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1.
Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K is the fixed point set of an involutive automorphism σ. This induces an involutive automorphism τ of the based loop space Ω(G). There exists a maximal torus T ⊂ G such that the canonical action of T × S
1 on Ω(G) is compatible with τ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat’s convexity theorem. Namely,
the images of Ω(G) and Ω(G)
τ
(fixed point set of τ) under the T × S
1 moment map on Ω(G) are equal. The space Ω(G)
τ
is homotopy equivalent to the loop space Ω(G/K) of the Riemannian symmetric space G/K. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in
\mathbbZ2 {\mathbb{Z}_2} of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [BS] had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved.
The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe [HHP]. 相似文献
2.
Letk be any field andG a finite group. Given a cohomology class α∈H
2(G,k
*), whereG acts trivially onk
*, one constructs the twisted group algebrak
αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z
n×Zn). This paper has two main results: First we prove that ifD=k
α
G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and
, the division algebraD=k
α
G is a product of cyclic algebras. Furthermore, ifD
p is ap-primary factor ofD, thenD
p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and
, the same result holds forD
p, p odd. Ifp=2 we show thatD
2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z
2×Z2n. 相似文献
3.
Wang Yanming 《数学学报(英文版)》1991,7(1):62-65
By using the classification theorem of finite simple groups, we have shown that “IfG is a finite group,H is a coprime operator group ofG, C
G(H)≤S(G), thenG is solvable.” As a direct corollary, we have completely proved the long-standing conjecture on fixed-point-free automorphism
group.
The author is grateful to Professor Chen Zhongmu for his supervision. 相似文献
4.
In this paper, we study a tower {A
n
G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA
n
G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA
n
G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the
‘generic case’. Finally we determine the Bratteli diagram of the tower {A
n
G(d): n} ≥ 1 (in the generic case). 相似文献
5.
6.
Consider a non-connected algebraic group G = G ⋉ Γ with semisimple identity component G and a subgroup of its diagram automorphisms Γ. The identity component G acts
on a fixed exterior component Gτ, id ≠ τ ∈ Γ by conjugation. In this paper we will describe the conjugacy classes and the
invariant theory of this action. Let T be a τ -stable maximal torus of G and its Weyl group W. Then the quotient space Gτ//G
is isomorphic to (T/(1 − τ )(T))/Wτ. Furthermore, exploiting the Jordan decomposition, the reduced fibres of this quotient map are naturally associated bundles
over semisimple G-orbits. Similar to Steinberg's connected and simply connected case [22] and under additional assumptions
on the fundamental group of G, a global section to this quotient map exists. The material presented here is a synopsis of
the Ph.D thesis of the author, cf. [15]. 相似文献
7.
Edward A. Bertram 《Israel Journal of Mathematics》1991,75(2-3):243-255
We prove first that if G is a finite solvable group of derived length d ≥ 2, then k(G) > |G|1/(2d−1), where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G(i): G(i+1)]
1
d−1
, where G(i) is theith derived group, leads to a |G|1/(2d−1) lower bound for k(G), from which we derive a |G|c/log
2log2|G| lower bound, independent of d(G). Finally, “almost logarithmic” lower bounds are found for solvable groups with a nilpotent
maximal subgroup, and for all Frobenius groups, solvable or not. 相似文献
8.
Julia Weber 《K-Theory》2005,36(1-2):169-207
We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K0 of the category of “ ϕ -endomorphisms of finitely generated free RΠ(G, X)-modules”. We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.
Received: February 2006 相似文献
9.
R. Chavosh Khatamy 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(1):60-65
The paper considers the associated bundle ξ = (G × KG/K, ρ
ξ
, G/K, G/K) and the tangent bundle τ
G/K
= (T
G/K
, π
G/K
, G/K, R
m
), and gives special examples of odd dimensional solvable Lie groups equipped with left invariant Riemannian metric. Some
conditions about existence of homogeneous geodesic vectors on the fiber space of ξ and τ
G/K
are proved. 相似文献
10.
V. D. Mazurov 《Algebra and Logic》2006,45(2):117-123
Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every
two non-commuting members of X generate a subgroup isomorphic to A4 or to A5. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they
generate a subgroup isomorphic to A4. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X)
and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A5 then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer
in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2)
The subgroup 〈XG〉 is a direct product of subgroups 〈C
α〉-generated by some connected components C
α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup
isomorphic to A5. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A5 or is cyclic of order 5.
Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental
Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President)
and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.
__________
Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006. 相似文献
11.
For a compactly generated LCA group G, it is shown that the setH(G) of all generalized characters on G equipped with the compact-open topology is a LCA group andH(G) = Ĝ (the dual group ofG) if and only ifG is compact. Both results fail for arbitrary LCA groups. Further, ifG is second countable, then the Gel’fand space of the commutative convolution algebraC
c
(G) equipped with the inductive limit topology is topologically homeomorphic toH(G). 相似文献
12.
Reinhard Bürger 《Monatshefte für Mathematik》1980,90(2):101-115
In [9]H. Reiter introduced functions of translation type on a locally compact group to show some functorial properties of the spaceS (G) of Schwartz-Bruhat functions. We shall investigate functorial properties of the Segal algebraP
1 (G) which is defined by means of a more general version of functions of translation type without using a norm. Furthermore we shall introduce a new Segal algebraE
1
(G) that is closely related to these functorial properties. 相似文献
13.
Annunziata Esposito 《Rendiconti del Circolo Matematico di Palermo》2004,53(3):437-442
Let u be harmonic in a simply connected domainG ⊂ ℝ2 and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth
functionu
1 and a second order uniformly elliptic operatorL with smooth coefficients in ℝ2, satisfying:u
1=u inK, Lu
1=0 in ℝ2. A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved. 相似文献
14.
Raphael Yuster 《Graphs and Combinatorics》2001,17(3):579-587
We prove that for every ε>0 and positive integer r, there exists Δ0=Δ0(ε) such that if Δ>Δ0 and n>n(Δ,ε,r) then there exists a packing of K
n
with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn
2 edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let αΔ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥αΔ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K
n
).
Received: March 15, 1999?Final version received: October 22, 1999 相似文献
15.
LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI
3(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I
2(G)I(H)+I(G)I(H)I(G)+I(H)I2(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH
2(G/H′, T)≤1, are computed. the subgroup ofG determined byI
n(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained 相似文献
16.
Ian M. Musson 《Israel Journal of Mathematics》1997,100(1):285-308
We describe the skew primitive elements in a multiparameter enveloping algebraU=U
q,p
−1 (g) and the links between cofinite maximal ideals in the corresponding quantum function algebra ℂ
q
[G]. These results are applied to determine the coradical filtration forU, and to obtain a moduli space for multiparameter Drinfeld doubles.
Research partially supported by NSA grant MDA 904-93-H3016. 相似文献
17.
Let M = G/K be a homogeneous differentiable manifold. We consider the homogeneous bundle = (G, π, G/K, K) and the tangent bundle τ
G/K of M = G/K, and give some results about the existence of homogeneous vectors on the fiber space of τ
G/K, for both cases of G semisimple and weakly semisimple.
相似文献
18.
V. M. Kopytov 《Algebra and Logic》2009,48(5):344-356
We create a method which allows an arbitrary group G with an infrainvariant system ℒ(G) of subgroups to be embedded in a group G* with an infrainvariant system ℒ(G*) of subgroups, so that G
α* ∩G ∈ ℒ(G) for every subgroup G
α* ∩G ∈ ℒ(G*) and each factor B/A of a jump of subgroups in ℒ(G*) is isomorphic to a factor of a jump in ℒ(G), or to any specified group H. Using this method, we state new results on right-ordered groups. In particular, it is proved that every Conrad right-ordered
group is embedded with preservation of order in a Conrad right-ordered group of Hahn type (i.e., a right-ordered group whose
factors of jumps of convex subgroups are order isomorphic to the additive group ℝ); every right-ordered Smirnov group is embedded
in a right-ordered Smirnov group of Hahn type; a new proof is given for the Holland–McCleary theorem on embedding every linearly
ordered group in a linearly ordered group of Hahn type. 相似文献
19.
Aschbacher’s localC(G; T) theorem asserts that ifG is a finite group withF*(G)=O
2(G), andTεSyl2(G), thenG=C(G; T)K(G), whereC(G; T)=〈N
G
(T
0)|1≠T
0 charT〉 andK(G) is the product of all near components ofG of typeL
2(2
n
) orA
2
n
+1. Near components are also known asχ-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher’s theorem in the case thatG is aK-group, i.e., in the case that every simple section ofG is isomorphic to one of the known simple groups. Our proof relies on a result of Meierfrankenfeld and Stroth [MS] on quadratic
four-groups and on the Baumann-Glauberman-Niles theorem, for which Stellmacher [St2] has given an amalgam-theoretic proof.
Apart from those results, our proof is essentially self-contained.
For John Thompson
Supported in part by NSF grant #DMS 89-03124, by DIMACS, an NSF Science and Technology Center, funded under contract STC-88-09648,
and by NSA grant #MDA-904-91-H-0043. Prof. Gorenstein died on August 26, 1992. 相似文献
20.
LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT
a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T
a, G/Γ,θ) is weak-mixing. (2) (T
a, G/Γ) is topologically weak-mixing. (3) (G
u, G/Γ) is uniquely ergodic. (4) (G
u, G/Γ,θ) is ergodic. (5) (G
u, G/Γ) is point transitive. (6) (G
u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T
a, G/Γ,θ) is ergodic” may be added to the above list.
The authors were partially supported by NSF grant MCS 75-05250. 相似文献