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We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G F (n,N) in the Lie group G F (N) defined naturally makes G F (n,N) a totally geodesic submanifold; (2) The imbedding S 7SO(8) defined by octonians makes S 7a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G F (N) in the sphere ScN^2-1(√N) of gl(N,F)is minimal. Therefore the natural imbedding G F (N)→gl(N,F )is formed by the eigenfunctions of the Laplacian on G F (N).

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In this paper we characterize commutative Fréchet-Lie groups using the exponential map. In particular we prove that if a commutative Fréchet-Lie groupG has an exponential map, which is a local diffeomorphism, thenG is the limit of a projective system of Banach-Lie groups.

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Let Fn, n≧ 1, denote the sequence of generic filiform (connected, simply connected) Lie groups. Here we study, for each Fn, the infinite dimensional simple quotients of the group C*-algebra of (the most obvious) one of its discrete cocompact subgroups Dn. For Dn, the most attractive concrete faithful representations are given in terms of Anzai flows, in analogy with the representations of the discrete Heisenberg group H 3 G 3 on L 2(T) that result from the irrational rotation flows on T; the representations of Dn generate infinite-dimensional simple quotients An of the group C*-algebra C*(Dn). For n>1, there are other infinite-dimensional simple quotients of C*(Dn) arising from non-faithful representations of Dn. Flows for these are determined, and they are also characterized and represented as matrix algebras over simple affine Furstenberg transformation group C*-algebras of the lower dimensional tori.

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Let us consider a triangular array of random vectors (X (n) j; Y (n) j), n = 1;2;: : :, 1 5 j 5 kn, such that the first coordinates X (n) j take their values in a non-compact Lie group and the second coordinates Y (n) j in a compact group. Let the random vectors (X (n) j; Y (n) j) be independent for fixed n, but we do not assume any (independence type) condition about the relation between the components of these vectors. We show under fairly general conditions that if both random products Sn = kn Q j=1 X (n) j and Tn = kn Q j=1 Y (n) j have a limit distribution, then also the random vectors (Sn; Tn) converge in distribution as n !1 . Moreover, the non-compact and compact coordinates of a random vector with this limit distribution are independent.

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almost para contact manifolds . J. Univ. Kuwait Science 16 ( 1989 ), 215 – 220 . [10] Warner , F. W . Foundations of Differentiable Manifolds and Lie Groups . Springer-Verlag , New York , 1971 .

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LetG be a real reductive Lie group, i.e., a Lie group whose Lie algebra is the direct product of a commutative and a semi-simple algebra. LetG 0 be the unit component ofG. We analyze the following question: if Φ is a continuous linear representation ofG over a finite dimensional complex vector spaceV then when can we find a scalar product onV so that the group Φ(G) become*-invariant with respect to it? In particular, ifG/G 0 is finite then we show that this is the case if and only if the same holds for the connected subgroup corresponding to the center ζ of the Lie algebra ofG and the latter condition is very easy to describe in terms ofdΦ/ζ. We discuss some related questions such as the relation between Cartan decompositions ofG and polar decompositions of Φ(G), the description of the closure of Φ(G), etc.

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