On Some Algebras Related to Simple Lie Triple Systems

Abstract

Lie triple system T over a field F of characteristic zero. It turns out that it contains nontrivial elements if and only if T is related to a simple Jordan algebra.In particular this provides a new proof of the determination by Laquer of the invariant affine connections in the simply connected compact irreducible Riemannian symmetric spaces. ᮊ

Invariant affine connections on odd-dimensional spheres

“…Nomizu's Theorem [22] can be formulated as follows: Theorem 2.3. Let G/H be a reductive homogeneous space with a fixed reductive decomposition (5). Then, there is a bijective correspondence between the set of G-invariant affine connections ∇ on G/H and the vector space of bilinear maps α : m × m → m such that Ad(H) ⊂ Aut(m, α), that is, such that α Ad(σ)(A), Ad(σ)(B) = Ad(σ)(α(A, B)) for all A, B ∈ m and σ ∈ H. In case H is connected, this equation is equivalent to…”

“…Theorem 2.7. Let M = G/H be a reductive homogeneous space endowed with a G-invariant Riemannian metric g and with a fixed reductive decomposition (5). A G-invariant affine connection ∇ is metric if and only if the bilinear operation α ∇ related to ∇ by Theorem 2.3 satisfies…”

“…In general, the properties of the connection algebras are not very well-known, except for a few cases. For instance, under the assumption that M is a symmetric space and G is simple, then m can be identified with the traceless elements of a simple finite-dimensional Jordan algebra and, up to scalars, α corresponds with the projection of the Jordan product [5]. Also, for M = S 6 treated as G 2 -homogeneous space, the related connection algebra (m, α) turns out to be a color algebra (see details in [14]).…”

The Centroid of a Lie Triple Algebra

“…It turns out that result on the centroid of Lie algebras is a key ingredient in the classification of extended affine Lie algebras. The centroids of Lie triple systems were mentioned by Benito et al [10]. Now, some results on centroids of Lie triple system and -Lie algebras were developed in [11,12].…”

Centroids of Lie Supertriple Systems

“…It turns out that results on the centroids of Lie algebras are a key ingredient in the classification of extended affine Lie algebras. The centroids of Lie triple systems were mentioned by Benito et al in [10]. Results on centroids of Lie triple system, -Lie algebras, and Lie triple algebras were developed in [11][12][13].…”