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Search: All articles in the CJM digital archive with keyword graded Lie algebra

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1. CJM 2017 (vol 69 pp. 721)

Allison, Bruce; Faulkner, John; Smirnov, Oleg
Weyl Images of Kantor Pairs
Kantor pairs arise naturally in the study of $5$-graded Lie algebras. In this article, we introduce and study Kantor pairs with short Peirce gradings and relate them to Lie algebras graded by the root system of type $\mathrm{BC}_2$. This relationship allows us to define so called Weyl images of short Peirce graded Kantor pairs. We use Weyl images to construct new examples of Kantor pairs, including a class of infinite dimensional central simple Kantor pairs over a field of characteristic $\ne 2$ or $3$, as well as a family of forms of a split Kantor pair of type $\mathrm{E}_6$.

Keywords:Kantor pair, graded Lie algebra, Jordan pair
Categories:17B60, 17B70, 17C99, 17B65

2. CJM 2014 (vol 67 pp. 55)

Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina
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

3. CJM 2012 (vol 65 pp. 82)

FĂ©lix, Yves; Halperin, Steve; Thomas, Jean-Claude
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

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