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26. CJM 2002 (vol 54 pp. 595)

Nahlus, Nazih
Lie Algebras of Pro-Affine Algebraic Groups
We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field $K$ of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine algebraic group $G$ over $K$, which is discrete in the finite-dimensional case and linearly compact in general. As an example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in $\mathcal{L}(G)$. We also discuss the Hopf algebra of representative functions $H(L)$ of a residually finite dimensional Lie algebra $L$. As an example, we show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$ is connected, then the canonical Hopf algebra morphism from $K[G]$ into $H(L)$ is injective if and only if $L$ is algebraically dense in $\mathcal{L}(G)$.

Categories:14L, 16W, 17B45

27. CJM 1999 (vol 51 pp. 881)

Witherspoon, Sarah J.
The Representation Ring and the Centre of a Hopf Algebra
When $H$ is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of $H$ which generalizes the conjugacy class sums of a finite group, and express the class equation of $H$ in terms of this basis. We show that the representation ring and the centre of $H$ are dual character algebras (or signed hypergroups).

Categories:16W30, 20N20

28. CJM 1999 (vol 51 pp. 488)

Burgess, W. D.; Saorín, Manuel
Homological Aspects of Semigroup Gradings on Rings and Algebras
This article studies algebras $R$ over a simple artinian ring $A$, presented by a quiver and relations and graded by a semigroup $\Sigma$. Suitable semigroups often arise from a presentation of $R$. Throughout, the algebras need not be finite dimensional. The graded $K_0$, along with the $\Sigma$-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties. A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert $\Sigma$-series in the associated path incidence ring. The rationality of the $\Sigma$-Euler characteristic, the Hilbert $\Sigma$-series and the Poincar\'e-Betti $\Sigma$-series is studied when $\Sigma$ is torsion-free commutative and $A$ is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

Categories:16W50, 16E20, 16G20

29. CJM 1999 (vol 51 pp. 294)

Enochs, Edgar E.; Herzog, Ivo
A Homotopy of Quiver Morphisms with Applications to Representations
It is shown that a morphism of quivers having a certain path lifting property has a decomposition that mimics the decomposition of maps of topological spaces into homotopy equivalences composed with fibrations. Such a decomposition enables one to describe the right adjoint of the restriction of the representation functor along a morphism of quivers having this path lifting property. These right adjoint functors are used to construct injective representations of quivers. As an application, the injective representations of the cyclic quivers are classified when the base ring is left noetherian. In particular, the indecomposable injective representations are described in terms of the injective indecomposable $R$-modules and the injective indecomposable $R[x,x^{-1}]$-modules.

Categories:18A40, 16599

30. CJM 1999 (vol 51 pp. 69)

Reichstein, Zinovy
On a Theorem of Hermite and Joubert
A classical theorem of Hermite and Joubert asserts that any field extension of degree $n=5$ or $6$ is generated by an element whose minimal polynomial is of the form $\lambda^n + c_1 \lambda^{n-1} + \cdots + c_{n-1} \lambda + c_n$ with $c_1=c_3=0$. We show that this theorem fails for $n=3^m$ or $3^m + 3^l$ (and more generally, for $n = p^m$ or $p^m + p^l$, if 3 is replaced by another prime $p$), where $m > l \geq 0$. We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\UD (n)$. We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\UD(n)$.

Categories:12E05, 16K20

31. CJM 1998 (vol 50 pp. 356)

Gross, Leonard
Some norms on universal enveloping algebras
The universal enveloping algebra, $U(\frak g)$, of a Lie algebra $\frak g$ supports some norms and seminorms that have arisen naturally in the context of heat kernel analysis on Lie groups. These norms and seminorms are investigated here from an algebraic viewpoint. It is shown that the norms corresponding to heat kernels on the associated Lie groups decompose as product norms under the natural isomorphism $U(\frak g_1 \oplus \frak g_2) \cong U(\frak g_1) \otimes U(\frak g_2)$. The seminorms corresponding to Green's functions are examined at a purely Lie algebra level for $\rmsl(2,\Bbb C)$. It is also shown that the algebraic dual space $U'$ is spanned by its finite rank elements if and only if $\frak g$ is nilpotent.

Categories:17B35, 16S30, 22E30

32. CJM 1998 (vol 50 pp. 401)

Li, Yuanlin
The hypercentre and the $n$-centre of the unit group of an integral group ring
In this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most $2$. We then give a complete characterization of the $n$-centre of that unit group. The $n$-centre of the unit group is either the centre or the second centre (for $n \geq 2$).

Categories:16U60, 20C05

33. CJM 1998 (vol 50 pp. 312)

Dokuchaev, Michael A.; Singer, Maria Lucia Sobral
Units in group rings of free products of prime cyclic groups
Let $G$ be a free product of cyclic groups of prime order. The structure of the unit group ${\cal U}(\Q G)$ of the rational group ring $\Q G$ is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups of ${\cal U}(\Q G)$, up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in $\Z G$ is proved. A strong version of the Tits Alternative for ${\cal U}(\Q G)$ is obtained as a corollary of the structural result.

Keywords:Free Products, Units in group rings, Zassenhaus Conjecture
Categories:20C07, 16S34, 16U60, 20E06

34. CJM 1998 (vol 50 pp. 3)

Amberg, B.; Dickenschied, O.; Sysak, Ya. P.
Subgroups of the adjoint group of a radical ring
It is shown that the adjoint group $R^\circ$ of an arbitrary radical ring $R$ has a series with abelian factors and that its finite subgroups are nilpotent. Moreover, some criteria for subgroups of $R^\circ$ to be locally nilpotent are given.

Categories:16N20, 20F19

35. CJM 1997 (vol 49 pp. 1265)

Snaith, V. P.
Hecke algebras and class-group invariant
Let $G$ be a finite group. To a set of subgroups of order two we associate a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its units to the class-group of ${\bf Z}[G]$. We show that this homomorphism takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimensional algebraic $K$-groups of rings of algebraic integers often differ by elements in the image of $\psi$. As an application we show that two such constructions coincide.

Categories:16S34, 19A99, 11R65

36. CJM 1997 (vol 49 pp. 788)

Lichtman, A. I.
Trace functions in the ring of fractions of polycyclic group rings, II
We prove the existence of trace functions in the rings of fractions of polycyclic-by-finite group rings or their homomorphic images. In particular a trace function exists in the ring of fractions of $KH$, where $H$ is a polycyclic-by-finite group and $\char K > N$, where $N$ is a constant depending on $H$.

Categories:20C07, 16A08, 16A39

37. CJM 1997 (vol 49 pp. 772)

Jie, Xiao
Finite dimensional representations of $U_t\bigl(\rmsl (2)\bigr)$ at roots of unity
All finite dimensional indecomposable representations of $U_t (\rmsl (2))$ at roots of $1$ are determined.

Categories:16G10, 16G70, 17B37
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