Poincaré's inequality and Sobolev spaces with monomial weights

In this paper, we use a weighted version of Poincaré's inequality to study density and extension properties of weighted Sobolev spaces over some open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$. Additionally, we study the specific case of monomial weights $w(x_1,\text{…},x_N)={\prod }_{i=1}^N{\left|x_i\right|}^{a_i},a_i\ge 0$, showing the validity of a weighted Poincaré inequality together with some embedding properties of the associated weighed Sobolev spaces.     

Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.     

二次截面Nakayama代数的McCoy性

Zhou Yuye;Cheng Zhi(School of Mathematics and Statistics,Anhui Normal University,Wuhu 241003,China)     

Normalization of monomial ideals and Hilbert functions

We study the normalization of a monomial ideal, and show how to compute its Hilbert function (using Ehrhart polynomials) if the ideal is zero dimensional. A positive lower bound for the second coefficient of the Hilbert polynomial is shown.

     

The Noetherian property in some quadratic algebras

We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations--namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2--and on the Ufnarovskii graph . The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.

     

Proper Laurent monomial property of acyclic cluster algebras

Using the unfolding method, following the idea of [4 Cerulli Irelli, G., Keller, B., Labardini-Fragoso, D., Plamondon, P.-G. (2013). Linear independence of cluster monomials for skew-symmetric cluster algebras. Compos Math. 149(10):1753–1764. DOI:10.1112/S0010437X1300732X.[Crossref], [Web of Science ®] , [Google Scholar]] and [5 Cerulli Irelli, G., Labardini-Fragoso, D. (2012). Quivers with potentials associated to triangulated surfaces, part III: Tagged triangulations and cluster monomials. Compos. Math. 148(06):1833–1866. DOI:10.1112/S0010437X12000528.[Crossref] , [Google Scholar]], we prove that all acyclic skew-symmetrizable cluster algebras have proper Laurent monomial property. As corollaries we prove that cluster monomials are linearly independent and they form an atomic base in case the cluster algebra is of finite type.     

Hyper-reflexivity of free semigroupoid algebras

As a generalization of the free semigroup algebras considered by Davidson and Pitts, and others, the second author and D.W. Kribs initiated a study of reflexive algebras associated with directed graphs. A free semigroupoid algebra is generated by a family of partial isometries, and initial projections, which act on a generalized Fock space spawned by the directed graph . We show that if the graph is finite, then is hyper-reflexive.

     

Monomial Hopf algebras over fields of positive characteristic

In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on Cd(n) are completely classified. Moreover, we construct a Hopf algebras filtration on Cd(n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.     

Trees, parking functions, syzygies, and deformations of monomial ideals

For a graph , we construct two algebras whose dimensions are both equal to the number of spanning trees of . One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to -parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

     

Elliptic hydrodynamics and quadratic algebras of vector fields on a torus

We construct a quadratic Poisson algebra of Hamiltonian functions on a two-dimensional torus compatible with the canonical Poisson structure. This algebra is an infinite-dimensional generalization of the classical Sklyanin-Feigin-Odesskii algebras. It yields an integrable modification of the two-dimensional hydrodynamics of an ideal fluid on the torus. The Hamiltonian of the standard two-dimensional hydrodynamics is defined by the Laplace operator and thus depends on the metric. We replace the Laplace operator with a pseudodifferential elliptic operator depending on the complex structure. The new Hamiltonian becomes a member of a commutative bi-Hamiltonian hierarchy. In conclusion, we construct a Lie bialgebroid of vector fields on the torus. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 355–370, March, 2007.     

Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras

The aim of this article is to attach to the set of L-S paths of type in a canonical way a basis of the corresponding representation . This basis has some nice algebraic-geometric properties. For example, it is compatible with restrictions to Schubert varieties and has the ``standard monomial property'. As a consequence we get new simple proofs of the normality of Schubert varieties, the surjectivity of the multiplication map or the restriction map for sections of a line bundle on Schubert varieties. Other applications to the defining ideal of Schubert varieties and associated Groebner basis will be discussed in a forthcoming paper.

     

CW-resolutions of monomial ideals that are supported on face posets

Given a monomial ideal I with minimal free resolution ? supported in characteristic p>0 on a CW-complex X with regular 2-skeleton, in general it is not the case that the face poset of X, P(X), also supports ? in the sense of Clark and Tchernev. We construct a (not necessarily regular) CW-complex Y that also supports ? and such that the face poset P(Y) also supports ?.     

Simplicity of quadratic Lie conformal algebras

In this paper, simplicity of quadratic Lie conformal algebras is investigated. From the view point of the corresponding Gel’fand–Dorfman bialgebras, some su?cient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several new classes of infinite simple Lie conformal algebras. These results will be useful for classification purposes.     

Filtered deformations of the Frank algebras

In this paper we prove the rigidity of simple Lie algebras of the Frank series of characteristic 3 with the standard grading of depth 2 with respect to filtered deformations.     

Level algebras with bad properties

This paper can be seen as a continuation of the works contained in the recent article (J. Alg., 305 (2006), 949-956) of the second author, and those of Juan Migliore (math. AC/0508067). Our results are:

1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for , we will construct a codimension three, type two -vector of socle degree such that all the level algebras with that -vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each .

2). There exist reduced level sets of points in of type two whose artinian reductions all fail to have the WLP. Indeed, the examples constructed here have the same -vectors we mentioned in 1).

3). For any integer , there exist non-unimodal monomial artinian level algebras of codimension . As an immediate consequence of this result, we obtain another proof of the fact (first shown by Migliore in the above-mentioned preprint, Theorem 4.3) that, for any , there exist reduced level sets of points in whose artinian reductions are non-unimodal.