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26. CMB 2009 (vol 52 pp. 39)

Cimpri\v{c}, Jakob
 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 geometryCategories:16W80, 46L05, 46L89, 14P99

27. CMB 2008 (vol 51 pp. 460)

Smoktunowicz, Agata
 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)

Novelli, Jean-Christophe; Thibon, Jean-Yves
 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)

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 toriCategory:16S35

30. CMB 2008 (vol 51 pp. 291)

Spinelli, Ernesto
 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 lengthCategories:16S34, 17B30

31. 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, homotopyCategories:20J06, 20K01, 16W22, 18G35

32. 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 involutionCategories:14P10, 16S30, 16W10

33. 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

34. 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$. Keywords:Clean rings, linear transformations, endomorphism ringsCategories:16S50, 16E50 35. CMB 2005 (vol 48 pp. 587) Lopes, Samuel A.  Separation of Variables for$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{+}$provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules. Categories:17B37, 16W35, 17B10, 16D60 36. CMB 2005 (vol 48 pp. 445) Patras, Frédéric; Reutenauer, Christophe; Schocker, Manfred  On the Garsia Lie Idempotent The orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group$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$. Categories:17B01, 05A99, 16S30, 17B60 37. CMB 2005 (vol 48 pp. 355) Chebotar, M. A.; Ke, W.-F.; Lee, P.-H.; Shiao, L.-S.  On Maps Preserving Products Maps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view. Categories:16W20, 16N50, 16N60 38. CMB 2005 (vol 48 pp. 317) Yousif, Mohamed F.; Zhou, Yiqiang; Zeyada, Nasr  On Pseudo-Frobenius Rings It is proved here that a ring$R$is right pseudo-Frobenius if and only if$R $is a right Kasch ring such that the second right singular ideal is injective. Categories:16D50, 16L60 39. CMB 2005 (vol 48 pp. 275) Smith, Patrick F.  Krull Dimension of Injective Modules Over Commutative Noetherian Rings Let$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 40. 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 41. 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 42. 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 wordsCategories:16N40, 16S15, 20M05, 20M25, 68R15 43. 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 44. 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. Category:16W50 45. 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$\rmC$. Category:16W50 46. 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. Category:16W50 47. CMB 2002 (vol 45 pp. 388) Gille, Philippe  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 galoisienneCategories:16S35, 12G05, 20G15 48. 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. Category:16L30 49. 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 50. 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
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