1. CJM 2017 (vol 69 pp. 721)
 Allison, Bruce; Faulkner, John; Smirnov, Oleg

Weyl Images of Kantor Pairs
Kantor pairs arise naturally in the study of
$5$graded Lie algebras. In this article, we introduce
and study Kantor pairs with short Peirce gradings and relate
them to Lie algebras
graded by the root system of type
$\mathrm{BC}_2$.
This relationship
allows us to define so called Weyl images
of short Peirce graded Kantor pairs. We use Weyl images to construct
new examples of Kantor pairs, including a class of infinite
dimensional
central simple Kantor pairs over a field of characteristic $\ne
2$ or $3$, as well as a family of forms of a split
Kantor pair of type
$\mathrm{E}_6$.
Keywords:Kantor pair, graded Lie algebra, Jordan pair Categories:17B60, 17B70, 17C99, 17B65 

2. CJM 2016 (vol 69 pp. 453)
 Marquis, Timothée; Neeb, KarlHermann

Isomorphisms of Twisted Hilbert Loop Algebras
The closest infinite dimensional relatives of compact Lie algebras are HilbertLie algebras, i.e. real Hilbert spaces with a Lie
algebra
structure for which the scalar product is invariant.
Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras
over simple HilbertLie algebras $\mathfrak{k}$, also called
affinisations of $\mathfrak{k}$.
They possess a root space decomposition
whose corresponding root system is a locally affine root system
of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$,
$D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some
infinite set $J$. To each of these types corresponds a ``minimal"
affinisation of some simple HilbertLie algebra $\mathfrak{k}$,
which we call standard.
In this paper, we give for each affinisation $\mathfrak{g}$ of
a simple HilbertLie algebra $\mathfrak{k}$ an explicit isomorphism
from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from
the classification
of locally affine root systems, but
for representation theoretic purposes it is crucial to obtain
it explicitly
as a deformation between two twists which is compatible
with the root decompositions.
We illustrate this by applying our isomorphism theorem to the
study of positive energy highest weight representations of $\mathfrak{g}$.
In subsequent work, the present paper will be used to obtain
a complete classification
of the positive energy highest weight representations of affinisations
of $\mathfrak{k}$.
Keywords:locally affine Lie algebra, HilbertLie algebra, positive energy representation Categories:17B65, 17B70, 17B22, 17B10 

3. CJM 2015 (vol 68 pp. 258)
 Calixto, Lucas; Moura, Adriano; Savage, Alistair

Equivariant Map Queer Lie Superalgebras
An equivariant map queer Lie superalgebra is the Lie superalgebra
of regular maps from an algebraic variety (or scheme) $X$ to
a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect
to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$.
In this paper, we classify all irreducible finitedimensional
representations of the equivariant map queer Lie superalgebras
under the assumption that $\Gamma$ is abelian and acts freely
on $X$. We show that such representations are parameterized
by a certain set of $\Gamma$equivariant finitely supported maps
from $X$ to the set of isomorphism classes of irreducible finitedimensional
representations of $\mathfrak{q}$. In the special case where $X$ is the
torus, we obtain a classification of the irreducible finitedimensional
representations of the twisted loop queer superalgebra.
Keywords:Lie superalgebra, queer Lie superalgebra, loop superalgebra, equivariant map superalgebra, finitedimensional representation, finitedimensional module Categories:17B65, 17B10 

4. CJM 2009 (vol 62 pp. 382)
 Lü, Rencai; Zhao, Kaiming

Verma Modules over Quantum Torus Lie Algebras
Representations of various onedimensional central
extensions of quantum tori (called quantum torus Lie algebras) were
studied by several authors. Now we define a central extension of
quantum tori so that all known representations can be regarded as
representations of the new quantum torus Lie algebras $\mathfrak{L}_q$. The
center of $\mathfrak{L}_q$ now is generally infinite dimensional.
In this paper, $\mathbb{Z}$graded Verma modules $\widetilde{V}(\varphi)$ over $\mathfrak{L}_q$
and their corresponding irreducible highest weight modules
$V(\varphi)$ are defined for some linear functions $\varphi$.
Necessary and sufficient conditions for $V(\varphi)$ to have all
finite dimensional weight spaces are given. Also necessary and
sufficient conditions for Verma modules $\widetilde{V}(\varphi)$ to
be irreducible are obtained.
Categories:17B10, 17B65, 17B68 

5. CJM 2008 (vol 60 pp. 892)
 Neeb, KarlHermann; Wagemann, Friedrich

The Second Cohomology of Current Algebras of General Lie Algebras
Let $A$ be a unital commutative associative algebra over a field of
characteristic zero, $\k$ a Lie algebra, and
$\zf$ a vector space, considered as a trivial module of the Lie algebra
$\gf := A \otimes \kf$. In this paper, we give a
description of the cohomology space $H^2(\gf,\zf)$
in terms of easily accessible data associated with $A$ and $\kf$.
We also discuss the topological situation, where
$A$ and $\kf$ are locally convex algebras.
Keywords:current algebra, Lie algebra cohomology, Lie algebra homology, invariant bilinear form, central extension Categories:17B56, 17B65 

6. CJM 2006 (vol 58 pp. 225)
 Azam, Saeid

Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori
We investigate a class of Lie algebras which we call {\it generalized reductive
Lie algebras}. These are generalizations of semisimple, reductive, and affine
KacMoody Lie algebras. A generalized reductive Lie algebra which has an irreducible
root system is said to be {\it irreducible\/} and we note that this class of algebras
has been under intensive investigation in recent years. They have also been called
{\it extended affine Lie algebras}. The larger class of generalized reductive Lie
algebras has not been so intensively investigated. We study them in this paper and note
that one way they arise is as fixed point subalgebras of finite order automorphisms. We
show that the core modulo the center of a generalized reductive Lie algebra is a direct
sum of centerless Lie tori. Therefore one can use the results known about the
classification of centerless Lie tori to classify the cores modulo centers of
generalized reductive Lie algebras.
Categories:17B65, 17B67, 17B40 

7. 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 

8. 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 

9. 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 

10. 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 

11. 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 

12. CJM 1997 (vol 49 pp. 119)