automorphisms given by $H_p$, the Heisenberg group of order $p^3$ with $p$ prime. After reviewing results in dimensions 3 and 4, we will determine the graded Clifford algebras that are AS-regular algebras of global dimension 5 and generalize certain results to arbitrary dimension $p$. More specifically, we prove that there is a nice correspondence between $\mathbb{P}^1_{\mathbb{F}_p}$ and the algebras isomorphic to the quantum space $\mathbb{C}_{-1}[x_0,\ldots,x_{p-1}]$.
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…
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, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a characterization of prime partial skew groupoid rings, and in particular of prime groupoid rings, thereby generalizing a classical result by Connell and partially generalizing recent results by Steinberg.
Prime group graded rings with applications to partial crossed products and Leavitt path algebras
本文研究了素数群格环的性质，并将其应用于部分交叉积和莱维特路径代数中，扩展了相关经典结果。
Primeness property for regular gradings
本文研究了正则分级的素性性质，证明了G-正则代数不满足素性，而Z_2-正则代数在普通意义上满足素性，且最小性对正则性无要求。
The analogue of Goldie's Theorem for prime rings is proved for rings graded by abelian groups, eliminating unnecessary additional hypotheses used in earlier versions.
Let $\mathcal G$ be an ultragraph and let $K$ be a field. We describe prime and primitive ideals in the ultragraph Leavitt path algebra $L_K(\mathcal G)$. We identify the graded prime ideals in terms of downward directed sets and then we characterize the non-graded prime ideals. We show that the non-graded prime ideals of $L_K(\mathcal G)$ are always primitive.
Generalized Homogeneous Derivations on Graded Rings
本文探讨了广义齐次导出在分层环中的扩展，研究了gr-广义导出的性质，并将素环中的重要结果推广到gr-素环中，同时探讨了这些映射的代数与模结构及范畴框架。
We present two criteria for a group $G$ to satisfy the following statements: any $G$-graded gr-prime (gr-semiprime) right gr-Goldie ring admits a gr-semisimple graded right classical quotient ring. The criterion for gr-semiprime rings is that the group $G$ is periodic. Actually, the sufficiency of periodicity was proved by the author in 2011 and the necessity of it follows from the well-known counterexample (1979). The main result of the paper concerns the gr-prime case. In this case, Goodearl and Stafford proved the graded version of Goldie's Theorem for rings graded by an Abelian group (2000…
Let F be characteristic zero field, G a residually finite group and W a G-prime and PI F-algebra. By constructing G-graded central polynomials for W, we prove the G-graded version of Posner's theorem. More precisely, if S denotes all non-zero degree e central elements of W, the algebra S^{-1}W is G-graded simple and finite dimensional over its center. Furthermore, we show how to use this theorem in order to recapture the result of Aljadeff and Haile stating that two G-simple algebras of finite dimension are isomorphix iff their ideals of graded identities coincide.
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 …
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 $E$ be an arbitrary (countable) graph and let $R$ be a unital commutative ring. We analyze the ideal structure of the Leavitt path algebra $\lr$ introduced by Mark Tomforde. We first modify the definition of basic ideals and we then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of $\lr$. Then by applying these results we determine prime graded basic ideals and left (or right) primitive graded ideals of $\lr$. In particular, we show that when $E$ satisfies Condition (K) and $R$ is a field, the set …
Ore Localisation for differential graded rings; Towards Goldie's theorem for differential graded algebras
本文研究微分分次代数的 Ore 局部化，定义 dg-素环、dg-半素环、dg-幂零根、dg-本质子模与 dg-一致维数，并推广 Goldie 定理至素 dg-环。
Jordan 3-graded Lie algebras with polynomial identities
本文借助 Tits-Kantor-Koecher 构造将 PI 条件转化为相关 Jordan 对的性质，证明强素情形下的 Posner-Rowen 型定理，并利用 Kostrikin 根刻画一般 PI Jordan 3-分次 Lie 代数。
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).
In this paper, prime as well as primitive Kumjian-Pask algebras $\mathrm{KP}_R(Λ)$ of a row-finite $k$-graph $Λ$ over a unital commutative ring $R$ are completely characterized in graph-theoretic and algebraic terms. By applying quotient $k$-graphs, these results describe prime and primitive graded basic ideals of Kumjian-Pask algebras. In particular, when $Λ$ is strongly aperiodic and $R$ is a field, all prime and primitive ideals of a Kumjian-Pask algebra $\mathrm{KP}_R(Λ)$ are determined.
The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra $M_n(K)$ over a field $K$ endowed with its canonical $\mathbb{Z}_n$-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field $\mathbb Q$ of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on $n\times n$ matrices, where $n$ is a prime number. We confirm such conjecture for polynomials of degree …
The maximal graded subalgebras for four families of Lie superalgebras of Cartan type over a field of prime characteristic are studied. All maximal reducible graded subalgebras are described completely and their isomorphism classes, dimension formulas are found. The classification of maximal irreducible graded subalgebras is reduced to the classification of the maximal irreducible subalgebras for the classical Lie superalgebras $\frak{gl}(m,n)$, $\frak{sl}(m,n)$ and $\frak{osp}(m,n)$.
There is an extensive recent literature on the graded, non-graded, prime, primitive, maximal ideals of Leavitt path algebras. In this introductory level survey, we will be giving an overview of different types of ideals and the correspondence between the lattice of ideals and the lattice of hereditary and saturated subsets of the graph over which the Leavitt path algebra is constructed.
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…
We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the nonzero prime ideals $P$ such that $A/P$ has GK dimension 2 is non-empty, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either…
Classification of Nottingham algebras
本文通过证明存在性和唯一性结果，完成了Nottingham代数的分类，确定了所有此类代数在同构意义下的全部情况。
Graded Lie algebras of maximal class of type $p$
本文研究了特征为 $p$ 的域上由特定次数元素生成的无限维极大类分次李代数，将 Caranti 和 Vaughan-Lee 在 $p=2$ 时的分类结果推广至任意素数特征，并证明了其中的主要步骤。
We investigate a class of algebras that provides multiparameter versions of both quantum symplectic space and quantum Euclidean $2n$-space. These algebras encompass the graded quantized Weyl algebras, the quantized Heisenberg space, and a class of algebras introduced by Oh. We describe the structure of the prime and primitive ideals of these algebras. Other structural results include normal separation and catenarity.
Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group …
We study the center of several types of path algebras. We start with the path algebra $KE$ and prove that if the number of vertices is infinite then the center is zero. Otherwise, it coincides with the field $K$ except when the graph $E$ is a cycle in which case the center is $K[x]$, the polynomial algebra in one indeterminate. Then we compute the centers of prime Cohn and Leavitt path algebras. A lower and an upper bound for the center of a Leavitt path algebra are given by introducing the graded Baer radical for graded algebras.
On the cyclic homology of certain universal differential graded algebras
该研究针对奇素数$p$和无$p$-挠交换$\bZ_{(p)}$-代数$R$，计算了由$R$泛零化$p$得到的泛微分分次代数$R//p$的周期循环同调，以及其无穷多个次数的循环同调与负循环同调。
Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies Condition (K). Uniqueness theorems in representing an ideal of $L$ as an irredundant intersection and also as an irredundant product of finitely many prime ideals are established. Leavitt path algebras containing only finitely many prime ideals and those in which every ideal is prime are described. Powers of a single ideal $I$ are considered and it is shown…
For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl.
Graded irreducible representations of Leavitt path algebras: a new type and complete classification
本文构造了Leavitt路径代数的一类新的分次不可约表示，其表示空间不同构于任何简单Chen模，并利用底层图性质刻画分次本原理想，从而完成了分次简单模的完全分类，同时指出已有文献中本原理想刻画的一个问题。
The domain and prime properties for Koszul rings and algebras
本文通过Koszul对偶信息建立证明Koszul分次环为素环或整域的技术，并应用于维数2的扭曲Calabi-Yau代数及预投射代数，得到素分段整域结论。
Given a partial action $α$ of a groupoid $G$ on a ring $R$, we study the associated partial skew groupoid ring $R \rtimes_α G$, which carries a natural $G$-grading. We show that there is a one-to-one correspondence between the $G$-invariant ideals of $R$ and the graded ideals of the $G$-graded ring $R \rtimes_αG.$ We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of $R \rtimes_αG.$ We show that every ideal of $R \rtimes_αG$ is graded if, and only if, $α$ has the residual intersection property. Furthermore, if $α$ is induced by a topological …
Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a polynomial ring over a perfect field of characteristic p>0 and if the inclusionof R into S splits, then D_k(R) is a simple ring. In the last section of the paper, we discuss how one might try to deduce …
We define Gerstenhaber's graded Lie bracket directly on complexes other than the bar complex, under some conditions. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic group of prime order.
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