A Riemann-Lie algebra is a Lie algebra G such that its dual G∗ carries a Riemannian metric compatible (in the sense introduced by the author in C. R. Acad. Sci. Paris, t. 333, Série I, (2001) 763–768) with the canonical linear Poisson structure of G∗ . The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds (see Differential Geometry and its A...

In honor of the 100th birthday of the article, " Sur les groupes continus " Abstract: A methodical analysis of the research related to the article , " Sur les groupes continus " , of Henri Poincaré reveals many historical misconceptions and inaccuracies regarding his contribution to Lie theory. A thorough reading of this article confirms the precedence of his discovery of many important concept...

We show that the representation theory for the toroidal extended affine Lie algebra is controlled by a VOA which is a tensor product of four VOAs: a sub-VOA V + Hyp of a hyperbolic lattice VOA, affine ˙ g and sl N VOAs and a Virasoro VOA. A tensor product of irre-ducible modules for these VOAs admits the structure of an irreducible module for the toroidal extended affine Lie algebra. We also sh...

In this paper, first we introduce the notions of relative Rota-Baxter operators nonzero weight on $3$-Lie algebras and $3$-post-Lie algebras. A 3-post-Lie algebra consists a 3-Lie structure ternary operation such that some compatibility conditions are satisfied. We show operator induces naturally. Conversely, gives rise to new algebra, which is called subadjacent an action original algebra. The...

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Riemannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a compatible pseudo-metric if and only if the Lie algebra is a pseudo-Riemannian Li...