The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

Tubular Structures of Compact Symmetric Spaces Associated with the Exceptional Lie Group F<Subscript>4</Subscript>

In the present paper we discuss in detail cohomogeneity one isometric actions on the compact symmetric spaces F₄ and . By means of isometric actions we show that these symmetric spaces have got well-defined tubular structures around the totally geodesic submanifolds Spin(9) and , respectively. Therefore these symmetric spaces can be thought of as compact tubes. The radii of the tubes and the principal curvatures of the tubular hypersurfaces are determined in explicit form. Moreover, we apply these results to compute the volumes of the principal orbits and the volumes of the symmetric spaces associated with the compact Lie group F₄.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Some group actions on K(x 1 , x 2 , x 3 )

LetK be any field which may not be algebraically closed,K(x ₁ ,x ₂ ,x ₃ ) be the rational function field of three variables overK, and σ:K(x ₁ ,x ₂ ,x ₃ ) → K(x ₁ ,x ₂ ,x ₃ ) be aK-automorphism defined by wherea _i ,b _i ,c _i ,d _i ∈K anda _i d _i −b _i c _i ≠0. Let ,f _i (T)=T ² −(a _i +d _i )T+(a _i d _i −b _i c _i )∈K[T] be the “characteristic polynomial” of σ _i . Theorem:Assume that charK≠2.Then the fixed field K(x ₁ ,x ₂ ,x ₃ ) ^<σ> is not rational (=purely transcendental) over K if and only if (i) for each 1≤i≤3, f _i (T) is irreducible; (ii) the Galois group of f ₁ (T)f ₂ (T)f ₃ (T) over K is of order 8; and (iii) for each 1≤i≤3,ord (σ _[itn] )is an even integer.     

SU(3)-manifolds of cohomogeneity one

In this article, we classify 7- and 8-dimensional manifolds M admitting an SU(3) action of cohomogeneity one such that (i) M is simply connected and the orbit space M/G is isomorphic to [0, 1], and (ii) \({M/G\cong S^{1}}\) and the principal orbits are simply connected. We discuss applications to the study of the group manifold SU(3) and to 8-dimensional quaternion-Kähler spaces, and links between dimension 7 and 8 given by circle actions.     

The G2 sphere of a 4-manifold

We bring to light a G ₂ structure existing on the unit sphere tangent bundle S _M of any given orientable Riemannian 4-manifold M. The associated 3-form φ is co-calibrated if, and only if, M is an Einstein manifold—a result which leads to new examples of co-calibrated G ₂ spaces. We hope to be contributing both to the knowledge of special geometries and to the study of 4-manifolds.     

Quasirecognition by the set of element orders of the groups E 6( q ) and 2 E 6( q )

We prove that if L is one of the simple groups E ₆(q) and ² E ₆(q) and G is some finite group with the same spectrum as L, then the commutant of G/F(G) is isomorphic to L and the quotient G/G′ is a cyclic {2,3}-group. Original Russian Text Copyright ? 2007 Kondrat’ev A. S. The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00463) and the RFBR-NSFC (Grant 05-01-39000). __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1250–1271, November–December, 2007.     

The Local Index Formula for SU q (2)

We discuss the local index formula of Connes–Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula. (Received: January 2005)     

Graphs satisfying inequality θ(G)θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G²)θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T²) for a tree T.     

SU(3)-Structures on Hypersurfaces of Manifolds With G 2-Structure

We study SU(3)-structures induced on orientable hypersurfaces of seven-dimensional manifolds with G₂-structure. Taking Gray-Hervella types for both structures into account, we relate the type of SU(3)-structure and the type of G₂-structure with the shape tensor of the hypersurface. Additionally, we show how to compute the intrinsic SU(3)-torsion and the intrinsic G₂-torsion by means of the exterior algebra.     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

Cyclic codes over the rings Z 2?+ uZ 2 and Z 2?+ uZ 2?+ u 2 Z 2

In this paper, we study cyclic codes over the rings Z ₂ + uZ ₂ and Z ₂ + uZ ₂ + u ² Z ₂ . We find a set of generators for these codes. The rank, the dual, and the Hamming distance of these codes are studied as well. Examples of cyclic codes of various lengths are also studied.      

Some Aspects of Lw1^r （G）∩ L（p,q, w2dμ）（G）

Let G be a locally compact Abelian group with Haar measure. The authors discuss some basic properties of Lw1^r （G）∩ L（p, q, w2dμ）（G） spaces. Then the necessary conditions for compact embeddings of the spaces Lw1^r （R^d）∩ L（p, q, w2dμ）（R^d） are showed.     

Conformally parallel G 2 structures on a class of solvmanifolds

Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G₂-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G₂ structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained in G₂. In the process we also find a new metric with exceptional holonomy. Received: 20 September     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.     

Holomorphic H-spherical distribution vectors in principal series representations

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L²(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown . In this article we construct a minimal G-invariant subdomain _H of with G/H as Shilov boundary. Let be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of , which admit a holomorphic extension to _H, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L²(G/H)_mc in a space of holomorphic functions.     

The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)