Expand all Collapse all | Results 1 - 25 of 41 |
1. CMB Online first
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 |
2. CMB Online first
Finite Semisimple Loop Algebras of Indecomposable $RA$ Loops There are at the most seven classes of finite indecomposable $RA$ loops upto isomorphism. In this paper, we completely characterize the structure of the unit loop of loop algebras of these seven classes of loops over finite fields of characteristic greater than $2$.
Keywords:unit loop, loop algebra, indecomposable $RA$ loops Categories:20N05, 17D05 |
3. CMB 2014 (vol 58 pp. 69)
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 |
4. CMB 2014 (vol 57 pp. 735)
On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras |
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 |
5. CMB 2012 (vol 56 pp. 606)
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 |
6. CMB 2011 (vol 55 pp. 870)
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 |
7. CMB 2011 (vol 55 pp. 579)
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 |
8. CMB 2011 (vol 55 pp. 351)
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 |
9. CMB 2011 (vol 55 pp. 67)
An $E_8$ Correspondence for Multiplicative Eta-Products We describe an $E_8$ correspondence for the multiplicative
eta-products of weight at least $4$.
Keywords:We describe an E_{8} correspondence for the multiplicative eta-products of weight at leastÂ 4. Categories:11F20, 11F12, 17B60 |
10. CMB 2011 (vol 54 pp. 442)
Nondegeneracy for Lie Triple Systems and Kantor Pairs We study the transfer of nondegeneracy
between Lie triple systems and their standard Lie algebra envelopes
as well as between Kantor pairs, their associated Lie triple systems,
and their Lie algebra envelopes. We also show that simple Kantor
pairs and Lie triple systems in characteristic $0$ are
nondegenerate.
Keywords:Kantor pairs, Lie triple systems, Lie algebras Categories:17A40, 17B60, 17B99 |
11. CMB 2011 (vol 54 pp. 519)
Erratum: Cartan Subalgebras of $\mathrm{gl}_\infty$
We correct an oversight in the the paper
Cartan Subalgebras of
$\mathrm{gl}_\infty$, Canad. Math. Bull. 46(2003), no. 4,
597-616.
doi: 10.4153/CMB-2003-056-1
Categories:17B65, 17B20 |
12. CMB 2011 (vol 54 pp. 297)
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 |
13. CMB 2010 (vol 54 pp. 44)
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 |
14. CMB 2010 (vol 53 pp. 425)
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 |
15. CMB 2009 (vol 52 pp. 245)
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 |
16. CMB 2008 (vol 51 pp. 291)
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 |
17. CMB 2008 (vol 51 pp. 298)
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 |
18. CMB 2007 (vol 50 pp. 603)
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 |
19. CMB 2007 (vol 50 pp. 469)
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 |
20. CMB 2006 (vol 49 pp. 492)
Conjugacy Classes of Subalgebras of the Real Sedenions By applying the Cayley--Dickson process to the division algebra
of real octonions, one obtains a 16-dimensional 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 non-division 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 |
21. CMB 2005 (vol 48 pp. 587)
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 Alev--Dumas 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 Joseph--Letzter
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 infinite-dimensional modules with good properties,
including a grading that is inherited by submodules.
Categories:17B37, 16W35, 17B10, 16D60 |
22. CMB 2005 (vol 48 pp. 445)
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 $S_{n-1}$. 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 |
23. CMB 2005 (vol 48 pp. 394)
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 $3m-1$ root spaces,
which we explicitly determine.
Categories:11E39, 15A63, 17B20 |
24. CMB 2005 (vol 48 pp. 460)
$B$-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 |
25. CMB 2003 (vol 46 pp. 597)
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 one-dimensional 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 self-normalizing 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 |