Showing papers in "Tohoku Mathematical Journal in 1972"

A class of almost contact riemannian manifolds

Abstract: Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain

Approximation of operator semigroups in a Banach space

Abstract: {U(t); t >0} of linear operators defined on a linear manifold D(•1⁄4X) such that for each x • ̧ D the function u(t) = U(t)x is a solution of the problem. To find the solution operators of our problem, we proceed as follows. Let {An} be a sequence of closed linear operators in X which are \"regular\" in comparison with A and approximate A in an appropriate sense , and let us consider a sequence of approximatingequations

On a semi-direct product decomposition of affine groups over a field of characteristic $0$

Abstract: PROOF. The Jacobson radical of a ring is the intersection of its maximal left (or right) ideals. Dually the coradical of a coalgebra C over a field is identical with the socle of C as a right (or left) C-comodule. Since k is perfect, the coradical of fc(g)fc A is k®kR, where k is the algebraic closure of k. Hence we can assume that k — k. Moreover A can be assumed to be finitely generated as a fc-algebra. Let Vi9 i = 1, 2 be two finite dimensional right A-comodules. Becauce G(A°) = Algfc(̂ 4., k) is dense in A* = Hom^A, k) [6, Lem. 3.6], V* is a semisimple A-comodule iff Vi is a semisimple left G(A°)-module. Hence by the remark above if V4 are semisimple, then Vι ® V2 is also semi-simple. This means that R (x) R is a semisimple right A-comodule. Since the multiplication μ: A (x) A —> A is a right A-comodule map, R R is contained in R. Clearly R is stable under the antipode of A. Hence R is a sub-Hopf algebra of A.