Weighted Projective Lines as Fine Moduli Spaces of Quiver Representations

We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their canonical bundles, and show that they are rarely tilting. We also give a moduli construction for these total spaces for weighted projective lines with three orbifold points.     

Quasicrystals,aperiodic order,and groupoid von Neumann algebras

We introduce tight binding operators for quasicrystals that are parametrized by Delone sets. These operators can be regarded in a natural operator algebra framework that encodes the long range aperiodic order. This algebraic point of view allows us to study spectral theoretic properties. In particular, the integrated density of states of the tight binding operators is related to a canonical trace on the associated von Neumann algebra. To cite this article: D. Lenz, P. Stollmann, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1131–1136.     

Canonical measures on the moduli spaces of compact Riemann surfaces

We study some explicit relations between the canonical line bundle and the Hodge bundle over moduli spaces for low genus. This leads to a natural measure on the moduli space of every genus which is related to the Siegel symplectic metric on Siegel upper half-space as well as to the Hodge metric on the Hodge bundle.     

Canonical basis linearity regions arising from quiver representations

In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author [Math. Z. 237 (2001) 639] arising from representations of quivers. We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig's approach to the canonical basis and the string parametrization of the canonical basis, are given on a large cone by linear functions arising from these monomial bases for a quantized enveloping algebra.     

On the noncommutative Donaldson-Thomas invariants arising from brane tilings

Given a brane tiling, that is a bipartite graph on a torus, we can associate with it a quiver potential and a quiver potential algebra. Under certain consistency conditions on a brane tiling, we prove a formula for the Donaldson-Thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra. We relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling. We prove that the same consistency conditions imply that the quiver potential algebra is a 3-Calabi-Yau algebra. We also formulate a rationality conjecture for the generating functions of the Donaldson-Thomas type invariants.     

Flatness Properties of S-Posets

We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence we show that a weakly shod algebra which is not quasi-tilted of canonical type is simply connected if and only if its first Hochschild cohomology group vanishes.     

Canonical characters on simple graphs

A multiplicative functional on a graded connected Hopf algebra is called the character. Every character decomposes uniquely as a product of an even character and an odd character. We apply the character theory of combinatorial Hopf algebras to the Hopf algebra of simple graphs. We derive explicit formulas for the canonical characters on simple graphs in terms of coefficients of the chromatic symmetric function of a graph and of canonical characters on quasi-symmetric functions. These formulas and properties of characters are used to derive some interesting numerical identities relating multinomial and central binomial coefficients.     

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen’s metric and interprete it as a local index theorem for the family of \(\overline\partial\)-operators in the associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kähler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein–Maass series. The cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten’s volume computation.     

Geometric aspects of the moduli space of Riemann surfaces

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore, the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.     

Operator pencils on the algebra of densities

We continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role played by the geometry of the extended manifold where the algebra of densities is a special class of functions. Firstly we consider basic examples. We give a projective line of diff(M)-equivariant pencil liftings for first order operators and describe the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO) pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO pencil lifting to describe all regular proj-equivariant pencil liftings. In particular, the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian.     

On commutators of Wick products on CCR and CAR algebras

In this paper we study the q-commutator of Wick products on the CCR (canonical commutation relation) algebra and on the CAR (canonical anticommutation relation) algebra. We obtain a formula on q-commutator of Wick products in terms of Wick products of monomials of lower order. Moreover, we give a relation of double q-commutator.     

Congruence-preserving functions on Stone and Kleene algebras

We investigate local polynomial functions on Stone algebras and on Kleene algebras. We find a generating set for the clone of all local polynomial functions. We also represent local polynomial functions on a given algebra by polynomial functions of some canonical extension of this algebra.     

A framization of the Hecke algebra of type B

We introduce a framization of the Hecke algebra of type B. For this framization, we construct a faithful tensorial representation and two linear bases. We also construct a Markov trace on such an algebra, and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.     

Moduli Spaces of Curves with Quasi-Symmetric Weierstrass GAp Seqnences

In this paper we give an explicit construction of the moduli space of the pointed complete Gorenstein curves of arithmetic genus g with a given quasi-symmetric Weierstrass semigroup, that is, a Weierstrass semigroup whose last gap is equal to 2g – 2. We identify such a curve with its image under the canonical embedding in the (g – 1)-dimensional projective space. By normalizing the coefficients of the quadratic relations and by constructing Gröbner bases of the canonical ideal, we obtain the equations of the moduli space in terms of Buchberger's criterion. Moreover, by analyzing syzygies of the canonical ideal we establish criteria that make the computations less expensive.     

Indexed algebras associated to a log structure and a theorem of p-descent on log schemes

After discussing gradings by sheaves of degrees, we associate to any log scheme a canonical invertible sheaf endowed with a certain multiplicative structure, which we call its associated graded algebra. In the relative case we construct a canonical connection on this algebra. In the log smooth case over a base of positive characteristic p, we study integrable and p-integrable graded modules over this algebra, and establish a Cartier type p-descent theorem, generalizing previous results of Ogus. We apply it to give an alternate proof of a result of Tsuji on closed forms fixed by the Cartier operator Received: 15 January 1999 / Revised version: 30 September 1999     

A crossed product of the canonical anticommutation relations algebra in the Cuntz algebra

We show that the Cuntz algebra can be represented as a crossed product of the canonical anticommutation relations algebra by an endomorphism.     

Irreducible Morphisms Between Modules over a Repetitive Algebras

We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra.     

A remark on the liftable derivation of moduli algebras of isolated hypersurface singularities

An example is given to show that not every derivation in the nilradical of the Lie algebra of derivations of moduli algebras can be liftable and the dimension of the nilradical of the Lie algebra of derivations of moduli algebras is not a topological invariant for an isolated hypersurface singularity.

     

平均场倒向重随机微分方程及其应用

研究了平均场倒向随重机微分方程, 得到了平均场倒向重随机微分方程解的存在唯一性.基于平均场倒向重随机微分方程的解, 给出了一类非局部随机偏微分方程解的概率解释.讨论了平均场倒向重随机系统的最优控制问题, 建立了庞特利亚金型的最大值原理.最后讨论了一个平均场倒向重随机线性二次最优控制问题, 展示了上述最大值原理的应用.     

扭Heisenberg-Virasoro代数上Hom-李代数结构

Hom-李代数是一类满足反对称和Hom-Jacobi等式的非结合代数.扭Heisenberg-Virasoro代数是次数不超过1的微分算子代数的中心扩张,它是一类重要的无限维李代数,与一些曲线的模空间有关.文章主要研究扭Heisenberg-Virasoro代数上Hom-李代数结构,确定了扭Heisenberg-Virasoro代数上存在非平凡的Hom-李代数结构.