On a Group Graded Version of BGG

A major result in Algebraic Geometry is the theorem of Bernstein–Gelfand–Gelfand that states the existence of an equivalence of triangulated categories: gr _Λ ? 𝒟^b(Coh ?ⁿ), where gr _Λ denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and 𝒟^b(Coh ?ⁿ) is the derived category of bounded complexes of coherent sheaves on projective space ?ⁿ.

Generalizations of this result were obtained in Martínez-Villa and Saorín (2004 Martínez-Villa , R. , Saorín , M. ( 2004 ). Koszul equivalences and dualities . Pacific J. Math. 214 ( 2 ): 359 – 378 . [Google Scholar]) and from a different point of view, the theorem has been extended by Yanagawa (2004 Yanagawa , K. ( 2004 ). Derived category of squarefree modules and local cohomology with monomial ideal support . J. Math. Soc. Japan 56 : 289 – 308 . [Google Scholar]) to ?ⁿ-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra.

The aim of the article is to extend the results of Martínez-Villa and Saorín (2004 Martínez-Villa , R. , Saorín , M. ( 2004 ). Koszul equivalences and dualities . Pacific J. Math. 214 ( 2 ): 359 – 378 . [Google Scholar]) to group graded algebras in order to obtain a generalization of Yanagawa's results having in mind the application to other settings (Geigle and Lenzing, 1987 Geigle , W. , Lenzing , H. ( 1987 ). A class of weghted projective curves in representation theory of finite dimensional algebras, in singularities, representations of algebras, and vector bundles . Lecture Notes in Math. 1273 : 265 – 297 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Koszul Differential Graded Algebras and BGG Correspondence II

The concept of Koszul differential graded （DG for short） algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module Ak is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinite-dimensional, little is known about A. As shown in [15] （see also Proposition 2.2）, Ak is not compact if H（A） is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H（A） is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.     

Recursive properties of branching and BGG resolution

Recurrence relations for branching coefficients are based on a certain decomposition of the singular element. We show that this decomposition can be used to construct parabolic Verma modules and to obtain the generalized Weyl-Verma formulas for characters. We also demonstrate that the branching coefficients determine the generalized Bernstein-Gelfand-Gelfand resolution.     

Comparison of polynomials and almost hypoellipticity

The paper investigates the hypoellipticity of polynomials in terms of comparisons.     

Nonlinear eigenvalues and analytic hypoellipticity

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non-self-adjoint operators has a nonzero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.     

Remarks about global analytic hypoellipticity

We present a characterization of the operators

which are globally analytic hypoelliptic on the torus. We give information about the global analytic hypoellipticity of certain overdetermined systems and of sums of squares.

     

Macdonald polynomials and BGG reciprocity for current algebras

We study the category $\mathcal I _{\mathrm{gr }}$ of graded representations with finite-dimensional graded pieces for the current algebra $\mathfrak{g }\otimes \mathbf{C }[t]$ where $\mathfrak{g }$ is a simple Lie algebra. This category has many similarities with the category $\mathcal O $ of modules for $\mathfrak{g }$ , and in this paper, we prove an analog of the famous BGG duality in the case of $\mathfrak{sl }_{n+1}$ .     

On global hypoellipticity of degenerate elliptic operators

We consider a class of degenerate elliptic operators on a torus and prove that global hypoellipticity is equivalent to an algebraic condition involving Liouville vectors and simultaneous approximability. For another class of operators we show that the zero order term may influence global hypoellipticity. Received August 13, 1997     

Non-analytic hypoellipticity in the presence of symplecticity

Here we construct non-analytic solutions to a class of hypoelliptic operators with symplectic characteristic set and in the form of a sum of squares of real analytic vector fields.

     

Hyperfunctions and (analytic) hypoellipticity

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known “sum of squares” operators, which satisfy Hörmander’s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).     

Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations

This paper studies the Gevrey regularity of weak solutions of a class of linear and semi-linear Fokker-Planck equations.     

Compactness in kinetic transport equations and hypoellipticity

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L¹ satisfying some appropriate transport relation

     

An analogue of the BGG resolution for locally analytic principal series

Let G be a connected reductive quasi-split algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for G(L). This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity.     

Rough hypoellipticity for the heat equation in Dirichlet spaces

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms, which satisfy mild assumptions concerning (1) the existence of cut-off functions, (2) a local ultracontractivity hypothesis, and (3) a weak off-diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussions on the existence of locally bounded heat kernel; $L^{\infty }$ structure results for ancient (local weak) solutions of the heat equation.     

分次环的分次Excellent扩张

本文引进了分次环的分次Excellent扩张概念,设S=⊕_(g∈G)S_g是R=⊕_(g∈G)R_g的分次Excellent扩张,证明了S是分次右V-环当且仅当R是分次右V-环,S是分次PS-环当且仅当R是分次PS-环,S是分次Von Neumann正则环当且仅当R是分次Von Neumann正则环。