Expand all Collapse all | Results 26 - 50 of 71 |
26. 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 |
27. 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 |
28. 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 |
29. CMB 2008 (vol 51 pp. 261)
On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups An $n$-dimensional quantum torus is a twisted group algebra of the
group $\Z^n$. It is called rational if all invertible commutators are roots
of unity. In the present note we describe a normal form for rational
$n$-dimensional quantum
tori over any field. Moreover, we show that for
$n = 2$ the natural exact sequence
describing the automorphism group of the quantum torus splits over any
field.
Keywords:quantum torus, normal form, automorphisms of quantum tori Category:16S35 |
30. CMB 2008 (vol 51 pp. 291)
Group Algebras with Minimal Strong Lie Derived Length Let $KG$ be a non-commutative strongly Lie solvable group algebra of a
group $G$ over a field $K$ of positive characteristic $p$. In this
note we state necessary and sufficient conditions so that the
strong Lie derived length of $KG$ assumes its minimal value, namely
$\lceil \log_{2}(p+1)\rceil $.
Keywords:group algebras, strong Lie derived length Categories:16S34, 17B30 |
31. 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 |
32. 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 |
33. CMB 2006 (vol 49 pp. 347)
Affine Completeness of Generalised Dihedral Groups In this paper we study affine completeness of generalised dihedral
groups. We give a formula for the number of unary compatible
functions on these groups, and we characterise for every $k \in~\N$
the $k$-affine complete generalised dihedral groups. We find that
the direct product of a $1$-affine complete group with itself need not
be $1$-affine complete. Finally, we give an example of a nonabelian
solvable affine complete group. For nilpotent groups we find a
strong necessary condition for $2$-affine completeness.
Categories:08A40, 16Y30, 20F05 |
34. CMB 2006 (vol 49 pp. 265)
Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in
C[x]$, Camillo and Sim\'{o}n called a ring $g(x)$-clean if every element is
the sum of a unit and a root of $g(x)$. If $V$ is a vector space of
countable dimension over a division ring $D,$ they showed that
$\end {}_{D}V$ is
$g(x)$-clean provided that $g(x)$ has two roots in $C(D)$. If $g(x)=x-x^{2}$
this shows that $\end {}_{D}V$ is clean, a result of Nicholson and Varadarajan.
In this paper we remove the countable condition, and in fact prove that
$\Mend {}_{R}M$ is $g(x)$-clean for any semisimple module $M$ over an arbitrary
ring $R$ provided that $g(x)\in (x-a)(x-b)C[x]$ where $a,b\in C$ and both $b$
and $b-a$ are units in $R$.
Keywords:Clean rings, linear transformations, endomorphism rings Categories:16S50, 16E50 |
35. 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 |
36. CMB 2005 (vol 48 pp. 445)
On the Garsia Lie Idempotent The orthogonal projection of the free associative algebra onto the
free Lie algebra is afforded by an idempotent in the rational group
algebra of the symmetric group $S_n$, in each homogenous degree
$n$. We give various characterizations of this Lie idempotent and show
that it is uniquely determined by a certain unit in the group algebra
of $S_{n-1}$. The inverse of this unit, or, equivalently, the Gram
matrix of the orthogonal projection, is described explicitly. We also
show that the Garsia Lie idempotent is not constant on descent classes
(in fact, not even on coplactic classes) in $S_n$.
Categories:17B01, 05A99, 16S30, 17B60 |
37. 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 |
38. CMB 2005 (vol 48 pp. 317)
On Pseudo-Frobenius Rings It is proved here that a ring $R$ is right pseudo-Frobenius
if and only if $R $ is a right Kasch ring such that the second
right singular ideal is injective.
Categories:16D50, 16L60 |
39. CMB 2005 (vol 48 pp. 275)
Krull Dimension of Injective Modules Over Commutative Noetherian Rings Let $R$ be a commutative Noetherian
integral domain with field of fractions $Q$. Generalizing a
forty-year-old theorem of E. Matlis, we prove that the $R$-module
$Q/R$ (or $Q$) has Krull dimension if and only if $R$ is semilocal
and one-dimensional. Moreover, if $X$ is an injective module over
a commutative Noetherian ring such that $X$ has Krull dimension,
then the Krull dimension of $X$ is at most $1$.
Categories:13E05, 16D50, 16P60 |
40. CMB 2005 (vol 48 pp. 80)
Trivial Units for Group Rings with $G$-adapted Coefficient Rings For each finite group $G$ for which the integral group ring
$\mathbb{Z}G$ has only trivial units, we give ring-theoretic
conditions for a commutative ring $R$ under which the group ring
$RG$ has nontrivial units. Several examples of rings satisfying
the conditions and rings not satisfying the conditions are given.
In addition, we extend a well-known result for fields by showing
that if $R$ is a ring of finite characteristic and $RG$ has only
trivial units, then $G$ has order at most 3.
Categories:16S34, 16U60, 20C05 |
41. CMB 2004 (vol 47 pp. 445)
Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators For a locally compact group $G$, the convolution product on
the space $\nN(L^p(G))$ of nuclear operators was defined by Neufang
\cite{Neuf_PhD}. We study homological properties of the convolution algebra
$\nN(L^p(G))$ and relate them to some properties of the group $G$,
such as compactness, finiteness, discreteness, and amenability.
Categories:46M10, 46H25, 43A20, 16E65 |
42. CMB 2004 (vol 47 pp. 343)
Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras We construct new examples of non-nil algebras with any number of
generators, which are direct sums of two
locally nilpotent subalgebras. Like all previously known examples, our examples
are contracted semigroup algebras and the underlying semigroups are unions
of locally nilpotent subsemigroups.
In our constructions we make more
transparent
than in the past the close relationship between the considered problem
and combinatorics of words.
Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words Categories:16N40, 16S15, 20M05, 20M25, 68R15 |
43. 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 |
44. 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 |
45. 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 |
46. 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 |
47. CMB 2002 (vol 45 pp. 388)
AlgÃ¨bres simples centrales de degrÃ© 5 et $E_8$ As a consequence of a theorem of Rost-Springer, we establish that the
cyclicity problem for central simple algebra of degree~5 on fields
containg a fifth root of unity is equivalent to the study of
anisotropic elements of order 5 in the split group of type~$E_8$.
Keywords:algÃ¨bres simples centrales, cohomologie galoisienne Categories:16S35, 12G05, 20G15 |
48. CMB 2002 (vol 45 pp. 448)
Erratum: A Characterization of Left Perfect Rings An error in {\it A characterization of left perfect rings},
Canad. Math. Bull. (3) {\bf 38}(1995), 382--384, is indicated
and the consequences identified.
Category:16L30 |
49. 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 |
50. CMB 2001 (vol 44 pp. 27)
Normal Subloops in the Integral Loop Ring of an $\RA$ Loop We show that an $\RA$ loop has a torsion-free normal complement in the
loop of normalized units of its integral loop ring. We also
investigate whether an $\RA$ loop can be normal in its unit loop.
Over fields, this can never happen.
Categories:20N05, 17D05, 16S34, 16U60 |