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Search: MSC category 16W25 ( Derivations, actions of Lie algebras )

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1. CMB Online first

Filippis, Vincenzo De
Annihilators and power values of generalized skew derivations on Lie ideals
Let $R$ be a prime ring of characteristic different from $2$, $Q_r$ be its right Martindale quotient ring and $C$ be its extended centroid. Suppose that $F$ is a generalized skew derivation of $R$, $L$ a non-central Lie ideal of $R$, $0 \neq a\in R$, $m\geq 0$ and $n,s\geq 1$ fixed integers. If \[ a\biggl(u^mF(u)u^n\biggr)^s=0 \] for all $u\in L$, then either $R\subseteq M_2(C)$, the ring of $2\times 2$ matrices over $C$, or $m=0$ and there exists $b\in Q_r$ such that $F(x)=bx$, for any $x\in R$, with $ab=0$.

Keywords:generalized skew derivation, prime ring
Categories:16W25, 16N60

2. CMB 2015 (vol 58 pp. 263)

De Filippis, Vincenzo; Mamouni, Abdellah; Oukhtite, Lahcen
Generalized Jordan Semiderivations in Prime Rings
Let $R$ be a ring, $g$ an endomorphism of $R$. The additive mapping $d\colon R\rightarrow R$ is called Jordan semiderivation of $R$, associated with $g$, if $$d(x^2)=d(x)x+g(x)d(x)=d(x)g(x)+xd(x)\quad \text{and}\quad d(g(x))=g(d(x))$$ for all $x\in R$. The additive mapping $F\colon R\rightarrow R$ is called generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if $$F(x^2)=F(x)x+g(x)d(x)=F(x)g(x)+xd(x)\quad \text{and}\quad F(g(x))=g(F(x))$$ for all $x\in R$. In the present paper we prove that if $R$ is a prime ring of characteristic different from $2$, $g$ an endomorphism of $R$, $d$ a Jordan semiderivation associated with $g$, $F$ a generalized Jordan semiderivation associated with $d$ and $g$, then $F$ is a generalized semiderivation of $R$ and $d$ is a semiderivation of $R.$ Moreover, if $R$ is commutative then $F=d$.

Keywords:semiderivation, generalized semiderivation, Jordan semiderivation, prime ring

3. CMB 2012 (vol 57 pp. 51)

Fošner, Ajda; Lee, Tsiu-Kwen
Jordan $*$-Derivations of Finite-Dimensional Semiprime Algebras
In the paper, we characterize Jordan $*$-derivations of a $2$-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R=Uoplus V$ such that both $U$ and $V$ are invariant under $delta$. Moreover, $*$ is the identity map of $U$ and $delta,|_U$ is a derivation, and the Jordan $*$-derivation $delta,|_V$ is inner. We also prove the theorem: Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, $operatorname{char},R e 2$, and let $delta$ be a nonzero Jordan $*$-derivation of $R$. If $delta$ is an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and $delta$ is inner.

Keywords:semiprime algebra, involution, (inner) Jordan $*$-derivation, elementary operator
Categories:16W10, 16N60, 16W25

4. CMB 2011 (vol 55 pp. 579)

Ndogmo, J. C.
Casimir Operators and Nilpotent Radicals
It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariants
Categories:16W25, 17B45, 16S30

5. CMB 1999 (vol 42 pp. 401)

Swain, Gordon A.; Blau, Philip S.
Lie Derivations in Prime Rings With Involution
Let $R$ be a non-GPI prime ring with involution and characteristic $\neq 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$ into itself. Then $\delta=\rho+\epsilon$ where $\epsilon$ is an additive map into the skew elements of the extended centroid of $R$ which is zero on $[K,K]$, and $\rho$ can be extended to an ordinary derivation of $\langle K\rangle$ into $RC$, the central closure.

Categories:16W10, 16N60, 16W25

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