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11
Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces
 Amer. J. of Math
"... Abstract. We show that the skewsymmetrized 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 repres ..."
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Cited by 53 (2 self)
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Abstract. We show that the skewsymmetrized 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 “omniLie algebras ” recently introduced by the second author. 1.
On twisted subgroups and Bol loops of odd order
 Rocky Mountain J. of Math
"... Abstract. In the spirit of Glauberman’s fundamental work in Bloops 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 t ..."
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Cited by 11 (6 self)
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Abstract. In the spirit of Glauberman’s fundamental work in Bloops 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 FeitThompson 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 socalled Aschbacher radical. 1.
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"... The picture on the cover represents the lattice of subgroups of the Smarandache loop L15(8). The lattice of subgroups of the commutative loop L15(8) is a nonmodular lattice with 22 elements. This is a Smarandache loop which satisfies the Smarandache Lagrange criteria. But for the Smarandache concep ..."
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The picture on the cover represents the lattice of subgroups of the Smarandache loop L15(8). The lattice of subgroups of the commutative loop L15(8) is a nonmodular lattice with 22 elements. This is a Smarandache loop which satisfies the Smarandache Lagrange criteria. But for the Smarandache concepts one wouldn't have studied the collection of subgroups of
(1.1)
, 1999
"... Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space ..."
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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 powerassociative Kikkawa left loop with twosided 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.
(1.1)
, 2000
"... Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space ..."
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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 powerassociative Kikkawa left loop with twosided 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.
Gyrogroups, the Grouplike Loops in the Service of Hyperbolic Geometry and Einstein’s Special Theory of Relativity
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