G. Aranda-Pino, E. Pardo, M. Siles-Molina
In this paper a bijection between the set of prime ideals of a Leavitt path algebra $L_K(E)$ and a certain set which involves maximal tails in $E$ and the prime spectrum of $K[x,x^{-1}]$ is established. Necessary and sufficient conditions on the graph $E$ so that the Leavitt path algebra $L_K(E)$ is primitive are also found.
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In this paper we describe three different variations of prime ideals: strongly irreducible ideals, strongly prime ideals and insulated prime ideals in the context of Leavitt path algebras. We give necessary and sufficient conditions under which a proper ideal of a Leavitt path algebra $L$ is a product as well as an intersection of finitely many of these different types of prime ideals. Such factorizations, when they are irredundant, are shown to be unique except for the order of the factors. We also characterize the Leavitt path algebras $L$ in which every ideal admits such factorizations and …
Semiprime ideals of an arbitrary Leavitt path algebra L are described in terms of their generators. This description is then used to show that the semiprime ideals form a complete sublattice of the lattice of ideals of L, and they enjoy a certain gap property identified by Kaplansky in prime spectra of commutative rings. It is also shown that the totally ordered sets that can be realized as prime spectra of Leavitt path algebras are precisely those that have greatest lower bounds on subchains and enjoy the aforementioned gap property. Finally, it is shown that a conjecture of Kaplansky regardi…
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.
Grothendieck group of the Leavitt path algebra over power graphs of prime-power cyclic groups
本文通过计算素数幂循环群幂图的邻接矩阵导出的矩阵的Smith标准形，研究了其Leavitt路代数的Grothendieck群。
Products and Intersections of Prime-Power Ideals in Leavitt Path Algebras
本文研究任意Leavitt路代数的乘法理想理论，证明不同素理想之幂构成的无冗余有限乘积或交分解的唯一性，并刻画完全不可约理想及其可分解理想。
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…
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 …
In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization of the prime and primitive ideals of an ultragraph Leavitt path algebra. We also show that every primitive ideal of an ultragraph Leavitt path algebra is exactly the annihilator of a Chen…
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.
The Dixmier-Moeglin Equivalence (for short, the DM-equivalence) is the equivalence of three distinguishing properties of prime ideals in a non-commutative algebra A. These properties are of (i) being primitive, (ii) being rational, and (iii) being locally closed in the Zariski topology of Spec(A). The DM-equivalence holds in many interesting algebras over a field. Recently, it was shown that the prime ideals of a Leavitt path algebra of a finite graph satisfy the DM-equivalence, In this note, we investigate the occurrence of the DM-equivalence in a Leavitt path algebra L of an arbitrary direct…
In this paper we give some sufficient and some necessary conditions for an étale groupoid algebra to be a prime ring. As an application we recover the known primeness results for inverse semigroup algebras and Leavitt path algebras. It turns out that primeness of the algebra is connected with the dynamical property of topological transitivity of the groupoid. We obtain analogous results for semiprimeness.
We associate in a natural way to any partially ordered set $(P,\leq)$ a directed graph $E_P$ (where the vertices of $E_P$ correspond to the elements of $P$, and the edges of $E_P$ correspond to related pairs of elements of $P$), and then describe the prime spectrum of the resulting Leavitt path algebra $L_K(E_P)$. This construction allows us to realize a wide class of partially ordered sets as the prime spectra of rings. More specifically, any partially ordered set in which every downward directed subset has a greatest lower bound, and where these greatest lower bounds satisfy certain compatib…
Ideals in Leavitt path algebras have been shown to share many properties with those of integral domains. Since studying factorizations of ideals in integral domains into special types of ideals (particularly, prime, prime-power, primary, irreducible, semiprime, and quasi-primary ones) has proved fruitful, we conduct an analogous investigation in the context of Leavitt path algebras. Specifically, we classify the proper ideals in these rings that admit factorizations into products of each of the above types of ideals. We also classify the Leavitt path algebras where every proper ideal admits a …
Given an arbitrary graph E and a field K, the prime ideals as well as the primitive ideals of the Leavitt path algebra L_K(E) are completely described in terms of their generators. The stratification of the prime spectrum of L_K(E) is indicated with information on its individual stratum. Necessary and sufficient conditions are given on the graph E under which every prime ideal of L_K(E) is primitive. Leavitt path algebras of Krull dimension zero are characterized and those with various prescribed Krull dimension are constructed. The minimal prime ideals of L_K(E) are are described in terms of …
We describe the centroid of some Leavitt path algebras. More precisely, we show that for Leavitt path algebras over a field $K$ that are simple its centroid is isomorphic to $K$, and for prime Leavitt path algebras its centroid is isomorphic to $K$ except if the graph is a row-finite comet, in which case the centroid is isomorphic to $K[x,x^{-1}]$.
It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the two-sided ideals of L form a distributive lattice. It is also shown that L is a multiplication ring, that is, given any two ideals A,B in L with A inside B, there is always an ideal C such that A = BC, an indication of a possible rich multiplicative ideal theory for L. Existence and uniqueness of factorization of the ideals of L as products of special types of ideals such as prime, irreducible or primary ideals is investigated. It turns out that the irreducible id…
Irreducible representations of both Leavitt and Cohn path algebras of an arbitrary digraph with coefficients in a commutative field is classified. They are constructed in several ways using both infinite paths on the right as well as direct limits or factors of one-sided projective ideals of the ordinary quiver algebra, respectively. Furthermore, their defining relations are described, too, whence criterions are easily given when they are finitely presented or finite dimensional. Moreover, their endomorphism rings, annihilator primitive ideals are also computed directly. In particular, the car…
Let $E$ be an arbitrary graph and $K$ be any field. We construct various classes of non-isomorphic simple modules over the Leavitt path algebra $L_{K}(E)$ induced by vertices which are infinite emiters, closed paths which are exclusive cycles and paths which are infinite, and call these simple modules Chen modules. It is shown that every primitive ideal of $L_{K}(E)$ can be realized as the annihilator ideal of some Chen module. Our main result establishes the equivalence between a graph theoretic condition and various conditions concerning the structure of simple modules over $L_K(E)$.
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。
Leavitt path algebras in which every Lie ideal is an ideal and applications
本文分类了所有每个李理想均为理想的Leavitt路径代数，并利用该性质构造出一类局部有限无限维李代数，完全确定其局部可解根，从而得到素特征域上一类新的半单李代数。
Let $E$ be any directed graph, and $K$ any field. We classify those graphs $E$ for which the Leavitt path algebra $L_K(E)$ is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive.
Given an arbitrary graph E and any field K, a new class of simple left modules over the Leavitt path algebra L of the graph E over K is constructed by using vertices that emit infinitely many edges. The corresponding annihilating primitive ideals are described and is used to show that these new class of simple L-modules are different from(that is non-isomorphic to) any of the previously known simple modules. Using a Boolean subring of idempotents induced by paths in E, bounds for the cardinality of the set of distinct isomorphism classes of simple L-modules are given. We also append other info…
Graded homotopy classification of Leavitt path algebras
本文通过分次Grothendieck群和代数K理论，证明了本原图幺元Leavitt路代数在分次同伦等价下的分类。
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.
Let $K$ be a field, let $E$ be a finite directed graph, and let $L_K(E)$ be the Leavitt path algebra of $E$ over $K$. We show that for a prime ideal $P$ in $L_K(E)$, the following are equivalent: \begin{enumerate} \item $P$ is primitive; \item $P$ is rational; \item $P$ is locally closed in ${\rm Spec}(L_K(E))$. \end{enumerate} We show that the prime spectrum ${\rm Spec}(L_K(E))$ decomposes into a finite disjoint union of subsets, each of which is homeomorphic to ${\rm Spec}(K)$ or to ${\rm Spec}(K[x,x^{-1}])$. In the case that $K$ is infinite, we show that $L_K(E)$ has a rational $K^{\times}$…
Let L be the Leavitt path algebra of an arbitrary directed graph E over a field K. This survey article describes how this highly non-commutative ring L shares a number of the characterizing properties of a Dedekind domain or a Prüfer domain expressed in terms of their ideal lattices. Special types of ideals such as the prime, the primary, the irreducible and the radical ideals of L are described by means of the graphical properties of E. The existence and the uniqueness of the factorization of a non-zero ideal of L as an irredundant product of prime or primary or irreducible ideals is establis…
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。
In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra \cite{CMMSS1}. We will go further into the structure itself of the Leavitt path algebr…
Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras $L_K(n, n + k)$ constructed by Leavitt. Using Bergman's Diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contra…
Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra $L$, in spite of being non-commutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [8] it was shown that the ideals of $L$ satisfy the distributive law, a property of Prüfer domains and that $L$ is a multiplication ring, a property of Dedekind domains. In this paper, we first show that $L$ satisfies two more chara…
Maximal Commutative Subalgebras of Steinberg Algebras
本文在素Steinberg代数中构造了极大交换子代数的大类，推广了Leavitt路代数的已知结果。
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.
We establish necessary and sufficient conditions on a (not necessarily countable) graph E for the graph C*-algebra C*(E) to be primitive. Along with a known characterization of the graphs E for which C*(E) is prime, our main result provides us with a systematic method for easily producing large classes of (necessarily nonseparable) C*-algebras that are prime but not primitive. We also compare and contrast our results with similar results for Leavitt path algebras.
Generalized Bassian Modules over Non-primitive Dedekind Prime Rings
本文研究了非原始迪克因素数环上的广义巴西安模块性质，并探讨了其在环论中的应用。
Bartholdi and Smoktunowicz constructed finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type. Moreover, using a construction of an inverse system of monomial ideals which arise from this construction we are able to further construct finitely generated primitive algebras with many regular enough growth types. Then, inspired by an example by Zelmanov, we show how our prime algebras can be constructed suc…
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.
We give a brief survey of primitivity in ring theory and in particular look at characterizations of primitive ideals in the prime spectrum for various classes of rings.
We introduce primitive hyperideals of a hyperring R and show relations with R itself, and with maximal and prime hyperideals of R. We endow a Jacobson topology on the set of primitive hyperideals of R and study topological properties of the corresponding hyperstructure space.
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。
Representation of partially ordered sets over Von Neumann regular algebras. More prime, non-primitive regular rings
本文对每个具有有限共尾子集的偏序集 I 构造局部矩阵型单位正则代数 B(I)，证明其理想格、Grothendieck 群偏序分别与 I 的下子集格和限制 Hahn 幂同构，并刻画了其素性、本原性与半 Artin 性的偏序条件。
Let $k$ be an uncountable field. We show that a finitely generated prime Goldie $k$-algebra of quadratic growth is either primitive or satisfies a polynomial identity, answering a question of Small in the affirmative.
We construct a non-separable C*-algebra that is prime but not primitive.
Let $k$ be a field and let $R$ be a countable dimensional prime von Neumann regular $k$-algebra. We show that $R$ is primitive, answering a special case of a question of Kaplansky.
An almost PI algebra is a generalisation of a just infinite algebra which does not satisfy a polynomial identity. An almost PI algebra has some nice properties: It is prime, has a countable cofinal subset of ideals and when satisfying ACC(semiprimes), it has only countably many height 1 primes. Consider an affine prime Goldie non-simple non-PI $k$-algebra $R$ of GK dimension $<3$, where $k$ is an uncountable field. $R$ is an almost PI algebra. We give some possible additional conditions which make such an algebra primitive. This gives a partial answer to Small's question: Let $R$ be an affine …
On a family of liftings of the Jordan plane
研究无限循环群上Jordan平面的一族Hopf代数提升，确定其中心、素谱、本原谱和自同构群，证明每个素理想是完全素理想且非零理想与中心非平凡相交，构造对应于所有本原理想的显式单模并分类有限维单模，最后证明这些Hopf代数满足Dixmier-Moeglin等价性。
We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. For primitive ideals we obtain an analogue of Duflo's Theorem, which says that every primitive ideal is the annihilator of a simple highest weight module. This depends on an extension of Lusztig's tensor product theorem.
Let q be a nonzero complex number that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O_q(M_{m,n}) of quantum matrices. Next, we describe all height one primes of O_q(M_{m,n}); these two problems are actually interlinked since it turns out that (0) is a primitive ideal of O_q(M_{m,n}) whenever O_q(M_{m,n}) has only finitely many height one primes. Finally, we compute the automorphism group of O_q(M_{m,n}) in the case where m is not equal to n. In order to do this, we first study the action of this group on the prime spectrum of O_q(M_{m,n}). Then, by…
On induced completely prime primitive ideals in enveloping algebras of classical Lie algebras
本文证明经典单李代数包络代数中的 Losev-Premet 理想均可由某个 Levi 子代数中关联簇为刚性幂零轨道闭包的 Losev-Premet 理想经抛物诱导得到，从而完成刚性 Losev-Premet 理想的分类。
This paper contains a survey of some ring-theoretic aspects of quantized coordinate rings, with primary focus on the prime and primitive spectra. For these algebras, the overall structure of the prime spectrum is governed by a partition into strata determined by the action of a suitable group of automorphisms of the algebra. We discuss this stratification in detail, as well as its use in determining the primitive spectrum -- under suitable conditions, the primitive ideals are precisely those prime ideals which are maximal within their strata. The discussion then turns to the global structure o…
Let G be a finite group, let p be a prime number, and let K be a field of characteristic 0 and k be a field of characteristic p, both large enough. In this note we state explicit formulae for the primitive idempotents of K\otimes pp_k(G), where pp_k(G) is the ring of p-permutation kG-modules.
We prove that any finite W-algebra U(g,e) admits a one-dimensional representation fixed by the action of the component group of the centraliser of e. As a consequence, for any nilpotent orbit O in g there exists a multiplicity-free (and hence completely prime) primitive ideal of the universal enveloping algebra U(g) whose associated variety coincides with the Zariski closure of O.
Graph monoids arise naturally in the study of non-stable K-theory of graph C*-algebras and Leavitt path algebras. They play also an important role in the current approaches to the realization problem for von Neumann regular rings. In this paper, we characterize when a finitely generated conical refinement monoid can be represented as a graph monoid. The characterization is expressed in terms of the behavior of the structural maps of the associated $I$-system at the free primes of the monoid.
Let p be a prime, K a p-adic field, G a nilpotent, uniform pro-p group. We prove that all faithful, primitive ideals in the Iwasawa algebra KG are controlled by the centraliser of the second term in the upper central series for G.
For the algebra $A$ in the title, its prime, primitive and maximal spectra are classified. The group of automorphisms of $A$ is determined. The simple unfaithful $A$-modules and the simple weight $A$-modules are classified.
Let F be the function field of a curve over a p-adic field. Let D/F be a central division algebra of prime exponent $\ell$ which is different from p. Assume that F contains a primitive ${\ell}^2$-th root of unity. Then the group $SK_1(D):=SL_1(D)/[D^*, D^*]$ is trivial.
The aim of this paper is to study the representation theory of quantum Schubert cells. Let $\g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $\g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(\g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits …
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