The uniformity of the decomposition law, for a family F of Lie algebras which includes the exceptional Lie algebras, of the tensor powers ad of their adjoint representations ad is now well-known. This paper uses it to embark on the development of a unified tensor calculus for the exceptional Lie algebras. It deals explicitly with all the tensors that arise at the n = 2 stage, obtaining a large ...

In this paper, a useful classification of all Lie subalgebras of a given Lie algebraup to an inner automorphism is presented. This method can be regarded as animportant connection between differential geometry and algebra and has many applications in different fields of mathematics. After main results, we have applied this procedure for classifying the Lie subalgebras of some examples of Lie al...

Solution by M.C. Nucci (Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy) Lie’s monumental work on transformation groups, [12], [13] and [14], and in particular contact transformations [15], has provided systematic techniques for obtaining exact solutions of differential equations [16]. Many books have been dedicated to this subject and its generalizations (...

Let  be a Banach algebra. Let  be linear mappings on . First we demonstrate a theorem concerning the continuity of double derivations; especially that all of -double derivations are continuous on semi-simple Banach algebras, in certain case. Afterwards we define a new vocabulary called “-higher double derivation” and present a relation between this subject and derivations and finally give some ...

let $mathcal{a}$ be a banach algebra and $mathcal{m}$ be a banach $mathcal{a}$-bimodule. we say that a linear mapping $delta:mathcal{a} rightarrow mathcal{m}$ is a generalized $sigma$-derivation whenever there exists a $sigma$-derivation $d:mathcal{a} rightarrow mathcal{m}$ such that $delta(ab) = delta(a)sigma(b) + sigma(a)d(b)$, for all $a,b in mathcal{a}$. giving some facts concerning general...

Let R be a 2-torsion free prime ring with center Z, right Utumi quotient ring U , generalized derivation F associated with a nonzero derivation d of R and L a Lie ideal of R. If F (uv)n = F (u)mF (v)l or F (uv)n = F (v)lF (u)m for all u, v ∈ L, where m,n, l are fixed positive integers, then L ⊆ Z. We also examine the case where R is 312 Ahmad N. Alkenani et al. a semiprime ring. Finally, as an ...

in this paper, by using of frobenius theorem a relation between lie subalgebras of the lie algebra of a top space t and lie subgroups of t(as a lie group) is determined. as a result we can consider these spaces by their lie algebras. we show that a top space with the finite number of identity elements is a c^{∞} principal fiber bundle, by this method we can characterize top spaces.

We introduce the notions of proto-complete, complete, complete⁎ and strong-complete objects in pointed categories. show under mild conditions on a exact protomodular category that every proto-complete (respectively complete) object is product an abelian object. This together with observation trivial group only complete recovers theorem Baer classifying groups. In addition we generalize several ...

From an octonion algebra $\mathbb{O}$ over a field $k$ of characteristic not two or three, we show that the fundamental representation ${\rm Im}(\mathbb{O})$ derivation Der}(\mathbb{O})$ and spinor $\mathfrak{so}({\rm Im}(\mathbb{O}))$ are special orthogonal representations. They have particular geometric properties coming from their similarities with binary cubics covariants these representati...