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1. CJM 2014 (vol 67 pp. 55)
On Varieties of Lie Algebras of Maximal Class We study complex projective varieties that parametrize
(finite-dimensional) filiform Lie algebras over ${\mathbb C}$,
using equations derived by Millionshchikov. In the
infinite-dimensional case we concentrate our attention on
${\mathbb N}$-graded Lie algebras of maximal class. As shown by A.
Fialowski
there are only
three isomorphism types of $\mathbb{N}$-graded Lie algebras
$L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$
and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the
structure of these algebras with the property $L=\langle L_1
\rangle$. In this paper we study those generated by the first and
$q$-th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under
some technical condition, there can only be one isomorphism type
of such algebras. For $q=3$ we fully classify them. This gives a
partial answer to a question posed by Millionshchikov.
Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification Categories:17B70, 14F45 |
2. CJM 2012 (vol 65 pp. 82)
The Ranks of the Homotopy Groups of a Finite Dimensional Complex Let $X$ be an
$n$-dimensional, finite, simply connected CW complex and set
$\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When
$0\lt \alpha_X\lt \infty$, we give upper and lower bound for $
\sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X) $ for $k$ sufficiently
large. We show also for any $r$ that $\alpha_X$ can be estimated
from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound
depending explicitly on $r$.
Keywords:homotopy groups, graded Lie algebra, exponential growth, LS category Categories:55P35, 55P62, , , , 17B70 |
3. CJM 2007 (vol 59 pp. 696)
AlgÃ¨bres de Lie d'homotopie associÃ©es Ã une proto-bigÃ¨bre de Lie On associe \`a toute structure de proto-big\`ebre de Lie sur un espace
vectoriel $F$ de dimension finie des structures d'alg\`ebre de Lie
d'homotopie d\'efinies respectivement sur la suspension de l'alg\`ebre
ext\'erieure de $F$ et celle de son dual $F^*$. Dans ces alg\`ebres,
tous les crochets $n$-aires sont nuls pour $n \geq 4$ du fait qu'ils
proviennent d'une structure de proto-big\`ebre de Lie. Plus
g\'en\'eralement, on associe \`a un \'el\'ement de degr\'e impair de
l'alg\`ebre ext\'erieure de la somme directe de $F$ et $F^*$, une
collection d'applications multilin\'eaires antisym\'etriques sur
l'alg\`ebre ext\'erieure de $F$ (resp.\ $F^*$), qui v\'erifient les
identit\'es de Jacobi g\'en\'eralis\'ees, d\'efinissant les alg\`ebres
de Lie d'homotopie, si l'\'el\'ement donn\'e est de carr\'e nul pour
le grand crochet de l'alg\`ebre ext\'erieure de la somme directe de
$F$ et de~$F^*$.
To any proto-Lie algebra structure on a finite-dimensional vector
space~$F$, we associate homotopy Lie algebra structures defined on
the suspension of the exterior algebra of $F$ and that of its dual
$F^*$, respectively. In these algebras, all $n$-ary brackets for $n
\geq 4$ vanish because the brackets are defined by the proto-Lie
algebra structure. More generally, to any element of odd degree in
the exterior algebra of the direct sum of $F$ and $F^*$, we associate
a set of multilinear skew-symmetric mappings on the suspension of the
exterior algebra of $F$ (resp.\ $F^*$), which satisfy the generalized
Jacobi identities, defining the homotopy Lie algebras, if the given
element is of square zero with respect to the big bracket of the
exterior algebra of the direct sum of $F$ and~$F^*$.
Keywords:algÃ¨bre de Lie d'homotopie, bigÃ¨bre de Lie, quasi-bigÃ¨bre de Lie, proto-bigÃ¨bre de Lie, crochet dÃ©rivÃ©, jacobiateur Categories:17B70, 17A30 |
4. CJM 2006 (vol 58 pp. 1291)
The General Structure of $G$-Graded Contractions of Lie Algebras I. The Classification We give the general structure of complex (resp., real) $G$-graded
contractions of Lie algebras where $G$ is an arbitrary finite Abelian
group. For this purpose, we introduce a number of concepts, such as
pseudobasis, higher-order identities, and sign invariants. We
characterize the equivalence classes of $G$-graded contractions by
showing that our set of invariants (support, higher-order identities,
and sign invariants) is complete, which yields a classification.
Keywords:Lie algebras, graded contractions Categories:17B05, 17B70 |
5. CJM 1999 (vol 51 pp. 658)
Nilpotency of Some Lie Algebras Associated with $p$-Groups Let $ L=L_0+L_1$ be a $\mathbb{Z}_2$-graded Lie algebra over a
commutative ring with unity in which $2$ is invertible. Suppose
that $L_0$ is abelian and $L$ is generated by finitely many
homogeneous elements $a_1,\dots,a_k$ such that every commutator in
$a_1,\dots,a_k$ is ad-nilpotent. We prove that $L$ is nilpotent.
This implies that any periodic residually finite $2'$-group $G$
admitting an involutory automorphism $\phi$ with $C_G(\phi)$
abelian is locally finite.
Categories:17B70, 20F50 |
6. CJM 1998 (vol 50 pp. 225)
Derivations and invariant forms of Lie algebras graded by finite root systems Lie algebras graded by finite reduced root systems have been
classified up to isomorphism. In this paper we describe the derivation
algebras of these Lie algebras and determine when they possess invariant
bilinear forms. The results which we develop to do this are much more
general and apply to Lie algebras that are completely reducible with
respect to the adjoint action of a finite-dimensional subalgebra.
Categories:17B20, 17B70, 17B25 |
7. CJM 1997 (vol 49 pp. 119)
Automorphisms of the Lie algebras $W^*$ in characteristic $0$ No abstract.
Categories:17B40, 17B65, 17B66, 17B68, 17B70 |