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A Lie triple system is a subspace of a Lie algebra closed under the ternary composition [[xy]z]; equivalently, it may be defined as the subspace of elements mapped into their negatives by an automorphism of order two (an involution) in a Lie algebra; or, finally, a Lie triple system may be defined by a set of identities. Lie triple systems were first noted by E. Cartan in his studies on totally geodesic submanifolds of Lie groups and on symmetric spaces [l] (see also [lO]): Lie triple systems are related to totally geodesic submanifolds in the same way that Lie algebras are related to analytic subgroups, and the symmetry in a symmetric space gives rise to the involution in the Lie algebra. Lie triple systems were studied from the algebraic point of view by Jacobson [6; 7] and Lister [9], the latter giving a complete structure theory including the classification of the simple finite-dimensional systems (in characteristic zero), the Levi decomposition, and (a special case of) the first Whitehead lemma. Simpler axioms were given by Yamaguti [14], who has also studied these and more general systems [15; 16]. In this paper we introduce cohomology groups for Lie triple systems and show that the usual interpretations (in terms of derivations and factor sets) hold for the first and second cohomology groups. In particular we obtain the two Whitehead lemmas (in characteristic zero). Further, these cohomology groups fit into the theory of [2 ] : they are the cohomology groups of a supplemented associative algebra (very closely related to the universal associative algebra of the Lie triple system). The groups may briefly be described as follows: if T is any Lie triple system and M a T-module we can construct the universal Lie algebra LU(T) of T and the "standard extension" M, which is an LU(T) module; an involution a operates on both LU(T) and Ms such that the elements mapped into their negatives are T and M respectively, and cr([/, «])= [cr(/), o(n)] for IELU(T), nEM.. Now (in a more general situation) if L is any Lie algebra with involution o and N any L-module also with involution 0, then <r also operates on the cohomology groups Hn(L, N) = Extu($, N) (i> the base field, U the universal associative algebra of L) ; we assume further that the characteristic is not 2, so that Hn(L, N) is a direct sum of two subspaces invariant under cr: say H"+(L, N), H1(L, N). We show that H\(L, A0 = Extx($, AO

Cohomology of Lie triple systems and Lie algebras with involution