Narrow Results

We show that for any finite group $G$, there exist infinitely many simple anticommutative algebras with multiplicative bases $(𝔄,\mathcal{ℬ})$ such that the group $\mathrm{\text{Aut}}((𝔄,\mathcal{ℬ}))$ is isomorphic to $G$.

In this paper, we prove some general results on Leibniz 2-cocyles for simple Leibniz algebras. Applying these results, we prove the triviality of the second Leibniz cohomology for a simple Leibniz algebra with coefficients in itself, whose associated Lie algebra is isomorphic to ${\mathrm{\text{sl}}}_2$.

Let $R$ be a simple algebra over its extended centroid $C$, and let $f(X_1,\dots ,X_n)$ be a noncommutative polynomial having zero constant term. We denote by $f{(R)}^{+}$ the additive subgroup of $R$ generated by all elements $f(x_1,\dots ,x_n)$ for $x_i\in R$. It is proved that if $\mathrm{\text{char}}R=0$, then $f{(R)}^{+}$ is equal to $\{0\}$, $C$, $[R,R]$, or $R$. As to the case $\mathrm{\text{char}}R=p>0$, an example of a polynomial $f$ satisfying $[R,R]⊊f{(R)}^{+}⊊R$ is given. Also, the polynomials $f$ with $f{(R)}^{+}=[R,R]$ are characterized if $\mathrm{\text{char}}R=0$ and $R\ne [R,R]$. Moreover, we work on the context of centrally closed prime algebras to get more general results.

For a given algebra A= 〈A,+,·〉, we can define its anti-symmetric algebra A^-= 〈A^-,+,[ , ]〉 using the commutator [ , ] of A, where the sets A and A^- are the same. We show that there are isomorphic algebras A₁ and A₂ such that their anti-symmetric algebras are not isomorphic. We define a special type Lie algebra and show that it is simple.

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal.

Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

Using a given ring, we define the Weyl-type non-associative rings in this work. If the ring is simple, then we show that the Weyl-type non-associative ring is simple in this work. All the derivations of a Weyl-type non-associative algebra are found in the papers (see [4, 6]). In this work, we find all the derivations of a Weyl-type algebra which includes the algebras in the papers (see [4, 6]). We show that the associators of the algebras and are zeroes.