1. CMB 2012 (vol 56 pp. 606)
 Mazorchuk, Volodymyr; Zhao, Kaiming

Characterization of Simple Highest Weight Modules
We prove that for simple complex finite dimensional
Lie algebras, affine KacMoody Lie algebras, the
Virasoro algebra and the HeisenbergVirasoro algebra,
simple highest weight modules are characterized
by the property that all positive root elements
act on these modules locally nilpotently. We
also show that this is not the case for higher rank
Virasoro and for Heisenberg algebras.
Keywords:Lie algebra, highest weight module, triangular decomposition, locally nilpotent action Categories:17B20, 17B65, 17B66, 17B68 

2. CMB 2011 (vol 55 pp. 870)
 Wang, Hui; Deng, Shaoqiang

Left Invariant EinsteinRanders Metrics on Compact Lie Groups
In this paper we study left invariant EinsteinRanders metrics on compact Lie
groups. First, we give a method to construct left invariant nonRiemannian EinsteinRanders metrics
on a compact Lie group, using the Zermelo navigation data.
Then we prove that this gives a complete classification of left invariant EinsteinRanders metrics on compact simple
Lie groups with the underlying Riemannian metric naturally reductive.
Further, we completely determine the identity component of the group of
isometries for this type of metrics on simple groups. Finally, we study some
geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature
of such metrics.
Keywords:EinsteinRanders metric, compact Lie groups, geodesic, flag curvature Categories:17B20, 22E46, 53C12 

3. CMB 2011 (vol 54 pp. 519)
4. CMB 2010 (vol 54 pp. 44)
 Cheung, WaiShun; Tam, TinYau

StarShapedness and $K$Orbits in Complex Semisimple Lie Algebras
Given a complex semisimple Lie algebra
$\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact
real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to
\mathfrak{h}$ be the orthogonal projection (with respect to the
Killing form) onto the Cartan subalgebra
$\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a
maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in
\mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, where $K$ is
the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show
that it is starshaped. The result extends a result of Tsing. We also
consider the generalized numerical range $f(\mathop{\textrm{Ad}}(K)x)$,
where $f$ is a linear functional on $\mathfrak{g}$. We establish the
starshapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras
of type $B$.
Categories:22E10, 17B20 

5. CMB 2005 (vol 48 pp. 460)
 Sommers, Eric N.

$B$Stable Ideals in the Nilradical of a Borel Subalgebra
We count the number of strictly positive $B$stable ideals in the
nilradical of a Borel subalgebra and prove that
the minimal roots of any $B$stable ideal are conjugate
by an element of the Weyl group to a subset of the simple roots.
We also count the number of ideals whose minimal roots are conjugate
to a fixed subset of simple roots.
Categories:20F55, 17B20, 05E99 

6. CMB 2005 (vol 48 pp. 394)
 Đoković, D. Ž.; Szechtman, F.; Zhao, K.

Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices
Let $n=2m$ be even and denote by $\Sp_n(F)$ the symplectic group
of rank $m$ over an infinite field $F$ of characteristic different
from $2$. We show that any $n\times n$ symmetric matrix $A$ is
equivalent under symplectic congruence transformations to the
direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal
and $C$ tridiagonal. Since the $\Sp_n(F)$module of symmetric
$n\times n$ matrices over $F$ is isomorphic to the adjoint module
$\sp_n(F)$, we infer that any adjoint orbit of $\Sp_n(F)$ in
$\sp_n(F)$ has a representative in the sum of $3m1$ root spaces,
which we explicitly determine.
Categories:11E39, 15A63, 17B20 

7. CMB 2003 (vol 46 pp. 597)
 Neeb, KarlHermann; Penkov, Ivan

Cartan Subalgebras of $\mathfrak{gl}_\infty$
Let $V$ be a vector space over a field $\mathbb{K}$ of characteristic
zero and $V_*$ be a space of linear functionals on $V$ which separate
the points of $V$. We consider $V\otimes V_*$ as a Lie algebra of
finite rank operators on $V$, and set $\mathfrak{gl} (V,V_*) :=
V\otimes V_*$. We define a Cartan subalgebra of $\mathfrak{gl}
(V,V_*)$ as the centralizer of a maximal subalgebra every element of
which is semisimple, and then give the following description of all
Cartan subalgebras of $\mathfrak{gl} (V,V_*)$ under the assumption
that $\mathbb{K}$ is algebraically closed. A subalgebra of
$\mathfrak{gl} (V,V_*)$ is a Cartan subalgebra if and only if it
equals $\bigoplus_j \bigl( V_j \otimes (V_j)_* \bigr) \oplus (V^0 \otimes
V_*^0)$ for some onedimensional subspaces $V_j \subseteq V$ and
$(V_j)_* \subseteq V_*$ with $(V_i)_* (V_j) = \delta_{ij} \mathbb{K}$
and such that the spaces $V_*^0 = \bigcap_j (V_j)^\bot \subseteq V_*$
and $V^0 = \bigcap_j \bigl( (V_j)_* \bigr)^\bot \subseteq V$ satisfy
$V_*^0 (V^0) = \{0\}$. We then discuss explicit constructions of
subspaces $V_j$ and $(V_j)_*$ as above. Our second main result claims
that a Cartan subalgebra of $\mathfrak{gl} (V,V_*)$ can be described
alternatively as a locally nilpotent selfnormalizing subalgebra whose
adjoint representation is locally finite, or as a subalgebra
$\mathfrak{h}$ which coincides with the maximal locally nilpotent
$\mathfrak{h}$submodule of $\mathfrak{gl} (V,V_*)$, and such that the
adjoint representation of $\mathfrak{h}$ is locally finite.
Categories:17B65, 17B20 

8. CMB 1999 (vol 42 pp. 486)
 Sawyer, P.

Spherical Functions on $\SO_0(p,q)/\SO(p)\times \SO(q)$
An integral formula is derived for the spherical functions on the
symmetric space $G/K=\break
\SO_0(p,q)/\SO(p)\times \SO(q)$. This formula
allows us to state some results about the analytic continuation of
the spherical functions to a tubular neighbourhood of the
subalgebra $\a$ of the abelian part in the decomposition $G=KAK$.
The corresponding result is then obtained for the heat kernel of the
symmetric space $\SO_0(p,q)/\SO (p)\times\SO (q)$ using the Plancherel
formula.
In the Conclusion, we discuss how this analytic continuation can be
a helpful tool to study the growth of the heat kernel.
Categories:33C55, 17B20, 53C35 
