We derive a new formula for the supersymmetric Schur polynomial sλ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for sλ(x/y). This new expression gives rise to a determinantal formula for sλ(x/y). I...

Let g be a semisimple complex Lie algebra and θ ∈ Aut g be an involution. If g = k⊕ p is the decomposition associated to θ , define a Lie subalgebra of End p by k̃ = {X : ∀f ∈ S(p∗)k, X.f = 0} . We prove that adp(k) = k̃ if, and only if, each irreducible factor of rank one of the symmetric pair (g, k) is isomorphic to (so(q + 1), so(q)) .

In this paper we take a close look at Lie derivatives on a Finsler bundle and give a geometric meaning to the vanishing of the mixed curvature of certain covariant derivatives on a Finsler bundle. As an application, we obtain some characterizations of Landsberg manifolds.

Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. For finite-dimensional ones we show that if they are semisimple as associative algebras then they are standard, on the other hand, if they are semisimple as Lie algebras then their associative products are trivial. We also give the descriptions of ...

The aim of this paper is to present some results about the finite dimensional p-Lie algebras and some properties of p-c-supplemented subalgebras of p-Lie algebras. In addition some relations between p-csupplemented Lie algebras and E-p-algebras are pointed out.

The discrete symmetries P, C and T are discussed in terms of Lie algebra extensions of the Poincare Lie algebra. This formulation leads to certain problems of Lie algebra theory which will be presented in this and succeeding papers. (To be submitted to Communications in Mathematical Physics) * Work supported by the U. S. Atomic Energy Commission.

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...

Previous work (Pradines, 1966, Aof and Brown, 1992) has given a setting for a holon-omy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for t...

We propose a recipe to construct matrix representations of Nambu– Lie 3-algebras in terms of irreducible representations of underlying Lie algebra. The case of Euclidean four-dimensional 3-algebra is considered in details. We find that representations of this 3-algebra are not possible in terms of only Hermitian matrices in spite of its Euclidean nature.

Let g be a complex simple Lie algebra. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The nilpotent radical of b is denoted by u. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). An ideal of b is called ad-nilpotent, if it is contained in [b, b]. The theory of ad-nilpotent ideals has attracted much recent attention in the work of Kostant, Cellini-Papi, Sommers, an...