An inductive approach to representations of complex reflection groups G(m, 1, n)

We propose an inductive approach to the representation theory of the chain of complex reflection groups G(m, 1, n). We obtain the Jucys-Murphy elements of G(m, 1, n) from the Jucys-Murphy elements of the cyclotomic Hecke algebra and study their common spectrum using representations of a degenerate cyclotomic affine Hecke algebra. We construct representations of G(m, 1, n) using a new associative algebra whose underlying vector space is the tensor product of the group ring ?G(m, 1, n) with a free associative algebra generated by the standard m-tableaux.     

Maximal solvable subalgebras of Lie superalgebras gl(m|n) and sl(m|n)

Moscow Institute for New Technologies in Equipment. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 2, pp. 92–94, April–June, 1994.     

On the functional equationS(m,n)F[n/(m,n)] =F(n)h[n/(m,n)]

In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σ _d|n ?(d)h(n/d) andS(m n) = Σd|(m, n) ?(d)h(n/d). The well-known Hölder's identity is a special case of this functional equation (?(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions.     

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, ${{\rm Sp}(2,\mathbb {R})}Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of K?hler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.     

(m,n)模糊超环

介绍(m,n)超环等一些相关概念,之后将(m,n)超环模糊化,给出(m,n)模糊超环的定义,初步探讨(m,n)模糊超环的结构和性质,分析(m,n)模糊超环在同态下的不变性。     

Intrinsic normalizations of seminonholonomic complexes SNGr(m,n, (m+1) (n−m)−ϱ)

V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 28, No. 2, pp. 299–314, April–June, 1988.     

广义(m,n)超环

本文研究了广义(m,n)超环,n元正则关系以及n元强正则关系等的一些性质.利用广义(m,n)超环间的同态关系以及正则和强正则关系,得到了(m,n)子超环和(m,n)超理想的不变性,广义(m,n)超环的商结构,以及构成商超环和商环的充分必要条件,推广了文献[5]的一些结果.     

Quasi-ALE Metrics with Holonomy SU(m) and Sp(m)

Let G be a finite subgroup of U(m),and X a resolution of ^m /G. We define aspecial class of Kähler metrics g on Xcalled Quasi Asymptotically Locally Euclidean (QALE) metrics. Thesesatisfy a complicated asymptotic condition, implying that gis asymptotic to the Euclidean metric on ^m /G away fromits singular set. When ^m /Ghas an isolated singularity,QALE metrics are just ALE metrics. Our main result is an existencetheorem for Ricci-flat QALE Kähler metrics: if G is afinite subgroup of SU(m) and X a crepant resolution of ^m /G, then there is a unique Ricci-flat QALE Kähler metric on X in each Kähler class.This is proved using a version of the Calabi conjecture for QALEmanifolds. We also determine the holonomy group of the metrics in termsof G.     

Maximal (semi-)lattices of fractions and injective hulls

We show that the maximal (semi-)lattice of fractions of a (semi-)lattice may be obtained as a canonical subsemilattice of the bicommutator of an injective hull of the (semi-)lattice under consideration, whithin a suitably defined category of semilattices. The construction is roughly similar to that of a maximal semigroup of quotients, but there are significant differences due to the commutativity and idempotency of (semi-)lattice operations. In particular, the construction is effective in the sense that it avoids the use of Zorn's lemma to obtain the required injective hulls.