In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras. We first discuss low-dimensional Lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is ...

This paper encloses a complete and explicit description of the derivations of the Lie algebra D(M) of all linear differential operators of a smooth manifold M , of its Lie subalgebra D 1 (M) of all linear first-order differential operators of M , and of the Poisson algebra S(M) = Pol(T * M) of all polynomial functions on T * M, the symbols of the operators in D(M). It turns out that, in terms o...

We present an explicit description of the 'fine group gradings' (i.e. group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras sl(4, C), sp(4, C), and o(4, C). All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed. The inclusions sl(4, C) ⊃ sp(4, C) ⊃ o(4, C) are an important tool in our presentation...

The algebraic cohoroology and the spectral sequences for a Z2-graded Lie algebra are briefly reviewed. The reducibility property of a strongly semisiiaple Lie superalgebra is established. The role of second and third cohomologies in the deformation of a Lie superalgebra is discussed. Using spectral sequences, the second cohomology of full BRS algebra is shown to be W and the third cohomology be...

We analyze certain subgroups of real and complex forms of the Lie group E8, and deduce that any “Theory of Everything” obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E8 lacks certain representationtheoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras along the lines of ...

A new class of infinite dimensional simple Lie algebras over a field with characteristic 0 are constructed. These are examples of non-graded Lie algebras. The isomorphism classes of these Lie algebras are determined. The structure space of these algebras is given explicitly.

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 ...