Vol. 68, No. 1, 1977

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Anti-commutative algebras and homogeneous spaces with multiplications

Arthur Argyle Sagle and J. R. Schumi

Vol. 68 (1977), No. 1, 255–269

As a generalization of certain results for Lie groups it is shown that an n-dimensional H-space (M,μ) with identity e has a coordinate system at e in which μ can be represented by a function F : Rn × Rn Rn which is analytic at (0,0) and that the second derivative of F induces a bilinear anti-commutative multiplication α on Rn. In this way an algebra (Rn) analogous to the Lie algebra of a Lie group is obtained and all such algebras are shown to be isomorphic. If M = G∕H is a reductive homogeneous space, then these results generalize the Lie group-Lie algebra correspondence and the algebra (Rn) induces a G-invariant connection on G∕H. Relative to this connection it is shown that an automorphism of (G∕H,μ) is an affine map and induces an algebra automorphism of (Rn). Also the connection is irreducible if (G∕H,μ) has no proper invariant subsystems (the analog of normal subgroups). In the case where G∕H has a Riemannian structure, it may happen that there are no local isometries among the coordinate maps which give rise to anti-commutative multiplications on Rn.

Mathematical Subject Classification 2000
Primary: 53C30
Secondary: 55D45
Received: 26 June 1976
Revised: 17 August 1976
Published: 1 January 1977
Arthur Argyle Sagle
J. R. Schumi