1. CMB Online first
 | Bavula, V. V.; Lu, T.
 |
The universal enveloping algebra of the Schrödinger algebra and its prime spectrum
The prime, completely prime, maximal and primitive spectra are
classified for the universal enveloping algebra of the Schrödinger
algebra. For all of these ideals their explicit generators are
given. A counterexample is constructed to the conjecture of Cheng
and Zhang about non-existence of simple singular Whittaker modules
for the Schrödinger 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 |
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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 |
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3. CMB 2017 (vol 61 pp. 16)
 | Bavula, V. V.; Lu, T.
 |
Classification of simple weight modules over the Schrödinger algebra
A classification of simple weight modules over the Schrödinger
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 Schrödinger (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 |
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4. CMB 2017 (vol 61 pp. 318)
 | Lee, Tsiu-Kwen
 |
Ad-nilpotent 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, ad-nilpotent element, Jordan element Categories:16N60, 16W10, 17B60 |
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5. CMB 2015 (vol 58 pp. 692)
 | Anona, F. M.; Randriambololondrantomalala, Princy; Ravelonirina, H. S. G.
 |
Sur les algèbres de Lie associées à 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 Chevalley-Eilenberg 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:algèbre de Lie, connexion, cohomologie de Chevalley-Eilenberg, champs dont la dérivée de Lie correspondante à une connexion est nulle, espace de nullité de la courbure Categories:17B66, 53B15 |
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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 $m-1$ variables,
depending upon whether $q$ is or is not a root of $1$.
Keywords:skew derivation, quantum group, invariants Categories:16T20, 17B37, 20G42 |
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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. 283-288. 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 quasi-nilpotent for each $a$ since it immediately
follows that $K$
is quasi-nilpotent. 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 |
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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
non-zero elements $\alpha,\beta\in F$?
Keywords:uniserial module, Lie algebra, associative algebra, primitive element Categories:17B10, 13C05, 12F10, 12E20 |
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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 Kac-Moody Lie algebras, the
Virasoro algebra and the Heisenberg-Virasoro 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 |
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11. CMB 2011 (vol 55 pp. 870)
 | Wang, Hui; Deng, Shaoqiang
 |
Left Invariant Einstein-Randers Metrics on Compact Lie Groups
In this paper we study left invariant Einstein-Randers metrics on compact Lie
groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics
on a compact Lie group, using the Zermelo navigation data.
Then we prove that this gives a complete classification of left invariant Einstein-Randers 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:Einstein-Randers metric, compact Lie groups, geodesic, flag curvature Categories:17B20, 22E46, 53C12 |
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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 |
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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 one-dimensional 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:non-associative algebra, homogeneous, automorphism Categories:17D99, 17A36 |
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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 Pseudo-Idempotents
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 |
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18. CMB 2010 (vol 54 pp. 44)
 | Cheung, Wai-Shun; Tam, Tin-Yau
 |
Star-Shapedness 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 star-shaped. 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
star-shapedness 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 Pre-Lie 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 pre-Lie 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 |
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20. CMB 2009 (vol 40 pp. 103)
 | Riley, David M.; Tasić, Vladimir
 |
The transfer of a commutator law from a nil-ring 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
centre-by-metabelian, in spite of the fact that $R$ is Lie
centre-by-metabelian
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 centre-by-metabelian nil-algebras
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
1-parameter 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
Clebsch-Gordan
formula applied to $V(2)_q \otimes V(2)_q$;
here $V(2)_q$ is the simple 3-dimensional
$U_q\bigl({\ss}(2)\bigr)$-module of
highest weight $q^2$.
Categories:17B37, 17A01 |
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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{skew-symmetric}
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 skew-symmetric elements of $RL$
commute if and only if
the characteristic of $R$ is $2$ or $4$.
Categories:20N05, 17D05 |
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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 type-free
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 |
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24. CMB 2008 (vol 51 pp. 291)
 | Spinelli, Ernesto
 |
Group Algebras with Minimal Strong Lie Derived Length
Let $KG$ be a non-commutative 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 |
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25. CMB 2007 (vol 50 pp. 603)
 | Penkov, Ivan; Zuckerman, Gregg
 |
Construction of Generalized Harish-Chandra 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 |
|