The Cohomology of the Complex of G-Invariant Forms onG -Manifolds. II

The cohomology of G-manifolds of the type M=P× _K (G/H), where G is a reductive Lie group, H and N are its closed subgroups, H is a normal subgroup of N, K=N/H, and P is a smooth principal K-bundle, are considered. In the case when the Lie algebras of H and N are reductive, the differential graded algebra C(M) introduced in the previous paper with the same title and having the same minimal model as one of the algebra of G-invariant forms on M is investigated. Moreover, the main theorem on the cohomology algebra of C(M) is proved under weaker conditions than those of the previous paper.     

Planar algebra of the subgroup-subfactor

We give an identification between the planar algebra of the subgroupsubfactor R ⋊ H ⊂ R ⋊ G and the G-invariant planar subalgebra of the planar algebra of the bipartite graph ★ _n , where n = [G: H]. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant of R ⋊ H ⊂ R ⋊ G in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor R ^G ⊂ R ^H and the G-invariant planar subalgebra of the planar algebra of the ‘flip’ of ★ _n .     

Associative conformal algebras with finite faithful representation

The classification of irreducible subalgebras of the associative conformal algebra Cend_N is presented in this paper. The structure theory of associative conformal algebras with finite faithful representation is developed.     

Invariant differential operators on symplectic grassmann manifolds

LetM _2n,r denote the vector space of real or complex2n×r matrices with the natural action of the symplectic group Sp _2n , and letG=G _n,r =Sp _2n ×M _2n,r denote the corresponding semi-direct product. For any integerp with 0≤p≤n−1, letH denote the subgroupG _p,r ×Sp _2n−2p ofG. We explicitly compute the algebra of left invariant differential operators onG/H, and we show that it is a free algebra if and only ifr≤2n−2p+1. We also give orthogonal analogues of these results, generalizing those of Gonzalez and Helgason [3]. Partially supported by NSF grant DMS-9101358 This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990.     

Straightening and bounded cohomology of hyperbolic groups

It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology Hⁿ_b(G,\Bbb R) H^n_b(G,{\Bbb R}) to Hⁿ(G,\Bbb R) H^n(G,{\Bbb R}) induced by inclusion is surjective for n 3 2 n \ge 2 . We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map Hⁿ_b(G,V) ? Hⁿ(G,V) H^n_b(G,V) \to H^n(G,V) is surjective for n 3 2 n \ge 2 when V is any bounded \Bbb QG {\Bbb Q}G -module and when V is any finitely generated abelian group.     

C*-index of observable algebras in <Emphasis Type="Italic">G</Emphasis>-spin model

In two-dimensional lattice spin systems in which the spins take values in a finite group G, one can define a field algebra F which carries an action of a Hopf algebra D(G), the double algebra of G and moreover, an action of D(G;H), which is a subalgebra of D(G) determined by a subgroup H of G, so that F becomes a modular algebra. The concrete construction of D(G;H)-invariant subspace A _H in F is given. By constructing the quasi-basis of conditional expectation γ _G of A _H onto A _G , the C*-index of γ _G is exactly the index of H in G.     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

A Canonical Ramsey Theorem

Say a graph H selects a graph G if given any coloring of H, there will be a monochromatic induced copy of G in H or a completely multicolored copy of G in H. Denote by s(G) the minimum order of a graph that selects G and set s(n) = max {s(G): |G| = n}. Upper and lower bounds are given for this function. Also, consider the Folkman function f_r(n) = max{min{|V(H)|: H → (G)¹_r}: |V(G)| = n}, where H → (G)¹_r indicates that H is vertex Ramsey to G, that is, any vertex coloring of H with r colors admits a monochromatic induced copy of G. The method used provides a better upper bound for this function than was previously known. As a tool, we establish a theorem for projective planes.     

Matrix Near-rings over Centralizer Near-rings

For a finite group G and a subgroup A of Aut(G), let M_A(G) denote the centralizer near-ring determined by A and G. The group G is an M_A(G)-module. Using the action of M_A(G) on G, one has the n × n generalized matrix near-ring Mat_n(M_A(G);G). The correspondence between the ideals of M_A(G) and those of Mat_n(M_A(G);G) is investigated. It is shown that if every ideal of M_A(G) is an annihilator ideal, then there is a bijection between the ideals of M_A(G) and those of Mat_n(M_A(G);G).1991 Mathematics Subject Classification: 16Y30     

A Class of Operators Similar to the Shift on H 2(G)

For a compact subset K in the complex plane, let A(K) denote the algebra of all functions continuous on K and analytic on K° and let R(K) denote the uniform closure of the rational functions with poles off K. Let G is a bounded open subset whose complement in the plane has a finite number of components. Suppose that and every function in H^∞(G) is the pointwise limit of a bounded sequence of functions in . The purpose of this paper is to characterize all subnormal operators similar to M_z, the operator of multiplication by the independent variable z on the Hardy space H²(G). We also characterize all bounded linear operators that are unitarily equivalent to M_z in the case when each of the components of G is simply connected. In particular, our similarity result extends a well-known result of W. Clary on the unit disk to multiply connected domains.     

The algebra ofG-relations

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).     

Abelian Group Gradings on Full Matrix Rings

For positive integers ? and n, several authors studied ?_?-gradings of the full matrix ring M _n (k) over a field k. In this article, we show that every (G × H)-grading of M _n (k) can be constructed by a pair of compatible G-grading and H-grading of M _n (k), where G and H are any finite groups. When G and H are finite cyclic groups, we characterize all (G × H)-gradings which are isomorphic to a good grading. Moreover, the results can be generalized for any finite abelian group grading of M _n (k).     

On ℑ<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-normal subgroups of finite groups

Given a class ℑ of finite groups, a subgroup H of a group G is called ℑ _n -normal in G if there exists a normal subgroup T of G such that HT is a normal subgroup of G and (H ∩ T)H _G /H _G is contained in the ℑ-hypercenter Z _∞^ℑ (G/H _G ) of G/H _G . We obtain some results about the ℑ _n -normal subgroups and use them to study the structure of some groups.     

The Galois Correspondence in Field Algebra of G-spin Model

Suppose that G is a finite group and D（G） the double algebra of G. For a given subgroup H of G, there is a sub-Hopf algebra D（G; H） of D（G）. This paper gives the concrete construction of a D（G; H）-invariant subspace AH in field algebra of G-spin model and proves that if H is a normal subgroup of G, then AH is Galois closed.     

Pan-H-Linked Graphs

Let H be a multigraph, possibly containing loops. An H-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrary multigraph of order k, size m, n ₀(H) isolated vertices and n ₁(H) vertices of degree one. In Gould and Whalen (Graphs Comb. 23:165–182, 2007) it was shown that if G is a simple graph of order n containing an H-subdivision H{\mathcal{H}} and d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}}, then G contains a spanning H-subdivision with the same ground set as H{\mathcal{H}} . As a corollary to this result, the authors were able to obtain Dirac’s famed theorem on hamiltonian graphs; namely that if G is a graph of order n ≥ 3 with d(G) 3 \fracn2{\delta(G)\ge\frac{n}{2}} , then G is hamiltonian. Bondy (J. Comb. Theory Ser. B 11:80–84, 1971) extended Dirac’s theorem by showing that if G satisfied the condition d(G) 3 \fracn2{\delta(G) \ge \frac{n}{2}} then G was either pancyclic or a complete bipartite graph. In this paper, we extend the result from Gould and Whalen (Graphs Comb. 23:165–182, 2007) in a similar manner. An H-subdivision H{\mathcal{H}} in G is 1-extendible if there exists an H-subdivision H^*{\mathcal{H}^{*}} with the same ground set as H{\mathcal{H}} and |H^*| = |H| + 1{|\mathcal{H}^{*}| = |\mathcal{H}| + 1} . If every H-subdivision in G is 1-extendible, then G is pan-H-linked. We demonstrate that if H is sufficiently dense and G is a graph of large enough order n such that d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}} , then G is pan-H-linked. This result is sharp.     

The fitting length of solvableH pn-groups

For a (finite) groupG and some prime powerp ⁿ, theH _p ⁿ -subgroupH _pn (G) is defined byH _p ⁿ (G)=〈xεG|x ^pn≠1〉. A groupH≠1 is called aH _p ⁿ -group, if there is a finite groupG such thatH is isomorphic toH _p ⁿ (G) andH _p ⁿ (G)≠G. It is known that the Fitting length of a solvableH _p ⁿ -group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH _p ⁿ -groups having Fitting lengthn, this result is “almost” best possible.     

A classification of graph capacity functions

Given two graphs G = (V(G), E(G)) and H = (V(H), E(H)), the sum of G and H, G + H, is the disjoint union of G and H. The product of G and H, G × H, is the graph with the vertex set V(G × H) that is the Cartesian product of V(G) and V(H), and two vertices (g₁, h₁), (g₂, h₂) are adjacent if and only if [g₁, g₂] (ELEMENT) E(G) and [h₁, h₂] (ELEMENT) E(H). Let G denote the set of all graphs. Given a graph G, the G-matching function, γ_G, assigns any graph H (ELEMENT) G to the maximum integer k such that kG is a subgraph of H. The graph capacity function for G, P_G: G → (RFRAKTUR), is defined as P_G(H) = lim_n→z[γ_G(Hⁿ)]^1/n, where Hⁿ denotes the n-fold product of H × H × … × H. Different graphs G may have different graph capacity functions, all of which are increasing. In this paper, we classify all graphs whose capacity functions are additive, multiplicative, and increasing; all graphs whose capacity functions are pseudo-additive, pseudo-multiplicative, and increasing; and all graphs whose capacity functions fall under neither of the above cases. © 1996 John Wiley & Sons, Inc.     

A Higgs bundle on a Hermitian symmetric space,II

Let M : = Γ\G/K be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice G ì G{\Gamma \subset G}. Let V be a complex irreducible representation of G. We give a Hodge decomposition of the cohomology of the Γ-module V in terms of the cohomologies of automorphic vector bundles on M associated to the Lie algebra cohomologies H*(\mathfrak p⁺ ,V){H*({\mathfrak p}^{+} ,V)}.