26. CJM 2003 (vol 55 pp. 1155)
 Đoković, Dragomir Ž.; Litvinov, Michael

The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair $(\SO_{p+q},\SO_p\times\SO_q)$
The main problem that is solved in this paper has the following simple
formulation (which is not used in its solution). The group $K =
\mathrm{O}_p ({\bf C}) \times \mathrm{O}_q ({\bf C})$ acts on the
space $M_{p,q}$ of $p\times q$ complex matrices by $(a,b) \cdot x =
axb^{1}$, and so does its identity component $K^0 = \SO_p ({\bf C})
\times \SO_q ({\bf C})$. A $K$orbit (or $K^0$orbit) in $M_{p,q}$ is said
to be nilpotent if its closure contains the zero matrix. The closure,
$\overline{\mathcal{O}}$, of a nilpotent $K$orbit (resp.\ $K^0$orbit)
${\mathcal{O}}$ in $M_{p,q}$ is a union of ${\mathcal{O}}$ and some
nilpotent $K$orbits (resp.\ $K^0$orbits) of smaller dimensions. The
description of the closure of nilpotent $K$orbits has been known for
some time, but not so for the nilpotent $K^0$orbits. A conjecture
describing the closure of nilpotent $K^0$orbits was proposed in
\cite{DLS} and verified when $\min(p,q) \le 7$. In this paper we
prove the conjecture. The proof is based on a study of two
prehomogeneous vector spaces attached to $\mathcal{O}$ and
determination of the basic relative invariants of these spaces.
The above problem is equivalent to the problem of describing the
closure of nilpotent orbits in the real Lie algebra $\mathfrak{so}
(p,q)$ under the adjoint action of the identity component of the real
orthogonal group $\mathrm{O}(p,q)$.
Keywords:orthogonal $ab$diagrams, prehomogeneous vector spaces, relative invariants Categories:17B20, 17B45, 22E47 

27. CJM 2003 (vol 55 pp. 856)
 Su, Yucai

Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to LocallyFinite Derivations
Xu introduced a class of nongraded Hamiltonian Lie algebras. These
Lie algebras have a Poisson bracket structure. In this paper, the
isomorphism classes of these Lie algebras are determined by employing
a ``sandwich'' method and by studying some features of these Lie
algebras. It is obtained that two Hamiltonian Lie algebras are
isomorphic if and only if their corresponding Poisson algebras are
isomorphic. Furthermore, the derivation algebras and the second
cohomology groups are determined.
Categories:17B40, 17B65 

28. 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 

29. 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 

30. CJM 2001 (vol 53 pp. 225)
 Britten, D. J.; Lemire, F. W.

Tensor Product Realizations of Simple Torsion Free Modules
Let $\calG$ be a finite dimensional simple Lie algebra over the
complex numbers $C$. Fernando reduced the classification of infinite
dimensional simple $\calG$modules with a finite dimensional weight
space to determining the simple torsion free $\calG$modules for
$\calG$ of type $A$ or $C$. These modules were determined by Mathieu
and using his work we provide a more elementary construction realizing
each one as a submodule of an easily constructed tensor product module.
Category:17B10 

31. CJM 2001 (vol 53 pp. 195)
 Mokler, Claus

On the Steinberg Map and Steinberg CrossSection for a Symmetrizable Indefinite KacMoody Group
Let $G$ be a symmetrizable indefinite KacMoody group over $\C$. Let
$\Tr_{\La_1},\dots,\Tr_{\La_{2nl}}$ be the characters of the
fundamental irreducible representations of $G$, defined as convergent
series on a certain part $G^{\tralg} \subseteq G$. Following
Steinberg in the classical case and Br\"uchert in the affine case, we
define the Steinberg map $\chi := (\Tr_{\La_1},\dots,
\Tr_{\La_{2nl}})$ as well as the Steinberg cross section $C$,
together with a natural parametrisation $\omega \colon \C^{n} \times
(\C^\times)^{\,nl} \to C$. We investigate the local behaviour of
$\chi$ on $C$ near $\omega \bigl( (0,\dots,0) \times (1,\dots,1)
\bigr)$, and we show that there exists a neighborhood of $(0,\dots,0)
\times (1,\dots,1)$, on which $\chi \circ \omega$ is a regular
analytical map, satisfying a certain functional identity. This
identity has its origin in an action of the center of $G$ on~$C$.
Categories:22E65, 17B65 

32. CJM 2000 (vol 52 pp. 503)
 Gannon, Terry

The Level 2 and 3 Modular Invariants for the Orthogonal Algebras
The `1loop partition function' of a rational conformal field theory
is a sesquilinear combination of characters, invariant under a natural
action of $\SL_2(\bbZ)$, and obeying an integrality condition.
Classifying these is a clearly defined mathematical problem, and at
least for the affine KacMoody algebras tends to have interesting
solutions. This paper finds for each affine algebra $B_r^{(1)}$ and
$D_r^{(1)}$ all of these at level $k\le 3$. Previously, only those at
level 1 were classified. An extraordinary number of exceptionals
appear at level 2the $B_r^{(1)}$, $D_r^{(1)}$ level 2
classification is easily the most anomalous one known and this
uniqueness is the primary motivation for this paper. The only level 3
exceptionals occur for $B_2^{(1)} \cong C_2^{(1)}$ and $D_7^{(1)}$.
The $B_{2,3}$ and $D_{7,3}$ exceptionals are cousins of the ${\cal
E}_6$exceptional and $\E_8$exceptional, respectively, in the
ADE classification for $A_1^{(1)}$, while the level 2 exceptionals
are related to the lattice invariants of affine~$u(1)$.
Keywords:KacMoody algebra, conformal field theory, modular invariants Categories:17B67, 81T40 

33. CJM 2000 (vol 52 pp. 141)
 Li, ChiKwong; Tam, TinYau

Numerical Ranges Arising from Simple Lie Algebras
A unified formulation is given to various generalizations of the
classical numerical range including the $c$numerical range,
congruence numerical range, $q$numerical range and von Neumann
range. Attention is given to those cases having connections with
classical simple real Lie algebras. Convexity and inclusion
relation involving those generalized numerical ranges are
investigated. The underlying geometry is emphasized.
Keywords:numerical range, convexity, inclusion relation Categories:15A60, 17B20 

34. CJM 1999 (vol 51 pp. 506)
35. CJM 1999 (vol 51 pp. 658)
 Shumyatsky, Pavel

Nilpotency of Some Lie Algebras Associated with $p$Groups
Let $ L=L_0+L_1$ be a $\mathbb{Z}_2$graded Lie algebra over a
commutative ring with unity in which $2$ is invertible. Suppose
that $L_0$ is abelian and $L$ is generated by finitely many
homogeneous elements $a_1,\dots,a_k$ such that every commutator in
$a_1,\dots,a_k$ is adnilpotent. We prove that $L$ is nilpotent.
This implies that any periodic residually finite $2'$group $G$
admitting an involutory automorphism $\phi$ with $C_G(\phi)$
abelian is locally finite.
Categories:17B70, 20F50 

36. CJM 1999 (vol 51 pp. 523)
 Fabbri, Marc A.; Okoh, Frank

Representations of VirasoroHeisenberg Algebras and VirasoroToroidal Algebras
Virasorotoroidal algebras, $\tilde\mathcal{T}_{[n]}$, are
semidirect products of toroidal algebras $\mathcal{T}_{[n]}$ and
the Virasoro algebra. The toroidal algebras are, in turn,
multiloop versions of affine KacMoody algebras. Let $\Gamma$ be
an extension of a simply laced lattice $\dot{Q}$ by a hyperbolic
lattice of rank two. There is a Fock space $V(\Gamma)$
corresponding to $\Gamma$ with a decomposition as a complex vector
space: $V(\Gamma) = \coprod_{m \in \mathbf{Z}}K(m)$. Fabbri and
Moody have shown that when $m \neq 0$, $K(m)$ is an irreducible
representation of $\tilde\mathcal{T}_{[2]}$. In this paper we
produce a filtration of $\tilde\mathcal{T}_{[2]}$submodules of
$K(0)$. When $L$ is an arbitrary geometric lattice and $n$ is a
positive integer, we construct a VirasoroHeisenberg algebra
$\tilde\mathcal{H}(L,n)$. Let $Q$ be an extension of $\dot{Q}$ by
a degenerate rank one lattice. We determine the components of
$V(\Gamma)$ that are irreducible $\tilde\mathcal{H}(Q,1)$modules
and we show that the reducible components have a filtration of
$\tilde\mathcal{H}(Q,1)$submodules with completely reducible
quotients. Analogous results are obtained for $\tilde\mathcal{H}
(\dot{Q},2)$. These results complement and extend results of
Fabbri and Moody.
Categories:17B65, 17B68 

37. CJM 1998 (vol 50 pp. 1323)
 Morales, Jorge

L'invariant de HasseWitt de la forme de Killing
Nous montrons que l'invariant de HasseWitt de la forme de Killing
d'une alg{\`e}bre de Lie semisimple $L$ s'exprime {\`a} l'aide de
l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de
$L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines
positives), et des invariants associ{\'e}s au groupe des
sym{\'e}tries du diagramme de Dynkin de $L$.
Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66 

38. CJM 1998 (vol 50 pp. 929)
 Broer, Abraham

Decomposition varieties in semisimple Lie algebras
The notion of decompositon class in a semisimple Lie algebra is a
common generalization of nilpotent orbits and the set of
regular semisimple elements. We prove that the closure of a
decomposition class has many properties in common with nilpotent
varieties, \eg, its normalization has rational singularities.
The famous Grothendieck simultaneous resolution is related to the
decomposition class of regular semisimple elements. We study the
properties of the analogous commutative diagrams associated to
an arbitrary decomposition class.
Categories:14L30, 14M17, 15A30, 17B45 

39. CJM 1998 (vol 50 pp. 972)
 Brüchert, Gerd

Trace class elements and crosssections in KacMoody groups
Let $G$ be an affine KacMoody group, $\pi_0,\dots,\pi_r,\pi_{\delta}$
its fundamental irreducible representations and $\chi_0, \dots,
\chi_r, \chi_{\delta}$ their characters. We determine the set of all
group elements $x$ such that all $\pi_i(x)$ act as trace class
operators, \ie, such that $\chi_i(x)$ exists, then prove that the
$\chi_i$ are class functions. Thus, $\chi:=(\chi_0, \dots, \chi_r,
\chi_{\delta})$ factors to an adjoint quotient $\bar{\chi}$ for $G$.
In a second part, following Steinberg, we define a crosssection $C$
for the potential regular classes in $G$. We prove that the
restriction $\chi_C$ behaves well algebraically. Moreover, we obtain
an action of $\hbox{\Bbbvii C}^{\times}$ on $C$, which leads to a
functional identity for $\chi_C$ which shows that $\chi_C$ is
quasihomogeneous.
Categories:22E65, 17B67 

40. CJM 1998 (vol 50 pp. 816)
41. CJM 1998 (vol 50 pp. 266)
 Britten, D. J.; Lemire, F. W.

The torsion free Pieri formula
Central to the study of simple infinite dimensional
$g\ell(n, \Bbb C)$modules having finite dimensional weight spaces are the
torsion free modules. All degree $1$ torsion free modules are known.
Torsion free modules of arbitrary degree can be constructed by tensoring
torsion free modules of degree $1$ with finite dimensional simple modules.
In this paper, the central characters of such a tensor product module are
shown to be given by a Pierilike formula, complete reducibility is
established when these central characters are distinct and an example
is presented illustrating the existence of a nonsimple indecomposable
submodule when these characters are not distinct.
Category:17B10 

42. 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 

43. CJM 1998 (vol 50 pp. 225)
 Benkart, Georgia

Derivations and invariant forms of Lie algebras graded by finite root systems
Lie algebras graded by finite reduced root systems have been
classified up to isomorphism. In this paper we describe the derivation
algebras of these Lie algebras and determine when they possess invariant
bilinear forms. The results which we develop to do this are much more
general and apply to Lie algebras that are completely reducible with
respect to the adjoint action of a finitedimensional subalgebra.
Categories:17B20, 17B70, 17B25 

44. CJM 1998 (vol 50 pp. 210)
 Zhao, Kaiming

Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$
In this paper, we determine when two simple generalized Cartan
type $W$ Lie algebras $W_d (A, T, \varphi)$ are isomorphic, and discuss
the relationship between the Jacobian conjecture and the generalized
Cartan type $W$ Lie algebras.
Keywords:Simple Lie algebras, the general Lie algebra, generalized Cartan type $W$ Lie algebras, isomorphism, Jacobian conjecture Categories:17B40, 17B65, 17B56, 17B68 

45. CJM 1997 (vol 49 pp. 1206)
 Letzter, Gail

Subalgebras which appear in quantum Iwasawa decompositions
Let $g$ be a semisimple Lie algebra. Quantum analogs of the
enveloping algebra of the fixed Lie subalgebra are introduced for
involutions corresponding to the negative of a diagram automorphism.
These subalgebras of the quantized enveloping algebra specialize to
their classical counterparts. They are used to form an Iwasawa type
decompostition and begin a study of quantum HarishChandra modules.
Category:17B37 

46. CJM 1997 (vol 49 pp. 772)
47. CJM 1997 (vol 49 pp. 820)
 Robart, Thierry

Sur l'intÃ©grabilitÃ© des sousalgÃ¨bres de Lie en dimension infinie
Une des questions fondamentales de la th\'eorie des groupes de
Lie de dimension infinie concerne l'int\'egrabilit\'e des
sousalg\`ebres de Lie topologiques $\cal H$ de l'alg\`ebre
de Lie $\cal G$ d'un groupe de Lie $G$ de dimension infinie
au sens de Milnor. Par contraste avec ce qui se passe en
th\'eorie classique il peut exister des sousalg\`ebres de Lie
ferm\'ees $\cal H$ de $\cal G$ nonint\'egrables en un
sousgroupe de Lie. C'est le cas des alg\`ebres de Lie de champs
de vecteurs $C^{\infty}$ d'une vari\'et\'e compacte qui ne
d\'efinissent pas un feuilletage de Stefan. Heureusement cette
``imperfection" de la th\'eorie n'est pas partag\'ee par tous les
groupes de Lie int\'eressants. C'est ce que montre cet article
en exhibant une tr\`es large classe de groupes de Lie de
dimension infinie exempte de cette imperfection. Cela permet de
traiter compl\`etement le second probl\`eme fondamental de
Sophus Lie pour les groupes de jauge de la
physiquemath\'ematique et les groupes formels de
diff\'eomorphismes lisses de $\R^n$ qui fixent l'origine.
Categories:22E65, 58h05, 17B65 

48. CJM 1997 (vol 49 pp. 119)