Canonical Representations and Overgroups for Hyperboloids of One Sheet and Lobachevsky Spaces

We define canonical representations for the hyperboloid of one sheet and for the Lobachevsky space =G/K where G=SO₀(1,n–1), H=SO₀(1,n–2) and K=SO(n–1), as the restriction to G of representations associated with a cone of overgroups and for and respectively. We determine explicitly the interaction of Lie operators of with operators intertwining canonical representations and representations of G associated with a cone.Mathematics Subject Classifications (2000) 43A85, 22E46, 53D55.V. F. Molchanov: Supported by grants of the Netherlands Organization for Scientific Research (NWO): 047-008-009, the Russian Foundation for Basic Research (RFBR): 01-01-00100-a, the Minobr RF: E00-1.0-156, the NTP Univ. Rossii: ur04.01.037.     

Canonical and Boundary Representations on the Lobachevsky Plane

Canonical representations on Hermitian symmetric spaces G/K were introduced by Vershik, Gelfand and Graev and Berezin. They are unitary. We study canonical representations in a wider sense. In this paper we restrict ourselves to a crucial example – the Lobachevsky plane: G=SU(1,1), K=U(1). Canonical representations are labelled by the complex parameter (Vershik–Gelfand–Graev's representations correspond to –3/2<<0). We decompose the canonical representations into irreducible components. The decomposition includes boundary representations generated by the canonical representations. So we study these boundary representations themselves. The decomposition of boundary representations is closely connected with the meromorphic structure of Poisson and Fourier transforms associated with canonical representations. In particular, second-order poles give second-order Jordan blocks. Finally, we give a full decomposition of the Berezin transform using generalized powers (Pochhammer symbols) instead of usual powers of .     

Canonical and Boundary Representations on a Hyperboloid of One Sheet

For the hyperboloid of one sheet X=G/H, G=SO₀(1,2), H=SO₀(1,1), canonical representations R _,, C, =0,1, are defined as the restrictions to G of representations of the overgroup =SO₀(2,2) associated with a cone. They act on the torus containing two copies of X as open G-orbits. We study boundary representations generated by R _,. For some , they contain Jordan blocks. The decomposition of R _, into irreducible constituents includes a finite number (depending on ) of irreducible parts of the boundary representations.     

Canonical Representations on Lobachevsky Spaces: An Interaction with an Overalgebra

We define canonical representations R _λ , , for the Lobachevsky space ℒ=G/K of dimension n−1 where G=SO₀(n−1,1), K=SO(n−1), as the restriction to G of maximal degenerate series representations of the overgroup . We determine explicitly the interaction of Lie operators of with operators intertwining canonical representations and representations of G associated with a cone. Supported by the Russian Foundation for Basic Research: grants No. 05-01-00074a and No. 05-01-00001a, the Netherlands Organization for Scientific Research (NWO): grant 047-017-015, the Scientific Program “Devel. Sci. Potent. High. School”: project RNP.2.1.1.351 and Templan No. 1.2.02.     

Lie Representations and an Algebra Containing Solomon's

We introduce and study a Hopf algebra containing the descent algebra as a sub-Hopf-algebra. It has the main algebraic properties of the descent algebra, and more: it is a sub-Hopf-algebra of the direct sum of the symmetric group algebras; it is closed under the corresponding inner product; it is cocommutative, so it is an enveloping algebra; it contains all Lie idempotents of the symmetric group algebras. Moreover, its primitive elements are exactly the Lie elements which lie in the symmetric group algebras.     

Lifts of Diffeomorphisms of Orbit Spaces for Representations of Compact Lie Groups

A structure of the group of diffeomorphisms for the orbit space of an orthogonal representation G> O(V) of a compact Lie group G is studied. For a finite G, it is proved that each diffeomorphism of the orbit space V/G has a smooth lift to V and the component group of the group of Diff(V/G is described, especially in the case when G is a finite Coxeter group. Similar results are obtained for the isotropy representations of some symmetric spaces.     

Projective Representations and Pure Pseudorepresentations of Hermitian Symmetric Simple Lie Groups

It is shown that an arbitrary irreducible continuous unitary projective representation of a simple Hermitian symmetric Lie group is generated by a strongly continuous pure unitary pseudorepresentation of the adjoint group of the Lie group.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 140–146.Original Russian Text Copyright © 2005 by A. I. Shtern.     

限制Hamiltonian型及Contact型李代数的不可约表示

设L=H（2r;1）或K（2r+1;1）是定义在特征p>2的代数封闭域F上的限制Hamiltonian型或Contact型李代数.在对广义Jacobson-Witt代数及特殊代数不可约表示的研究基础上,通过定义L的如下阶化:L=L_[q],I,其中I是{1,2,…,r}的子集,得到当p-特征函数χ是正则半单时,所有不可约U_χ（L）-模都是从不可约U_χ(L_[O].I)-模诱导的.     

效应代数的表示

本文讨论了抽象效应代数的表示问题. 对于一个抽象效应代数(E,⊕, 0, 1), 如果存在一个Hilbert 空间 H 和一个单态射 φ:E →ε(H), 那么称 E 为可表示的且称(φ,H) 是E 的一个表示, 其中ε(H) 表示 H 上所有正压缩算子构成的效应代数. 给出了一些可表示的和不可表示的效应代数的例子, 证 明了非空集 X 上的任一模糊集系统 F 和Boolean 代数B_X 都是可表示的效应代数.     

一类Witt代数的包络代数的表示

设C为复数域,P,q∈C,且pq是m次本原单位根．我们构造了一个Zm-分次模类V(a,b),它为Witt代数的包络代数的(P,q)变形U(Wpq)的Zm-分次模类,并证明了任何一个Zm-分次U(Wpq)-模都与某个V(a,b)同构．     

Automorphism group and representation of a twisted multi-loop algebra

Let G be the complexification of the real Lie algebra so（3） and A = C[t1^±1, t2^±1] be the Lau-ent polynomial algebra with commuting variables. Let L：（t1, t2, 1） = G c .A be the twisted multi-loop Lie algebra. Recently we have studied the universal central extension, derivations and its vertex operator representations. In the present paper we study the automorphism group and bosonic representations ofL（t1, t2, 1）.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

Pontryagin Classes of Generalized Symmetric Spaces

The real characteristic Pontryagin classes of generalized symmetric spaces with simple compact principal Lie groups are calculated in this paper.     

量子环面上斜导子李代数的表示

记L为量子环面上的斜导子李代数，本文构造了一族从sl2-模到L-模的函子F^αg，并对L-模F^αg（V）的结构进行了完全刻画，最后给出了L-模F^αg1（V）与F^βg2（W）同构的充分必要条件。     

Extremal Bases for the Adjoint Representations of the Simple Lie Algebras

We construct n distinct weight bases, which we call extremal bases, for the adjoint representation of each simple Lie algebra 𝔤 of rank n: One construction for each simple root. We explicitly describe actions of the Chevalley generators on the basis elements. We show that these extremal bases are distinguished by their “supporting graphs” in three ways. (In general, the supporting graph of a weight basis for a representation of a semisimple Lie algebra is a directed graph with colored edges that describe the supports of the actions of the Chevalley generators on the elements of the basis.) We show that each extremal basis constructed is essentially the only basis with its supporting graph (i.e., each extremal basis is solitary), and that each supporting graph is a modular lattice. Each extremal basis is shown to be edge-minimizing: Its supporting graph has the minimum number of edges. The extremal bases are shown to be the only edge-minimizing as well as the only modular lattice weight bases (up to scalar multiples) for the adjoint representation of 𝔤. The supporting graph for an extremal basis is shown to be a distributive lattice if and only if the associated simple root corresponds to an end node for a “branchless” simple Lie algebra, i.e., type A, B, C, F, or G. For each extremal basis, basis elements for the Cartan subalgebra are explicitly expressed in terms of the h _i Chevalley generators.