Tensoring with Infinite-Dimensional Modules in $\mathcal {O}_0$

We show that the principal block O₀\mathcal {O}_0 of the BGG category O\mathcal {O} for a semisimple Lie algebra \frak g\frak g acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category O\mathcal {O}. We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on O₀\mathcal {O}_0. Furthermore, all this generalises to parabolic subcategories of O₀\mathcal {O}_0. As an example, we present some explicit computations for the algebra \fraksl₃\frak{sl}_3.     

Maximum scattered linear sets of pseudoregulus type and the Segre variety \mathcal{S}_{n,n}

In this paper we study a family of scattered $\mathbb{F}_{q}$ -linear sets of rank tn of the projective space PG(2n?1,q ^t ) (n≥1, t≥3), called of pseudoregulus type, generalizing results contained in Lavrauw and van de Voorde, Des. Codes Crypt. 20(1) (2013) and in Marino et al. J. Combin. Theory, Ser. A 114:769–788 (2007). As an application, we characterize, in terms of the associated linear sets, some classical families of semifields: the Generalized Twisted Fields and the 2-dimensional Knuth semifields.     

$\mathcal{P}$-kernel normal systems for $\mathcal{P}$-inversive semigroups

As a generalization of Preston’s kernel normal systems, P\mathcal{P}-kernel normal systems for P\mathcal{P}-inversive semigroups are introduced, and strongly regular P\mathcal{P}-congruences on P\mathcal{P}-inversive semigroups in terms of their P\mathcal{P}-kernel normal systems are characterized. These results generalize the corresponding results for P\mathcal{P}-regular semigroups and P\mathcal{P}-inversive semigroups.     

Finite rank operators in Jacobson radical \mathcal{R}_{\mathcal{N} \otimes \mathcal{M}}

In this paper we investigate finite rank operators in the Jacobson radical of Alg( ), where are nests. Based on the concrete characterizations of rank one operators in Alg( ) and , we obtain that each finite rank operator in can be written as a finite sum of rank one operators in and the weak closure of equals Alg( ) if and only if at least one of is continuous.     

Towards a classification of modular compactifications of \mathcal {M}_{g,n}

The moduli space of smooth curves admits a beautiful compactification $\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,n}$ by the moduli space of stable curves. In this paper, we undertake a systematic classification of alternate modular compactifications of $\mathcal{M}_{g,n}$ . Let $\mathcal{U}_{g,n}$ be the (non-separated) moduli stack of all n-pointed reduced, connected, complete, one-dimensional schemes of arithmetic genus g. When g=0, $\mathcal{U}_{0,n}$ is irreducible and we classify all open proper substacks of $\mathcal{U}_{0,n}$ . When g≥1, $\mathcal{U}_{g,n}$ may not be irreducible, but there is a unique irreducible component $\mathcal{V}_{g,n} \subset\mathcal{U}_{g,n}$ containing $\mathcal{M}_{g,n}$ . We classify open proper substacks of $\mathcal {V}_{g,n}$ satisfying a certain stability condition.     

On Projective Modules in Category \mathcal{O}_{int} of Quantum \mathfrak{sl}_{2}

The restriction of a Verma module of ${\bf U}(\mathfrak{sl}_3)$ to ${\bf U}(\mathfrak{sl}_2)$ is isomorphic to a Verma module tensoring with all the finite dimensional simple modules of ${\bf U}(\mathfrak{sl}_2)$ . The canonical basis of the Verma module is compatible with such a decomposition. An explicit decomposition of the tensor product of the Verma module of highest weight 0 with a finite dimensional simple module into indecomposable projective modules in the category $\mathcal O_{\rm{int}}$ of quantum $\mathfrak{sl}_2$ is given.     

The reducible Specht modules for the Hecke algebra \mathcal{H}_{\mathbb{C},{-}1}(\mathfrak{S}_{n})

The reducible Specht modules for the Hecke algebra $\mathcal {H}_{\mathbb{F},q}(\mathfrak{S}_{n})$ have been classified except when q=?1. We prove one half of a conjecture which we believe classifies the reducible Specht modules when q=?1.     

Extremal effective divisors on \overline{\mathcal M}_{1,n}

For every \(n\ge 3\) , we exhibit infinitely many extremal effective divisors on \(\overline{\mathcal M}_{1,n}\) , the Deligne-Mumford moduli space parameterizing stable genus one curves with \(n\) ordered marked points.     

Classification of quasifinite\mathcal{W}_\infty -modules-modules

It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ₁ ^m …D ₁ ^m foru ∈F[Γ] (the group algebra), andm _i≥0 with , whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.     

强W-Gorenstein 复形

对自正交模类$\mathcal{W}$,引入了强$\mathcal{W}$-Gorenstein复形的概念.给出了强$\mathcal{W}$-Gorenstein复形的刻画,并将其应用到强Gorenstein内射复形.     

关于$\mathcal{F}$-S-可补子群

设$\mathcal{F}$是一个群类. 群$G$的子群$H$称为在$G$中$\mathcal{F}$-S-可补的，如果存在$G$的一个子群$K$,使得$G=HK$且$K/K\cap{H_G}\in\mathcal{F}$, 其中$H_G=\bigcap_{g\in G}H^g$是包含在$H$中的$G$的最大正规子群．本文利用子群的$\mathcal{F}$-S-可补性, 给出了有限群的可解性, 超可解性和幂零性的一些新的刻画. 应用这些结果, 我们可以得到一系列推论, 其中包括有关已知的著名结果.     

On the reflexivity of $$\mathcal {P}_{w}(^{n}E;F)$$

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.     

Construction of scrambled polynomial lattice rules over \mathbb{F}_{2} with small mean square weighted \mathcal{L}_{2} discrepancy

The \(\mathcal{L}_{2}\) discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Wo?niakowski to take into account the relative importance of the discrepancy of lower dimensional projections. As known under the name of quasi-Monte Carlo methods, point sets with small weighted \(\mathcal{L}_{2}\) discrepancy are useful in numerical integration. This study investigates the component-by-component construction of polynomial lattice rules over the finite field \(\mathbb{F}_{2}\) whose scrambled point sets have small mean square weighted \(\mathcal{L}_{2}\) discrepancy. An upper bound on this discrepancy is proved, which converges at almost the best possible rate of N ^?2+δ for all δ>0, where N denotes the number of points. Numerical experiments confirm that the performance of our constructed polynomial lattice point sets is comparable or even superior to that of Sobol’ sequences.     

Notes on the partition property of {\mathcal{P}_\kappa\lambda}

We investigate the partition property of ${\mathcal{P}_{\kappa}\lambda}$ . Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that ${\mathcal{P}_{\kappa}\lambda}$ carries a (λ ^κ , 2)-distributive normal ideal without the partition property, then λ is ${\Pi^1_n}$ -indescribable for all n?<?ω but not ${\Pi^2_1}$ -indescribable. (2) If cf(λ) ≥?κ, then every ineffable subset of ${\mathcal{P}_{\kappa}\lambda}$ has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over ${\mathcal{P}_{\kappa}\lambda}$ has the partition property.     

Representation theory of $\mathcal{W}$-algebras

We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10     

$\mathcal{C}^{2}$ surface diffeomorphisms have symbolic extensions

We prove that C²\mathcal{C}^{2} surface diffeomorphisms have symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. Following the strategy of Downarowicz and Maass (Invent. Math. 176:617–636, 2009) we bound the local entropy of ergodic measures in terms of Lyapunov exponents. This is done by reparametrizing Bowen balls by contracting maps in a approach combining hyperbolic theory and Yomdin’s theory.     

Accelerated dynamical approaches for finding the unique positive solution of K S $\mathcal {K}\mathcal {S}$ -tensor equations

Numerical Algorithms - A new class of tensors called $\mathcal {K}\mathcal {S}$ -tensors, which is a subset of non-singular $\mathcal {P}$ -tensors and generalization of ${\mathscr{H}}^{+}$...