26. 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 

27. 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 

28. CMB 2003 (vol 46 pp. 529)
29. CMB 2002 (vol 45 pp. 509)
 Benkart, Georgia; Elduque, Alberto

Lie Superalgebras Graded by the Root Systems $C(n)$, $D(m,n)$, $D(2,1;\alpha)$, $F(4)$, $G(3)$
We determine the Lie superalgebras that are graded by the root
systems of the basic classical simple Lie superalgebras of type
$C(n)$, $D(m,n)$, $D(2,1;\alpha)$ $(\alpha \in \mathbb{F} \setminus
\{0,1\})$, $F(4)$, and $G(3)$.
Category:17A70 

30. CMB 2002 (vol 45 pp. 525)
31. CMB 2002 (vol 45 pp. 672)
 Rao, S. Eswara; Batra, Punita

A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables
We study the representations of extended affine Lie algebras
$s\ell_{\ell+1} (\mathbb{C}_q)$ where $q$ is $N$th primitive root of
unity ($\mathbb{C}_q$ is the quantum torus in two variables). We
first prove that $\bigoplus s\ell_{\ell+1} (\mathbb{C})$ for a
suitable number of copies is a quotient of $s\ell_{\ell+1}
(\mathbb{C}_q)$. Thus any finite dimensional irreducible module for
$\bigoplus s\ell_{\ell+1} (\mathbb{C})$ lifts to a representation of
$s\ell_{\ell+1} (\mathbb{C}_q)$. Conversely, we prove that any finite
dimensional irreducible module for $s\ell_{\ell+1} (\mathbb{C}_q)$
comes from above. We then construct modules for the extended affine
Lie algebras $s\ell_{\ell+1} (\mathbb{C}_q) \oplus \mathbb{C} d_1
\oplus \mathbb{C} d_2$ which is integrable and has finite dimensional
weight spaces.
Categories:17B65, 17B66, 17B68 

32. CMB 2002 (vol 45 pp. 653)
33. CMB 2002 (vol 45 pp. 623)
34. CMB 2002 (vol 45 pp. 606)
 Gannon, Terry

Postcards from the Edge, or Snapshots of the Theory of Generalised Moonshine
We begin by reviewing Monstrous Moonshine. The impact of Moonshine on
algebra has been profound, but so far it has had little to teach
number theory. We introduce (using `postcards') a much larger context
in which Monstrous Moonshine naturally sits. This context suggests
Moonshine should indeed have consequences for number theory. We
provide some humble examples of this: new generalisations of Gauss
sums and quadratic reciprocity.
Categories:11F22, 17B67, 81T40 

35. CMB 2002 (vol 45 pp. 567)
 De Sole, Alberto; Kac, Victor G.

Subalgebras of $\gc_N$ and Jacobi Polynomials
We classify the subalgebras of the general Lie conformal algebra
$\gc_N$ that act irreducibly on $\mathbb{C} [\partial]^N$ and that
are normalized by the sl$_2$part of a Virasoro element. The
problem turns out to be closely related to classical Jacobi
polynomials $P_n^{(\sigma,\sigma)}$, $\sigma \in \mathbb{C}$. The
connection goes both wayswe use in our classification some
classical properties of Jacobi polynomials, and we derive from the
theory of conformal algebras some apparently new properties of
Jacobi polynomials.
Categories:17B65, 17B68, 17B69, 33C45 

36. CMB 2001 (vol 44 pp. 27)
37. CMB 2000 (vol 43 pp. 459)
 Ndogmo, J. C.

Properties of the Invariants of Solvable Lie Algebras
We generalize to a field of characteristic zero certain properties of
the invariant functions of the coadjoint representation of solvable
Lie algebras with abelian nilradicals, previously obtained over the
base field $\bbC$ of complex numbers. In particular we determine
their number and the restricted type of variables on which they
depend. We also determine an upper bound on the maximal number of
functionally independent invariants for certain families of solvable
Lie algebras with arbitrary nilradicals.
Categories:17B30, 22E70 

38. CMB 2000 (vol 43 pp. 79)
39. CMB 2000 (vol 43 pp. 3)
 Adin, Ron; Blanc, David

Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and
free Lie algebras in the category of nonassociative algebras. These
resolutions derive in both cases from geometric objects, which in turn
reflect the combinatorics of suitable collections of leaflabeled
trees.
Keywords:resolutions, homology, Lie algebras, associative algebras, nonassociative algebras, Jacobi identity, leaflabeled trees, associahedron Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50 

40. 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 

41. CMB 1999 (vol 42 pp. 412)
 Tai, YungSheng

Peirce Domains
A theorem of Kor\'anyi and Wolf displays any Hermitian symmetric
domain as a Siegel domain of the third kind over any of its
boundary components. In this paper we give a simple proof that an
analogous realization holds for any selfadjoint homogeneous cone.
Category:17C27 

42. CMB 1997 (vol 40 pp. 143)
 Bremner, Murray

Quantum deformations of simple Lie algebras
It is shown that every simple complex Lie algebra $\fg$ admits a
1parameter family $\fg_q$ of deformations outside the category of
Lie algebras.
These deformations are derived from a tensor product decomposition for
$U_q(\fg)$modules;
here $U_q(\fg)$ is the quantized enveloping algebra of $\fg$.
From this it follows that the multiplication on $\fg_q$ is
$U_q(\fg)$invariant.
In the special case $\fg = {\ss}(2)$, the structure constants for
the deformation ${\ss}(2)_q$ are obtained from the quantum
ClebschGordan
formula applied to $V(2)_q \otimes V(2)_q$;
here $V(2)_q$ is the simple 3dimensional
$U_q\bigl({\ss}(2)\bigr)$module of
highest weight $q^2$.
Categories:17B37, 17A01 

43. CMB 1997 (vol 40 pp. 103)
 Riley, David M.; Tasić, Vladimir

The transfer of a commutator law from a nilring to its adjoint group
For every field $F$ of characteristic $p\geq 0$,
we construct an example of a finite dimensional nilpotent
$F$algebra $R$ whose adjoint group $A(R)$ is not
centrebymetabelian, in spite of the fact that $R$ is Lie
centrebymetabelian
and satisfies the identities $x^{2p}=0$ when $p>2$ and
$x^8=0$ when $p=2$. The
existence of such algebras answers a question raised by
A.~E.~Zalesskii, and is in contrast to
positive results obtained by Krasilnikov, Sharma and Srivastava
for Lie metabelian rings
and by Smirnov for the class Lie centrebymetabelian nilalgebras
of exponent 4 over a field of characteristic 2 of cardinality at least 4.
Categories:16U60, 17B60 
