26. CJM 2003 (vol 55 pp. 766)
 Kerler, Thomas

Homology TQFT's and the AlexanderReidemeister Invariant of 3Manifolds via Hopf Algebras and Skein Theory
We develop an explicit skeintheoretical algorithm to compute the
Alexander polynomial of a 3manifold from a surgery presentation
employing the methods used in the construction of quantum invariants
of 3manifolds. As a prerequisite we establish and prove a rather
unexpected equivalence between the topological quantum field theory
constructed by Frohman and Nicas using the homology of
$U(1)$representation varieties on the one side and the
combinatorially constructed Hennings TQFT based on the quasitriangular
Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^*
\mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL
(2,\mathbb{R})$equivariant functors and, as such, are isomorphic.
The $\SL (2,\mathbb{R})$action in the Hennings construction comes
from the natural action on $\mathcal{N}$ and in the case of the
FrohmanNicas theory from the HardLefschetz decomposition of the
$U(1)$moduli spaces given that they are naturally K\"ahler. The
irreducible components of this TQFT, corresponding to simple
representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus
yield a large family of homological TQFT's by taking sums and products.
We give several examples of TQFT's and invariants that appear to fit
into this family, such as Milnor and Reidemeister Torsion,
SeibergWitten theories, Casson type theories for homology circles
{\it \`a la} Donaldson, higher rank gauge theories following Frohman
and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of
ReshetikhinTuraev theories over the cyclotomic integers $\mathbb{Z}
[\zeta_p]$. We also conjecture that the Hennings TQFT for
quantum$\mathfrak{sl}_2$ is the product of the ReshetikhinTuraev
TQFT and such a homological TQFT.
Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27 

27. CJM 2003 (vol 55 pp. 42)
 Benanti, Francesca; Di Vincenzo, Onofrio M.; Nardozza, Vincenzo

$*$Subvarieties of the Variety Generated by $\bigl( M_2(\mathbb{K}),t \bigr)$
Let $\mathbb{K}$ be a field of characteristic zero, and $*=t$ the
transpose involution for the matrix algebra $M_2 (\mathbb{K})$. Let
$\mathfrak{U}$ be a proper subvariety of the variety of algebras with
involution generated by $\bigl( M_2 (\mathbb{K}),* \bigr)$. We define
two sequences of algebras with involution $\mathcal{R}_p$,
$\mathcal{S}_q$, where $p,q \in \mathbb{N}$. Then we show that
$T_* (\mathfrak{U})$ and $T_* (\mathcal{R}_p \oplus \mathcal{S}_q)$
are $*$asymptotically equivalent for suitable $p,q$.
Keywords:algebras with involution, asymptotic equivalence Categories:16R10, 16W10, 16R50 

28. CJM 2002 (vol 54 pp. 1319)
 Yekutieli, Amnon

The Continuous Hochschild Cochain Complex of a Scheme
Let $X$ be a separated finite type scheme over a noetherian base ring
$\mathbb{K}$. There is a complex $\widehat{\mathcal{C}}^{\cdot} (X)$
of topological $\mathcal{O}_X$modules, called the complete Hochschild
chain complex of $X$. To any $\mathcal{O}_X$module
$\mathcal{M}$not necessarily quasicoherentwe assign the complex
$\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr)$ of continuous
Hochschild cochains with values in $\mathcal{M}$. Our first main
result is that when $X$ is smooth over $\mathbb{K}$ there is a
functorial isomorphism
$$
\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr) \cong \R
\mathcal{H}om_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M})
$$
in the derived category $\mathsf{D} (\Mod \mathcal{O}_{X^2})$, where
$X^2 := X \times_{\mathbb{K}} X$.
The second main result is that if $X$ is smooth of relative dimension
$n$ and $n!$ is invertible in $\mathbb{K}$, then the standard maps
$\pi \colon \widehat{\mathcal{C}}^{q} (X) \to \Omega^q_{X/
\mathbb{K}}$ induce a quasiisomorphism
$$
\mathcal{H}om_{\mathcal{O}_X} \Bigl( \bigoplus_q \Omega^q_{X/
\mathbb{K}} [q], \mathcal{M} \Bigr) \to
\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr).
$$
When $\mathcal{M} = \mathcal{O}_X$ this is the quasiisomorphism
underlying the Kontsevich Formality Theorem.
Combining the two results above we deduce a decomposition of the
global Hochschild cohomology
$$
\Ext^i_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) \cong
\bigoplus_q \H^{iq} \Bigl( X, \bigl( \bigwedge^q_{\mathcal{O}_X}
\mathcal{T}_{X/\mathbb{K}} \bigr) \otimes_{\mathcal{O}_X} \mathcal{M}
\Bigr),
$$
where $\mathcal{T}_{X/\mathbb{K}}$ is the relative tangent sheaf.
Keywords:Hochschild cohomology, schemes, derived categories Categories:16E40, 14F10, 18G10, 13H10 

29. CJM 2002 (vol 54 pp. 897)
 Fortuny Ayuso, Pedro

The Valuative Theory of Foliations
This paper gives a characterization of valuations that follow the
singular infinitely near points of plane vector fields, using the
notion of L'H\^opital valuation, which generalizes a well known classical
condition. With that tool, we give a valuative description of vector
fields with infinite solutions, singularities with rational quotient
of eigenvalues in its linear part, and polynomial vector fields with
transcendental solutions, among other results.
Categories:12J20, 13F30, 16W60, 37F75, 34M25 

30. CJM 2002 (vol 54 pp. 595)
 Nahlus, Nazih

Lie Algebras of ProAffine Algebraic Groups
We extend the basic theory of Lie algebras of affine algebraic groups
to the case of proaffine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the prodiscrete
topology on the Lie algebra $\mathcal{L}(G)$ of the proaffine
algebraic group $G$ over $K$, which is discrete in the
finitedimensional 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 

31. 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 

32. 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\'eBetti $\Sigma$series is studied
when $\Sigma$ is torsionfree 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 

33. 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 

34. 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 $\lambda^n + c_1 \lambda^{n1} +
\cdots + c_{n1} \lambda + c_n$ 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 

35. 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 

36. CJM 1998 (vol 50 pp. 401)
37. 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 

38. CJM 1998 (vol 50 pp. 3)
39. CJM 1997 (vol 49 pp. 1265)
 Snaith, V. P.

Hecke algebras and classgroup 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 classgroup 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
higherdimensional 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 

40. 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
polycyclicbyfinite group rings or their homomorphic images. In
particular a trace function exists in the ring of fractions of $KH$,
where $H$ is a polycyclicbyfinite group and $\char K > N$, where
$N$ is a constant depending on $H$.
Categories:20C07, 16A08, 16A39 

41. CJM 1997 (vol 49 pp. 772)