We show that the Manin triple characterization of Lie bialgebras in terms of the Drinfel’d double may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using double cotangent bundles, rather than the direct sum structures (Courant algebroids) utilized for similar purposes by Liu, Weinstein and Xu. This is achieved in terms of an abstract notion of double Lie algebroid (w...

We study Manin triples for a reductive Lie algebra, g. First, we generalize results of E. Karolinsky, on the classification of Lagrangian subalgebras (cf. KAROLINSKY E., A Classification of Poisson homogeneous spaces of a compact Poisson Lie group, Dokl. Ak. Nauk, 359 (1998), 13-15). Then we show that, if g is non commutative, one can attach , to each Manin triple in g , another one for a stric...

Two-dimensional hyporeductive triple algebras (h.t.a) are investigated. Using the K. Yamaguti's approach for the classification of two-dimensional Lie triple systems (L.t.s), a classification of two-dimensional h.t.a is suggested. MIRAMARE TRIESTE May 1998 Regular Associate of the ICTP. Fax: (229)212525

A result of Lewis on the extreme properties of the inner product of two vectors in a Cartan subspace of a semisimple Lie algebra is extended. The framework used is an Eaton triple which has a reduced triple. Applications are made for determining the minimizers and maximizers of the distance function considered by Chu and Driessel with spectral constraint.

Reductivity in the Ma’tsev algebras is inquired. This property relates the Mal’tsev algebras to the general Lie triple systems. 2000 MSC: 20N05, 17D10

This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting oper...

We introduce the notion of a symplectic Lie affgebroid and their Lagrangian submanifolds in order to describe time-dependent Mechanics in terms of this type of structures. Analogous to the autonomous case, we construct the Tulczyjew’s triple associated with a Lie affgebroid and a Hamiltonian section. We describe our theory with several examples.

The Lie superalgebras in the extended Freudenthal Magic Square in characteristic 3 are shown to be related to some known simple Lie superalgebras, specific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jordan algebras.

Let $ {\mathcal{A}} be a unital \ast -algebra containing nontrivial projection. Under some mild conditions on , it is shown that map \Phi:{\mathcal{A}}\rightarrow{\mathcal{A}} nonlinear mixed Jordan triple * -derivation if and only \Phi an additive -derivation. In particular, we apply the above result to prime -algebras, von Neumann algebras with no central summands of type I_{1} factor algebra...