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1.
Jü rgen Herzog Takayuki Hibi Ngô Viê t Trung Xinxian Zheng 《Transactions of the American Mathematical Society》2008,360(12):6231-6249
The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining these algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on Rees algebras of squarefree monomial ideals.
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3.
We extend the sortability concept to monomial ideals which are not necessarily generated in one degree and as an application we obtain normal Cohen-Macaulay toric rings attached to vertex cover ideals of graphs. Moreover, we consider a construction on a graph called a clique multi-whiskering which always produces vertex cover ideals with componentwise linear powers. 相似文献
4.
Seth Sullivant 《Journal of Algebra》2008,319(1):115-142
Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of secants of monomial ideals. We use Gröbner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case. Among the applications are a new, unified approach to the Gröbner bases of symbolic powers of determinantal and Pfaffian ideals. 相似文献
5.
Anurag K. Singh 《Mathematische Zeitschrift》2000,235(2):335-344
Let I be a divisorial ideal of a strongly F-regular ring A. K.-i. Watanabe raised the question whether the symbolic Rees algebra is Cohen-Macaulay whenever it is Noetherian. We develop the notion of multi-symbolic Rees algebras and use this to show that
is indeed Cohen-Macaulay whenever a certain auxiliary ring is finitely generated over A.
Received August 10, 1998 / in final form October 18, 1999 / Published online July 20, 2000 相似文献
6.
《Journal of Algebra》2006,295(2):458-472
In this paper we study the finite generation of Ext-algebras of a class of algebras called δ-resolution determined algebras. We characterize the δ-resolution determined algebras which are monomial algebras. If Λ is a graded algebra such that the associated monomial algebra is δ-resolution determined, we classify when the Ext-algebra of Λ is finitely generated. 相似文献
7.
Intan Muchtadi-Alamsyah 《代数通讯》2013,41(7):2544-2569
We construct an action of a braid group associated to a complete graph on the derived category of a certain symmetric Nakayama algebra which is also a Brauer star algebra with no exceptional vertex. We connect this action with the affine braid group action on Brauer star algebras defined by Schaps and Zakay–Illouz. We show that for Brauer star algebras with no exceptional vertex, the action is faithful. 相似文献
8.
Vittoria Bonanzinga Cesar A. Escobar Rafael H. Villarreal 《Results in Mathematics》2002,41(3-4):258-264
Let A be an integral matrix with non negative entries and let K be a field. We give a sufficient condition for the normality of the monomial subring determined by the columns of A over the field K. One of the main results proves that if A is totally unimodular, then the Rees algebra of the monomial ideal over K defined by the columns of A is a normal domain. 相似文献
9.
Be’eri Greenfeld 《Israel Journal of Mathematics》2017,220(1):161-174
Bartholdi and Smoktunowicz constructed in 2014 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 without further limitations on the growth type.Then, inspired by Zelmanov’s example in 1979, we show how our prime algebras can be constructed such that they contain non-zero locally nilpotent ideals; this is the very opposite of the primitive constructions. 相似文献
10.
S. A. Ilyasov 《Journal of Mathematical Sciences》2007,142(2):1933-1941
In this paper, we consider the problem of algorithmically constructing the left syzygy module for a finite system of elements
in an automaton monomial algebra. The class of automaton monomial algebras includes free associative algebras and finitely
presented algebras. In such algebras the left syzygy module for a finite system of elements is finitely generated. In general,
the left syzygy module in an automaton monomial algebra is not finitely generated. Nevertheless, the generators of the left
syzygy module have a recursive specification with the help of regular sets. This allows one to solve many algorithmic problems
in automaton monomial algebras. For example, one can solve linear equations, recognize the membership in a left ideal, and
recognize zero-divisors.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 101–113, 2005. 相似文献
11.
Sarah Rees 《Algebras and Representation Theory》2008,11(3):207-214
It is well known that the sets of strings that define all representations of string algebras and many representations of other
quotients of path algebras form a regular set, and hence are defined by finite state automata. This short article aims to
explain this connection between representation theory and automata theory in elementary terms; no technical background in
either representation theory or automata theory is assumed. The article describes the structure of the set of strings of a
monomial algebra as a locally testable and hence regular set, and describes explicitly the construction of the automaton,
illustrating the construction with an elementary example. Hence it explains how the sets of strings and bands of a monomial
algebra correspond to the sets of paths and closed (non-powered) circuits in a finite graph, and how the growth rate of the
set of bands is immediately visible from that graph.
Presented by C. Ringel. 相似文献
12.
We present criteria for the Cohen–Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen–Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen–Macaulayness of the second symbolic power or of all symbolic powers of a Stanley–Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen–Macaulay. In particular, all symbolic powers are Cohen–Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen–Macaulayness can pass from a symbolic power to another symbolic powers in different ways. 相似文献
13.
M. Siles Molina 《Journal of Algebra》2008,319(12):5265-5278
Leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras. Using this fact we obtain maximal algebras of right quotients from those (Leavitt) path algebras whose associated graph satisfies that every vertex connects to a line point (equivalently, the Leavitt path algebra has essential socle). We also introduce and characterize the algebraic counterpart of Toeplitz algebras. 相似文献
14.
S. A. Ilyasov 《Journal of Mathematical Sciences》2008,152(1):95-136
Abstract. The paper considers a new algebraic object, the completely automaton binomial algebras, which generalize certain existing classes of algebras. The author presents a classification of semigroup algebras taking into account completely automaton algebras and gives the corresponding examples. A number of standard algorithmic problems are solved for completely automaton binomial algebras: the recognition of a strict and nonstrict polynomial property, the recognition of the right and/or left finite processing, and the construction of the determining regular language for an algebra with finite processing and for monomial subalgebras of a free associative algebra and certain completely automaton algebras. For an automaton monomial algebra, the author constructs the left syzygy module of a finite system of elements and the Gröbner basis of a finitely generated left ideal; also, some algorithmic problems are solved. 相似文献
15.
For all boundary modules of the Koszul complex of a monomial sequence we construct complexes, which we call Taylor complexes.
For a monomial d-sequences these complexes provide free resolutions of the boundary modules. Let M be the ideal generated by a monomial d-sequence. We use the Taylor complexes to construct minimal free resolutions of the Rees algebra and the associated graded
ring of M.
Received: 13 November 1997 / Revised version: 6 March 1998 相似文献
16.
Finite vs affine W-algebras 总被引:1,自引:0,他引:1
In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct,
in the most general framework, the Zhu algebra ZhuΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra ZhuH V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal
enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras
(resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra ZhuH R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic
properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the
most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the
machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6
we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence
of three definitions of a finite W-algebra is established.
“I am an old man, and I know that a definition cannot be so complicated.” I.M. Gelfand (after a talk on vertex algebras in
his Rutgers seminar) 相似文献
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18.
D. I. Piontkovski 《Functional Analysis and Its Applications》2001,35(2):133-137
A family of examples is obtained which shows that, generally, it is impossible to decide for known Hilbert series of a qudratic algebra and its dual algebra whether or not this algebra has the Koszul property. The simplest example is given by two finitely generated algebras concentrated at the degrees not exceeding five; one of these algebras is monomial, while the other is not a Koszul algebra. This proves the conjecture of Positselskii. 相似文献
19.
In this paper we find necessary and sufficient conditions for an algebra to be a monomial algebra. These are conditions on finite abelian group gradings of the algebra and the first Hochschild cohomology group of the associated covering algebra of the grading. In particular, for a certain class of algebras, we show that an algebra Λ is a monomial algebra if and only if H 1(Λ,Λ) is the reduced Euler characteristic of the quiver of Λ. The proof of this uses the theory of noncommutat ive Gröbner bases. 相似文献
20.
The algebra of basic covers of a graph G, denoted by [`(A)](G)\bar{A}(G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension
of [`(A)](G)\bar{A}(G) in terms of the combinatorics of G. As a consequence, we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Furthermore,
we show that if the graph is bipartite, then [`(A)](G)\bar{A}(G) is a homogeneous algebra with straightening laws, and thus it is Koszul. Finally, we characterize the Cohen–Macaulay property
and the Castelnuovo–Mumford regularity of the edge ideal of a certain class of graphs. 相似文献