1. CJM 2014 (vol 67 pp. 55)
 Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina

On Varieties of Lie Algebras of Maximal Class
We study complex projective varieties that parametrize
(finitedimensional) filiform Lie algebras over ${\mathbb C}$,
using equations derived by Millionshchikov. In the
infinitedimensional case we concentrate our attention on
${\mathbb N}$graded Lie algebras of maximal class. As shown by A.
Fialowski
there are only
three isomorphism types of $\mathbb{N}$graded Lie algebras
$L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$
and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the
structure of these algebras with the property $L=\langle L_1
\rangle$. In this paper we study those generated by the first and
$q$th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under
some technical condition, there can only be one isomorphism type
of such algebras. For $q=3$ we fully classify them. This gives a
partial answer to a question posed by Millionshchikov.
Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification Categories:17B70, 14F45 

2. CJM 2013 (vol 66 pp. 1250)
 Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter

Symplectic Degenerate Flag Varieties
A simple finite dimensional module $V_\lambda$ of a simple complex
algebraic group $G$ is naturally endowed with a filtration induced by the PBWfiltration
of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module
for the group $G^a$, which can be roughly described as a semidirect product of a
Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy
to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$,
we call the closure
$\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$
of the $G^a$orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$.
In general this is a
singular variety, but we conjecture that it has many nice properties similar to
that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group.
The symplectic case is important for the conjecture
because it is the first known case where even for fundamental weights $\omega$ the varieties
$\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit
construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations,
similar to the BottSamelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$
are normal locally complete intersections with terminal and rational singularities.
We also show that these varieties are Frobenius split. Using the above mentioned results, we
prove an analogue of the BorelWeil theorem and obtain a $q$character formula
for the characters of irreducible $Sp_{2n}$modules via the AtiyahBottLefschetz fixed
points formula.
Keywords:Lie algebras, flag varieties, symplectic groups, representations Categories:14M15, 22E46 

3. CJM 2009 (vol 62 pp. 52)
 Deng, Shaoqiang

An Algebraic Approach to Weakly Symmetric Finsler Spaces
In this paper, we introduce a new algebraic notion, weakly symmetric
Lie algebras, to give an algebraic description of an
interesting class of homogeneous RiemannFinsler spaces, weakly symmetric
Finsler spaces. Using this new definition, we are able to give a
classification of weakly symmetric Finsler spaces with dimensions $2$
and $3$. Finally, we show that all the nonRiemannian reversible weakly
symmetric Finsler spaces we find are nonBerwaldian and with vanishing
Scurvature. This means that reversible nonBerwaldian Finsler spaces
with vanishing Scurvature may exist at large. Hence the generalized
volume comparison theorems due to Z. Shen are valid for a rather large
class of Finsler spaces.
Keywords:weakly symmetric Finsler spaces, weakly symmetric Lie algebras, Berwald spaces, Scurvature Categories:53C60, 58B20, 22E46, 22E60 

4. CJM 2006 (vol 58 pp. 1291)
 WeimarWoods, Evelyn

The General Structure of $G$Graded Contractions of Lie Algebras I. The Classification
We give the general structure of complex (resp., real) $G$graded
contractions of Lie algebras where $G$ is an arbitrary finite Abelian
group. For this purpose, we introduce a number of concepts, such as
pseudobasis, higherorder identities, and sign invariants. We
characterize the equivalence classes of $G$graded contractions by
showing that our set of invariants (support, higherorder identities,
and sign invariants) is complete, which yields a classification.
Keywords:Lie algebras, graded contractions Categories:17B05, 17B70 

5. 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 

6. CJM 1997 (vol 49 pp. 133)
 Reeder, Mark

Exterior powers of the adjoint representation
Exterior powers of the adjoint representation of a complex semisimple Lie
algebra are decomposed into irreducible representations, to varying
degrees of satisfaction.
Keywords:Lie algebras, adjoint representation, exterior algebra Categories:20G05, 20C30, 22E10, 22E60 
