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51. CMB 2008 (vol 51 pp. 261)

Neeb, Karl-Hermann
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

52. CMB 2008 (vol 51 pp. 81)

Kassel, Christian
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

53. 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

54. CMB 2006 (vol 49 pp. 347)

Ecker, Jürgen
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

55. CMB 2006 (vol 49 pp. 265)

Nicholson, W. K.; Zhou, Y.
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

56. CMB 2005 (vol 48 pp. 587)

Lopes, Samuel A.
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

57. CMB 2005 (vol 48 pp. 445)

Patras, Frédéric; Reutenauer, Christophe; Schocker, Manfred
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

58. CMB 2005 (vol 48 pp. 355)

Chebotar, M. A.; Ke, W.-F.; Lee, P.-H.; Shiao, L.-S.
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

59. CMB 2005 (vol 48 pp. 275)

Smith, Patrick F.
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

60. CMB 2005 (vol 48 pp. 317)

Yousif, Mohamed F.; Zhou, Yiqiang; Zeyada, Nasr
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

61. CMB 2005 (vol 48 pp. 80)

Herman, Allen; Li, Yuanlin; Parmenter, M. M.
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

62. CMB 2004 (vol 47 pp. 343)

Drensky, Vesselin; Hammoudi, Lakhdar
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

63. CMB 2004 (vol 47 pp. 445)

Pirkovskii, A. Yu.
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

64. 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

65. CMB 2002 (vol 45 pp. 499)

Bahturin, Yu. A.; Zaicev, M. V.
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.


66. CMB 2002 (vol 45 pp. 711)

Yoshii, Yoji
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 $\rmC$.


67. CMB 2002 (vol 45 pp. 451)

Allison, Bruce; Smirnov, Oleg
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.


68. CMB 2002 (vol 45 pp. 388)

Gille, Philippe
Algèbres simples centrales de degré 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:algèbres simples centrales, cohomologie galoisienne
Categories:16S35, 12G05, 20G15

69. CMB 2002 (vol 45 pp. 448)

Zhou, Yiqiang
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.


70. CMB 2002 (vol 45 pp. 11)

Bahturin, Yuri; Kochetov, Mikhail; Montgomery, Susan
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

71. CMB 2001 (vol 44 pp. 27)

Goodaire, Edgar G.; Milies, César Polcino
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

72. CMB 2000 (vol 43 pp. 413)

Chatters, A. W.
Non-Isomorphic Maximal Orders with Isomorphic Matrix Rings
We construct a countably infinite family of pairwise non-isomorphic maximal ${\mathbb Q}[X]$-orders such that the full $2$ by $2$ matrix rings over these orders are all isomorphic.

Categories:16S50, 16H05, 16N60

73. CMB 2000 (vol 43 pp. 3)

Adin, Ron; Blanc, David
Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedron
Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

74. CMB 2000 (vol 43 pp. 100)

Okon, James S.; Vicknair, J. Paul
A Gorenstein Ring with Larger Dilworth Number than Sperner Number
A counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension $3$ with larger Dilworth number than Sperner number. The Dilworth number of $A[Z/pZ\oplus Z/pZ]$ is computed when $A$ is an unramified principal Artin local ring.

Categories:13E15, 16S34

75. CMB 2000 (vol 43 pp. 79)

König, Steffen
Cyclotomic Schur Algebras and Blocks of Cyclic Defect
An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

Categories:20G05, 20C20, 16G30, 17B37, 57M25
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