Abstract. We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie algebra, is derived from the integrated adjoint representation. We apply this construction to realize the bracket operations on the sections of Courant algebroids and on the “omni-Lie algebras ” recently introduced by the second author. 1.

Abstract. In the spirit of Glauberman’s fundamental work in B-loops and Moufang loops [17] [18], we prove Cauchy and strong Lagrange theorems for Bol loops of odd order. We also establish necessary conditions for the existence of a simple Bol loop of odd order, conditions which should be useful in the development of a Feit-Thompson theorem for Bol loops. Bol loops are closely related to Aschbacher’s twisted subgroups [1], and we survey the latter in some detail, especially with regard to the so-called Aschbacher radical. 1.

CONTENTS Preface 5 1. General Fundamentals 1.1 Basic Concepts 7 1.2 A few properties of groups and graphs 8 1.3 Lattices and its properties 11 2. Loops and its properties 2.1 Definition of loop and examples 15 2.2 Substructures in loops 17 2.3 Special identities in loops 22 2.4 Special types of loops 23 2.5 Representation and isotopes of loops 29 2.6 On a new class of loops and its properties 31 2.7 The new class of loops and its application to proper edge colouring of the graph K 2n 40 3. Smarandache Loops 3.1 Definition of Smarandache loops with examples 47 3.2 Smarandache substructures in loops 51 3.3 Some new classical S-loops 56 3.4 Smarandache commutative and commutator subloops 61 3.5 Smarandache associativite and associator subloops 67 3.6 Smarandache identities in loops 71 3.7 Some special structures in S-loops 74 3.8 Smarandache mixed direct product loops 78 3.9 Smarandache cosets in loops 84 3.10 Some special type of Smarandache loops 91 4. Properti

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Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a power-associative Kikkawa left loop with two-sided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space structure of S. (S, ⊙) is not a loop, and the right translations which fail to be injective are easily characterized. (S, ⊙) satisfies the left power alternative and left Bol identities “almost everywhere ” but not everywhere. Left translations are everywhere analytic; right translations are analytic except at −e0 where they have a nonremovable discontinuity. The orthogonal group O(H) is a semidirect product of (S, ⊙) with its automorphism group. The left loop structure of (S, ⊙) gives some insight into spherical geometry.

Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a power-associative Kikkawa left loop with two-sided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space structure of S. (S, ⊙) is not a loop, and the right translations which fail to be injective are easily characterized. (S, ⊙) satisfies the left power alternative and left Bol identities “almost everywhere” but not everywhere. Left translations are everywhere analytic; right translations are analytic except at −e0 where they have an essential discontinuity. The orthogonal group O(H) is a semidirect product of (S, ⊙) with its automorphism group. The left loop structure of (S, ⊙) gives some insight into spherical geometry.

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