Narrow Results

If $𝔄$ is an alternative algebra containing a nontrivial symmetric idempotent, $\delta :𝔄\to 𝔄$ is a multiplicative ∗-reverse derivation and $𝔇:𝔄\to 𝔄$ is a multiplicative Jordan ∗-reverse derivation, then under a mild condition on $𝔄$ we prove that $\delta$ and $D$ are additives. Furthermore if $𝔇:𝔄\to 𝔄$ is a Jordan ∗-reverse derivation, then under a mild condition on $𝔄$ and $𝔇$ we prove that $𝔇$ is the form $d+\delta$, where $d$ is a ∗-reverse derivation of $𝔄$ and $\delta$ is a singular Jordan ∗-reverse derivation of $𝔄$. Moreover, $d$ and $\delta$ are uniquely determined.

A base of the free alternative superalgebra on one odd generator is constructed. As a corollary, a base of the alternative Grassmann algebra is given. We also find a new element of degree 5 from the radical of the free alternative algebra of countable rank.

A base of the free Malcev superalgebra on one odd generator is constructed. Some corollaries for skew-symmetric functions and central elements in free Malcev and free alternative algebras are obtained.

A base of the free alternative nil-superalgebra of index 3 on one odd generator is constructed. In particular, its index of solvability is computed. We consider also the corresponding Grassmann algebra and show that the well-known Dorofeev's example of solvable non-nilpotent alternative algebra is its homomorphic image.

We define a notion of associative representation for algebras. We prove the existence of faithful associative representations for any alternative, Mal’cev, and Poisson algebra, and prove analogs of Ado-Iwasawa theorem for each of these cases. We construct also an explicit associative representation of the Cayley–Dickson algebra in the matrix algebra $M_8(F).$

The aim of this paper is to study Kupershmidt-(dual-)Nijenhuis structures on alternative algebras with representations. The notion of a (dual-)Nijenhuis pair is introduced and it can generate a trivial deformation of an alternative algebra with a representation. We introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on an alternative algebra with a representation. Furthermore, we verify that Kupershmidt operators and Kupershmidt-(dual-)Nijenhuis structures can give rise to each other under some conditions. Finally, we study the notions of Rota–Baxter–Nijenhuis structures and alternative $r$-matrix-Nijenhuis structures. Meanwhile, we investigate the relation between them.

Every multiplicative $\delta$-derivation of an alternative algebra $A$ is additive if there exists an idempotent $e^{\prime }(e^{\prime }\ne 0,1)$ in $A$ satisfying the following conditions: (i) $e^{\prime }Au=0$ implies $A=0$; (ii) $e^{\prime }ueA(1-e)=0$ implies $e^{\prime }ue=0$; (iii) $uA=0$ implies $u=0$ for $e^{\prime }=\delta (e)$. In particular, every $\delta$-derivation of a prime alternative algebra with a nontrivial idempotent is additive. This generalizes the known result obtained by Rodrigues, Guzzo and Ferreira for $\delta$-derivations. As an application, we apply multiplicative $\delta$-derivation to an alternative complex algebra $M_n(ℂ)$ of all $n\times n$ complex matrices to see how it decomposes into a sum of $\delta$-inner derivation and a $\delta$-derivation on $M_n(ℂ)$ given by an additive derivation $\gamma$ on $ℂ$.