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1. CMB Online first
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 Category:16W25 |
2. CMB 2014 (vol 57 pp. 264)
On Semisimple Hopf Algebras of Dimension $pq^n$ Let $p,q$ be prime numbers with $p^2\lt q$, $n\in \mathbb{N}$, and $H$ a
semisimple Hopf algebra of dimension $pq^n$ over an algebraically
closed field of characteristic $0$. This paper proves that $H$ must
possess one of the following structures: (1) $H$ is semisolvable;
(2) $H$ is a Radford biproduct $R\# kG$, where $kG$ is the group
algebra of group $G$ of order $p$, and $R$ is a semisimple Yetter--Drinfeld
Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^n$.
Keywords:semisimple Hopf algebra, semisolvability, Radford biproduct, Drinfeld double Category:16W30 |
3. CMB 2013 (vol 57 pp. 506)
On Braided and Ribbon Unitary Fusion Categories We prove that every braiding over a unitary fusion category is
unitary and every unitary braided fusion category admits a unique
unitary ribbon structure.
Keywords:fusion categories, braided categories, modular categories Categories:20F36, 16W30, 18D10 |
4. CMB 2012 (vol 57 pp. 51)
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 |
5. CMB 2012 (vol 56 pp. 584)
On Automorphisms and Commutativity in Semiprime Rings Let $R$ be a semiprime ring with center
$Z(R)$. For $x,y\in R$, we denote by $[x,y]=xy-yx$ the commutator of
$x$ and $y$. If $\sigma$ is a non-identity automorphism of $R$ such
that
$$
\Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0
$$
for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed
positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$
such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when
$R$ is a prime ring, $R$ is commutative.
Keywords:automorphism, generalized polynomial identity (GPI) Categories:16N60, 16W20, 16R50 |
6. CMB 2011 (vol 56 pp. 344)
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 |
7. CMB 2011 (vol 55 pp. 271)
On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$-graded Algebras Let $F$ be an algebraically closed field of characteristic zero, and
let $A$ be an associative unitary $F$-algebra graded by a group of
prime order. We prove that if $A$ is finite dimensional then the
graded exponent of $A$ exists and is an integer.
Keywords:exponent, polynomial identities, graded algebras Categories:16R50, 16R10, 16W50 |
8. CMB 2011 (vol 55 pp. 579)
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 |
9. CMB 2011 (vol 55 pp. 260)
A Note on the Antipode for Algebraic Quantum Groups Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.
In this note, we show that this formula can be proved for any regular multiplier Hopf
algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a
finite-dimensional Hopf algebra, but also that of any
Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that
the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the
theory of regular multiplier Hopf algebras with integrals.
We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.
Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipode Categories:16W30, 46L65 |
10. CMB 2011 (vol 55 pp. 208)
Abelian Gradings on Upper Block Triangular Matrices Let $G$ be an arbitrary finite abelian group. We describe all
possible $G$-gradings on upper block triangular matrix algebras
over an algebraically closed field of characteristic zero.
Keywords:gradings, upper block triangular matrices Category:16W50 |
11. CMB 2009 (vol 52 pp. 564)
Group Actions on Quasi-Baer Rings A ring $R$ is called {\it quasi-Baer} if the right
annihilator of every right ideal of $R$ is generated by an
idempotent as a right ideal. We investigate the quasi-Baer
property of skew group rings and fixed rings under a finite group
action on a semiprime ring and their applications to
$C^*$-algebras.
Various examples to illustrate and
delimit our results are provided.
Keywords:(quasi-) Baer ring, fixed ring, skew group ring, $C^*$-algebra, local multiplier algebra Categories:16S35, 16W22, 16S90, 16W20, 16U70 |
12. CMB 2009 (vol 52 pp. 39)
A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings We present a new approach to noncommutative real algebraic geometry
based on the representation theory of $C^\ast$-algebras.
An important result in commutative real algebraic geometry is
Jacobi's representation theorem for archimedean quadratic modules
on commutative rings.
We show that this theorem is a consequence of the
Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras.
A noncommutative version of Gelfand--Naimark theory was studied by
I. Fujimoto. We use his results to generalize
Jacobi's theorem to associative rings with involution.
Keywords:Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry Categories:16W80, 46L05, 46L89, 14P99 |
13. CMB 2008 (vol 51 pp. 460)
On Primitive Ideals in Graded Rings Let $R=\bigoplus_{i=1}^{\infty}R_{i}$ be a graded nil ring. It is shown
that primitive ideals in $R$ are homogeneous. Let
$A=\bigoplus_{i=1}^{\infty}A_{i}$ be a graded non-PI just-infinite
dimensional algebra and let $I$ be a prime ideal in $A$. It is shown
that either $I=\{0\}$ or $I=A$. Moreover, $A$ is either primitive or
Jacobson radical.
Categories:16D60, 16W50 |
14. CMB 2008 (vol 51 pp. 424)
Noncommutative Symmetric Bessel Functions The consideration of tensor products of $0$-Hecke algebra modules
leads to natural analogs of the Bessel $J$-functions in the algebra
of noncommutative symmetric functions. This provides a simple explanation
of various combinatorial properties of Bessel functions.
Categories:05E05, 16W30, 05A15 |
15. CMB 2008 (vol 51 pp. 81)
Homotopy Formulas for Cyclic Groups Acting on Rings The positive cohomology groups of a finite group acting on a ring
vanish when the ring has a norm one element. In this note we give
explicit homotopies on the level of cochains when the group is cyclic,
which allows us to express any cocycle of a cyclic group
as the coboundary of an explicit cochain.
The formulas in this note are closely related to the effective problems considered in previous joint work
with Eli Aljadeff.
Keywords:group cohomology, norm map, cyclic group, homotopy Categories:20J06, 20K01, 16W22, 18G35 |
16. CMB 2007 (vol 50 pp. 105)
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 |
17. CMB 2005 (vol 48 pp. 587)
Separation of Variables for $U_{q}(\mathfrak{sl}_{n+1})^{+}$ Let $U_{q}(\SL)^{+}$ be the positive part of the quantized enveloping
algebra $U_{q}(\SL)$. Using results of Alev--Dumas and Caldero related
to the center of $U_{q}(\SL)^{+}$, we show that this algebra is free
over its center. This is reminiscent of Kostant's separation of
variables for the enveloping algebra $U(\g)$ of a complex semisimple
Lie algebra $\g$, and also of an analogous result of Joseph--Letzter
for the quantum algebra $\Check{U}_{q}(\g)$. Of greater importance to
its representation theory is the fact that $\U{+}$ is free over a
larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$
to $\U{+}$ provides infinite-dimensional modules with good properties,
including a grading that is inherited by submodules.
Categories:17B37, 16W35, 17B10, 16D60 |
18. CMB 2005 (vol 48 pp. 355)
On Maps Preserving Products Maps preserving certain algebraic properties of elements
are often studied in Functional Analysis and Linear Algebra. The
goal of this paper is to discuss the relationships among these
problems from the ring-theoretic point of view.
Categories:16W20, 16N50, 16N60 |
19. CMB 2003 (vol 46 pp. 14)
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 |
20. CMB 2002 (vol 45 pp. 499)
Group Gradings on Matrix Algebras Let $\Phi$ be an algebraically closed field of characteristic zero,
$G$ a finite, not necessarily abelian, group. Given a $G$-grading on
the full matrix algebra $A = M_n(\Phi)$, we decompose $A$ as the
tensor product of graded subalgebras $A = B\otimes C$, $B\cong M_p
(\Phi)$ being a graded division algebra, while the grading of $C\cong
M_q (\Phi)$ is determined by that of the vector space $\Phi^n$. Now
the grading of $A$ is recovered from those of $A$ and $B$ using a
canonical ``induction'' procedure.
Category:16W50 |
21. CMB 2002 (vol 45 pp. 711)
Classification of Quantum Tori with Involution Quantum tori with graded involution appear as coordinate algebras of
extended affine Lie algebras of type $\rmA_1$, $\rmC$ and $\BC$.
We classify them in the category of algebras with involution. From
this, we obtain precise information on the root systems of extended
affine Lie algebras of type $\rmC$.
Category:16W50 |
22. CMB 2002 (vol 45 pp. 451)
Coordinatization Theorems For Graded Algebras In this paper we study simple associative algebras with finite
$\mathbb{Z}$-gradings. This is done using a simple algebra $F_g$
that has been constructed in Morita theory from a bilinear form
$g\colon U\times V\to A$ over a simple algebra $A$. We show that
finite $\mathbb{Z}$-gradings on $F_g$ are in one to one
correspondence with certain decompositions of the pair $(U,V)$. We
also show that any simple algebra $R$ with finite
$\mathbb{Z}$-grading is graded isomorphic to $F_g$ for some
bilinear from $g\colon U\times V \to A$, where the grading on $F_g$
is determined by a decomposition of $(U,V)$ and the coordinate
algebra $A$ is chosen as a simple ideal of the zero component $R_0$
of $R$. In order to prove these results we first prove similar
results for simple algebras with Peirce gradings.
Category:16W50 |
23. CMB 2002 (vol 45 pp. 11)
Polycharacters of Cocommutative Hopf Algebras In this paper we extend a well-known theorem of M.~Scheunert on
skew-symmetric bicharacters of groups to the case of skew-symmetric
bicharacters on arbitrary cocommutative Hopf algebras over a field of
characteristic not 2. We also classify polycharacters on (restricted)
enveloping algebras and bicharacters on divided power algebras.
Categories:16W30, 16W55 |
24. CMB 1999 (vol 42 pp. 401)
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 |
25. CMB 1998 (vol 41 pp. 452)
Dependent automorphisms in prime rings For each $n\geq 4$ we construct a class of examples of a minimal
$C$-dependent set of $n$ automorphisms of a prime ring $R$, where $C$
is the extended centroid of $R$. For $n=4$ and $n=5$ it is shown that
the preceding examples are completely general, whereas for $n=6$ an
example is given which fails to enjoy any of the nice properties of
the above example.
Categories:16N60, 16W20 |