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Search: MSC category 16W10 ( Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx] )

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1. 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

2. CMB 2011 (vol 56 pp. 344)

Goodaire, Edgar G.; Milies, César Polcino
Involutions and Anticommutativity in Group Rings
Let $g\mapsto g^*$ denote an involution on a group $G$. For any (commutative, associative) ring $R$ (with $1$), $*$ extends linearly to an involution of the group ring $RG$. An element $\alpha\in RG$ is symmetric if $\alpha^*=\alpha$ and skew-symmetric if $\alpha^*=-\alpha$. The skew-symmetric elements are closed under the Lie bracket, $[\alpha,\beta]=\alpha\beta-\beta\alpha$. In this paper, we investigate when this set is also closed under the ring product in $RG$. The symmetric elements are closed under the Jordan product, $\alpha\circ\beta=\alpha\beta+\beta\alpha$. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

Categories:16W10, 16S34

3. CMB 2007 (vol 50 pp. 105)

Klep, Igor
On Valuations, Places and Graded Rings Associated to $*$-Orderings
We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups.

Keywords:$*$--orderings, valuations, rings with involution
Categories:14P10, 16S30, 16W10

4. CMB 2003 (vol 46 pp. 14)

Bahturin, Yu. A.; Parmenter, M. M.
Generalized Commutativity in Group Algebras
We study group algebras $FG$ which can be graded by a finite abelian group $\Gamma$ such that $FG$ is $\beta$-commutative for a skew-symmetric bicharacter $\beta$ on $\Gamma$ with values in $F^*$.

Categories:16S34, 16R50, 16U80, 16W10, 16W55

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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