1. CMB Online first
 Bavula, V. V.; Lu, T.

The universal enveloping algebra of the SchrÃ¶dinger algebra and its prime spectrum
The prime, completely prime, maximal and primitive spectra are
classified for the universal enveloping algebra of the SchrÃ¶dinger
algebra. For all of these ideals their explicit generators are
given. A counterexample is constructed to the conjecture of Cheng
and Zhang about nonexistence of simple singular Whittaker modules
for the SchrÃ¶dinger algebra (and all such modules are classified).
It is proved that the conjecture holds 'generically'.
Keywords:prime ideal, weight module, simple module, centralizer, Whittaker module Categories:17B10, 16D25, 16D60, 16D70, 16P50 

2. CMB Online first
 Criswell, Jackson; Salisbury, Ben; Tingley, Peter W.

PBW bases and marginally large tableaux in types B and C
We explicitly describe the isomorphism between two combinatorial
realizations of Kashiwara's infinity crystal in types B and C.
The first realization is in terms of marginally large tableaux
and the other is in terms of Kostant partitions coming from PBW
bases. We also discuss a stack notation for Kostant partitions
which simplifies that realization.
Keywords:crystal, Kostant partition, Lusztig data, marginally large tableau Categories:05E10, 17B37 

3. CMB 2017 (vol 61 pp. 16)
 Bavula, V. V.; Lu, T.

Classification of simple weight modules over the SchrÃ¶dinger algebra
A classification of simple weight modules over the SchrÃ¶dinger
algebra is given. The Krull and the global dimensions are found
for the centralizer $C_{\mathcal{S}}(H)$ (and some of its prime factor
algebras) of the Cartan element $H$ in the universal enveloping
algebra $\mathcal{S}$ of the SchrÃ¶dinger (Lie) algebra. The simple
$C_{\mathcal{S}}(H)$modules are classified. The Krull and the global
dimensions are found for some (prime) factor algebras of the
algebra $\mathcal{S}$ (over the centre). It is proved that some (prime)
factor algebras of $\mathcal{S}$ and $C_{\mathcal{S}}(H)$ are tensor homological/Krull
minimal.
Keywords:weight module, simple module, centralizer, Krull dimension, global dimension, tensor homological minimal algebra, tensor Krull minimal algebra Categories:17B10, 17B20, 17B35, 16E10, 16P90, 16P40, 16P50 

4. CMB 2017 (vol 61 pp. 318)
 Lee, TsiuKwen

Adnilpotent Elements of Semiprime Rings with Involution
Let $R$ be an $n!$torsion free semiprime ring with
involution $*$ and with extended centroid $C$, where $n\gt 1$ is
a positive integer. We characterize $a\in K$, the Lie algebra
of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This
generalizes both Martindale and Miers' theorem
and the theorem of Brox et al.
To prove it we
first prove that if $a, b\in R$ satisfy
$(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on
$R$, where either $n$ is even or $b=0$, then
$\big(a\lambda\big)^{[\frac{n+1}{2}]}=0$
for some $\lambda\in C$.
Keywords:Semiprime ring, Lie algebra, Jordan algebra, faithful $f$free, involution, skew (symmetric) element, adnilpotent element, Jordan element Categories:16N60, 16W10, 17B60 

5. 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 

6. CMB 2015 (vol 58 pp. 363)
7. 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 

8. 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 

9. 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 

10. 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 

11. 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 

12. 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 

13. 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 

14. CMB 2011 (vol 55 pp. 67)
15. CMB 2011 (vol 54 pp. 442)
16. CMB 2011 (vol 54 pp. 519)
17. 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 

18. 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 

19. 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 

20. CMB 2009 (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 

21. CMB 2009 (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 

22. 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 

23. 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 

24. 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 

25. 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 
