1. CMB 2015 (vol 58 pp. 692)
 Anona, F. M.; Randriambololondrantomalala, Princy; Ravelonirina, H. S. G.

Sur les algÃ¨bres de Lie associÃ©es Ã une connexion
Let $\Gamma$ be a connection on a smooth manifold
$M$, in this paper we give some properties of $\Gamma$ by studying
the corresponding Lie algebras. In particular, we compute the
first ChevalleyEilenberg cohomology space of the horizontal
vector fields Lie algebra on the tangent bundle of $M$, whose
the corresponding Lie derivative of $\Gamma$ is null, and of
the horizontal nullity curvature space.
Keywords:algÃ¨bre de Lie, connexion, cohomologie de ChevalleyEilenberg, champs dont la dÃ©rivÃ©e de Lie correspondante Ã une connexion est nulle, espace de nullitÃ© de la courbure Categories:17B66, 53B15 

2. CMB 2015 (vol 58 pp. 363)
3. CMB 2015 (vol 58 pp. 233)
 Bergen, Jeffrey

Affine Actions of $U_q(sl(2))$ on Polynomial Rings
We classify the affine actions of $U_q(sl(2))$ on commutative
polynomial rings in $m \ge 1$ variables.
We show that, up to scalar multiplication, there are two possible
actions.
In addition, for each action, the subring of invariants is a
polynomial ring in either $m$ or $m1$ variables,
depending upon whether $q$ is or is not a root of $1$.
Keywords:skew derivation, quantum group, invariants Categories:16T20, 17B37, 20G42 

4. CMB 2014 (vol 58 pp. 69)
 Fulp, Ronald Owen

Correction to "Infinite Dimensional DeWitt Supergroups and Their Bodies"
The Theorem below is a correction to Theorem
3.5 in the article
entitled " Infinite Dimensional DeWitt Supergroups and Their
Bodies" published
in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283288. Only part
(iii) of that Theorem
requires correction. The proof of Theorem 3.5 in the original
article failed to separate
the proof of (ii) from the proof of (iii). The proof of (ii)
is complete once it is established
that $ad_a$ is quasinilpotent for each $a$ since it immediately
follows that $K$
is quasinilpotent. The proof of (iii) is not complete
in the original article. The revision appears as the proof of
(iii) of the revised Theorem below.
Keywords:super groups, body of super groups, Banach Lie groups Categories:58B25, 17B65, 81R10, 57P99 

5. CMB 2014 (vol 57 pp. 735)
 Cagliero, Leandro; Szechtman, Fernando

On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras
We describe of all finite
dimensional uniserial representations of a commutative associative
(resp. abelian Lie) algebra over a perfect (resp. sufficiently
large perfect) field. In the Lie case the size of the field
depends on the answer to following question, considered and solved
in this paper. Let $K/F$ be a finite separable field extension
and
let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some
nonzero elements $\alpha,\beta\in F$?
Keywords:uniserial module, Lie algebra, associative algebra, primitive element Categories:17B10, 13C05, 12F10, 12E20 

6. 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 

7. 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 

8. CMB 2011 (vol 55 pp. 579)
 Ndogmo, J. C.

Casimir Operators and Nilpotent Radicals
It is shown that a Lie algebra having a nilpotent radical has a
fundamental set of invariants consisting of Casimir operators. A
different proof is given in the well known special case of an
abelian radical. A result relating the number of invariants to the
dimension of the Cartan subalgebra is also established.
Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariants Categories:16W25, 17B45, 16S30 

9. CMB 2011 (vol 55 pp. 351)
 MacDougall, J. A.; Sweet, L. G.

Rational Homogeneous Algebras
An algebra $A$ is homogeneous if the automorphism group of $A$
acts transitively on the onedimensional subspaces of $A$. The existence of homogeneous algebras depends critically on the choice of the scalar field. We examine the case where the scalar field is the rationals. We prove that if $A$ is a rational homogeneous algebra with $\operatorname{dim} A>1$, then $A^{2}=0$.
Keywords:nonassociative algebra, homogeneous, automorphism Categories:17D99, 17A36 

10. CMB 2011 (vol 55 pp. 67)
11. CMB 2011 (vol 54 pp. 442)
12. CMB 2011 (vol 54 pp. 519)
13. CMB 2011 (vol 54 pp. 297)
 Johnson, Marianne; Stöhr, Ralph

Lie Powers and PseudoIdempotents
We give a new factorisation of the classical Dynkin operator,
an element of the integral group ring of the symmetric group that
facilitates projections of tensor powers onto Lie powers.
As an application we show that the iterated Lie power $L_2(L_n)$ is
a module direct summand of the Lie power $L_{2n}$ whenever the
characteristic of the ground field does not divide $n$. An explicit
projection of the latter onto the former is exhibited in this case.
Categories:17B01, 20C30 

14. 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 

15. CMB 2010 (vol 53 pp. 425)
 Chapoton, Frédéric

Free PreLie Algebras are Free as Lie Algebras
We prove that the $\mathfrak{S}$module $\operatorname{PreLie}$ is a free Lie algebra in
the category of $\mathfrak{S}$modules and can therefore be written as the
composition of the $\mathfrak{S}$module $\operatorname{Lie}$ with a new $\mathfrak{S}$module
$X$. This implies that free preLie algebras in the category of
vector spaces, when considered as Lie algebras, are free on
generators that can be described using $X$. Furthermore, we define a
natural filtration on the $\mathfrak{S}$module $X$. We also obtain a
relationship between $X$ and the $\mathfrak{S}$module coming from the
anticyclic structure of the $\operatorname{PreLie}$ operad.
Categories:18D50, 17B01, 18G40, 05C05 

16. CMB 2009 (vol 52 pp. 245)
 Goodaire, Edgar G.; Milies, César Polcino

Involutions of RA Loops
Let $L$ be an RA loop, that is, a loop whose loop ring
over any coefficient ring $R$
is an alternative, but not associative, ring. Let
$\ell\mapsto\ell^\theta$ denote an involution on $L$ and extend
it linearly to the loop ring $RL$. An element $\alpha\in RL$ is
\emph{symmetric} if $\alpha^\theta=\alpha$ and \emph{skewsymmetric}
if $\alpha^\theta=\alpha$. In this paper, we show that
there exists an involution making
the symmetric elements of $RL$ commute if and only if
the characteristic of $R$ is $2$ or $\theta$ is the
canonical involution on $L$,
and an involution making the skewsymmetric elements of $RL$
commute if and only if
the characteristic of $R$ is $2$ or $4$.
Categories:20N05, 17D05 

17. CMB 2008 (vol 51 pp. 298)
 Tocón, Maribel

The Kostrikin Radical and the Invariance of the Core of Reduced Extended Affine Lie Algebras
In this paper we prove that the Kostrikin radical of an extended affine Lie algebra of
reduced type coincides with the center of its core, and use this characterization to get a typefree
description of the core of such algebras. As a consequence we get that the core of an extended affine
Lie algebra of reduced type is invariant under the automorphisms of the algebra.
Keywords:extended affine Lie algebra, Lie torus, core, Kostrikin radical Categories:17B05, 17B65 

18. CMB 2008 (vol 51 pp. 291)
 Spinelli, Ernesto

Group Algebras with Minimal Strong Lie Derived Length
Let $KG$ be a noncommutative strongly Lie solvable group algebra of a
group $G$ over a field $K$ of positive characteristic $p$. In this
note we state necessary and sufficient conditions so that the
strong Lie derived length of $KG$ assumes its minimal value, namely
$\lceil \log_{2}(p+1)\rceil $.
Keywords:group algebras, strong Lie derived length Categories:16S34, 17B30 

19. CMB 2007 (vol 50 pp. 603)
 Penkov, Ivan; Zuckerman, Gregg

Construction of Generalized HarishChandra Modules with Arbitrary Minimal $\mathfrak k$Type
Let $\mathfrak g$ be a semisimple complex Lie algebra and $\k\subset\g$ be
any algebraic subalgebra reductive in $\mathfrak g$. For any simple
finite dimensional $\mathfrak k$module $V$, we construct simple
$(\mathfrak g,\mathfrak k)$modules $M$ with finite dimensional $\mathfrak k$isotypic
components such that $V$ is a $\mathfrak k$submodule of $M$ and the Vogan
norm of any simple $\k$submodule $V'\subset M, V'\not\simeq V$, is
greater than the Vogan norm of $V$. The $(\mathfrak g,\mathfrak k)$modules
$M$ are subquotients of the fundamental series of
$(\mathfrak g,\mathfrak k)$modules.
Categories:17B10, 17B55 

20. CMB 2007 (vol 50 pp. 469)
 Tvalavadze, M. V.

Simple Decompositions of the Exceptional Jordan Algebra
This paper presents some
results on the simple exceptional Jordan algebra over an algebraically
closed field $\Phi$ of characteristic not $2$. Namely an example of
simple decomposition of $H(O_3)$ into the sum of two subalgebras
of the type $H(Q_3)$ is produced, and it is shown that this
decomposition is the only one possible in terms of simple
subalgebras.
Category:17C40 

21. CMB 2006 (vol 49 pp. 492)
 Chan, KaiCheong; Đoković, Dragomir Ž.

Conjugacy Classes of Subalgebras of the Real Sedenions
By applying the CayleyDickson process to the division algebra
of real octonions, one obtains a 16dimensional real algebra
known as (real) sedenions. We denote this algebra by $\bA_4$.
It is a flexible quadratic algebra (with unit element 1) but
not a division algebra.
We classify the subalgebras of $\bA_4$ up to conjugacy (\emph{i.e.,}
up to the action of the automorphism group $G$ of $\bA_4$)
with one exception: we leave aside the more complicated case
of classifying the quaternion subalgebras.
Any nonzero subalgebra contains 1 and we show that there are
no proper subalgebras of dimension 5, 7 or $>8$.
The proper nondivision subalgebras have dimensions
3, 6 and 8. We show that in each of these dimensions
there is exactly one conjugacy class of such subalgebras.
There are infinitely many conjugacy classes of subalgebras in
dimensions 2 and 4, but only 4 conjugacy classes in dimension 8.
Categories:17A45, 17A36, 17A20 

22. CMB 2005 (vol 48 pp. 587)
 Lopes, Samuel A.

Separation of Variables for $U_{q}(\mathfrak{sl}_{n+1})^{+}$
Let $U_{q}(\SL)^{+}$ be the positive part of the quantized enveloping
algebra $U_{q}(\SL)$. Using results of AlevDumas and Caldero related
to the center of $U_{q}(\SL)^{+}$, we show that this algebra is free
over its center. This is reminiscent of Kostant's separation of
variables for the enveloping algebra $U(\g)$ of a complex semisimple
Lie algebra $\g$, and also of an analogous result of JosephLetzter
for the quantum algebra $\Check{U}_{q}(\g)$. Of greater importance to
its representation theory is the fact that $\U{+}$ is free over a
larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$
to $\U{+}$ provides infinitedimensional modules with good properties,
including a grading that is inherited by submodules.
Categories:17B37, 16W35, 17B10, 16D60 

23. 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 

24. CMB 2005 (vol 48 pp. 445)
 Patras, Frédéric; Reutenauer, Christophe; Schocker, Manfred

On the Garsia Lie Idempotent
The orthogonal projection of the free associative algebra onto the
free Lie algebra is afforded by an idempotent in the rational group
algebra of the symmetric group $S_n$, in each homogenous degree
$n$. We give various characterizations of this Lie idempotent and show
that it is uniquely determined by a certain unit in the group algebra
of $S_{n1}$. The inverse of this unit, or, equivalently, the Gram
matrix of the orthogonal projection, is described explicitly. We also
show that the Garsia Lie idempotent is not constant on descent classes
(in fact, not even on coplactic classes) in $S_n$.
Categories:17B01, 05A99, 16S30, 17B60 

25. 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 
